A new formalism for the estimation of the CP-violation parameters
Maurice M. Courbage, Thomas T. Durt, Seyed Majid S.M. Saberi Fathi
aa r X i v : . [ h e p - ph ] J u l A new formalism for the estimation of the
C P -violationparameters
M. Courbage , T. Durt and S.M. Saberi Fathi Abstract
In this paper, we use the time super-operator formalism in the 2-level Friedrichs model[1] to obtain a phenomenological model of mesons decay. Our approach provides a fairlygood estimation of the CP symmetry violation parameter in the case of K, B and D mesons.We also propose a crucial test aimed at discriminating between the standard approach andthe time super-operator approach developed throughout the paper. PACS number:03.65.-w, 13.90.+i,13.20.Eb,13.20.He,13,20.Jf
There have been several theoretical approaches to CP violation in kaons (see e.g, the collectionof papers edited in [2]) and the question is partially open today. In this paper, we use aHamiltonian model, describing a two-level states coupled to a continuum of degrees of freedom,that makes is possible to simulate the phenomenology of neutral kaons. Then, the time super-operator formalism for the decay probability provides new numerical estimate of the parametersof CP violation.It is well known [3] that kaons appears in pair K and K each one being conjugated toeach other. The decay processes of K and K correspond to combinations of two orthogonaldecaying modes K and K , that are distinguished by their lifetime. The discovery of the small CP -violation effect was also accompanied by the non orthonormality of the short and long liveddecay modes, now denoted K S and K L , slightly different from K and K and depending ona CP -violation parameter ǫ . Lee, Oehme and Yang (LOY) [4] proposed a generalization ofthe Wigner-Weisskopf theory [5] in order to account the “exponential decay”. Later on, L. A.Khalfin [6] has pointed that, for a quantum system with energy spectrum bounded from below,the decay could not be exponential for large times. It was also observed [7] that short-timebehavior of decaying systems could not be exponential and this led to the so-called Zeno effect[8, 9]. The departure from the exponential type behavior has been experimentally observed(see references quoted in [10]). L.A. Khalfin also corrected the parameter ǫ at the lowest order Laboratoire Mati`ere et Syst`emes Complexes (MSC), UMR 7057 CNRS et Universit´e Paris 7- Denis Diderot,Case 7056, Bˆatiment Condorcet, 10, rue Alice Domon et Lonie Duquet 75205 Paris Cedex 13, FRANCE. email:[email protected] TENA-TONA Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium. email:[email protected] Laboratoire de Physique Th´eorique et Mod`elisation (LPTM), UMR 8089, CNRS et Universit´e de Cergy-Pontoise, Site Saint-Matrin, 2, rue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, FRANCE. email:[email protected]
1f perturbation. His estimation has been presented and reexamined in the reference [10] andapplied to other mesons.We show that our model allows us to obtain a better estimation of the CP -violation pa-rameter for kaon as well as B and D mesons. We also make new predictions that differ fromstandard predictions and that could be tested experimentally.This paper is organized as follows. In Section 2, we introduce the time super-operator fordecay probability density. Then, in Section 3 we present the 2-levels Friedrichs model. Kaonphenomenology is recalled in Section 4. In Section 5, we present the theory of CP -violation inthe Hilbert space and another derivation of the intensity formula for mesons that already hasbeen used in [11]. Finally, in Section 6, we derive the time super-operator intensity formula andwe compute CP -violation parameters for K, B, and D mesons. Then, we compare our resultswith the experimental data. T ) approach T ′ ) approach In the Wigner-Weisskopf approximation to time evolution of quantum unstable systems, theenergy spectrum of the Hamiltonian is extended from −∞ to + ∞ . In this approximation, adecay time operator T ′ is canonically conjugated to H . That is, Hψ ( ω ) = ωψ ( ω ) (2.1) T ′ ψ ( ω ) = − i ddω ψ ( ω ) (2.2)so that T ′ satisfies to the commutation relation [ H, T ′ ] = i I . The T ′ -representation is obtainedby a Fourier transform b ψ ( τ ) = 1 √ π Z ∞−∞ e − i τω ψ ( ω ) dω (2.3)and the unstable states are the those prepared such that the decay occurs in the future, that is, b ψ ( τ ) = 0 for τ <
0. Any state of the form ψ un ( ω ) = A/ ( ω − z ), ( z = a − i b, b > b ψ un ( τ ) = (cid:26) i A √ πe − i τz τ ≥ , τ < | b ψ un ( τ ) | = 2 π | A | e − bτ (2.5)This is an exponential distribution of decay times that is very common in particle physics. T ) formalism Rigourously speaking, when the Hamiltonian has a positive spectrum, it is forbidden in principleto define a time operator that satisfies the commutation relation [
H, T ′ ] = i I . This argumentwas elaborated by Pauli who showed that if one could find such an operator ˆ T ′ one could use itfor generating arbitrary translations in the energy eigenspace so that then the spectrum of ˆ H ought to be unbounded by below, which clearly constitutes a physical impossibility.In order to escape this contradiction one needs to go to the space of density matrices inorder to obtain a time operator that is conjugated to the evolution operator (the Liouville-von2eumann operator) because it is sufficient that the Hamiltonian is not upperly bounded so thatthe Liouville-von Neumann operator has a spectrum extending from −∞ to ∞ . In order todo so, let us consider the Liouville-von Neumann space which is the space of operators ρ on H equipped with the scalar product < ρ, ρ ′ > = T r ( ρ ∗ ρ ′ ) for which the time evolution is given by U t ρ = e − itH ρe itH (2.6) U t = e − itL is generated by the Liouville von-Neumann operator L given by: Lρ = Hρ − ρH (2.7)The time super-operator T is a self-adjoint super-operator on the Liouville-von Neumann spaceconjugated to L , i.e. [ T, L ] = i I . This definition is equivalent to the Weyl relation: e itL T e − itL = T + tI .The average of T in the state ρ is given by : h T i ρ = h ρ, T ρ i (2.8)The time of occurrence of a random event fluctuates and we speak of the probability of itsoccurrence in a time interval I =] t , t ]. The observable T ′ = − T is associated to such event.In fact, for a system in the initial state ρ the average time of occurrence h T ′ i ρ is to be relatedto the time parameter t and to the average time of occurrence in the state ρ t = e − i tL ρ by: h T ′ i ρ t = h T ′ i ρ − t (2.9)This equation follows from the Weyl relation.Let P τ denote the family of spectral projection operators of T : T = Z R τ d P τ (2.10)and let Q τ be the family of spectral projections of T ′ , then, in the state ρ , the probability ofoccurrence of the event in a time interval I is given, as in the usual formulations, by P ( I, ρ ) = kQ t ρ k − kQ t ρ k = k ( Q t − Q t ) ρ k := kQ ( I ) ρ k (2.11)The unstable “undecayed” states observed at t = 0 are the states ρ such that P ( I, ρ ) = 0 forany negative time interval I , that is: kQ τ ρ k = 0 , ∀ τ ≤ Q ρ = 0. It is straightforwardly checked that thespectral projections Q τ are related to the spectral projections P τ by the following relation: Q τ = 1 − P − τ (2.13)Let F τ be the subspace on which P τ projects . Thus, the unstable undecayed states are thosestates satisfying ρ = P ρ and they coincide with the subspace F . For these states, the The linearity that usually characterizes the relation between average values of observable A and densitymatrix M : tr( MA ) seems to be violated here, but one should not forget that (a) in the case of pure states thedensity matrix equals its square and (b) this paradox is easily solved in the case of mixtures by imposing that ρ is the square root of the density matrix M = ρ ∗ ρ , tr( MA ) = tr( ρ ∗ Aρ ). Therefore, a subspace F t is a set of decaying states prepared at time t . We call it an unstable space of T. ρ is found to decay some time duringthe interval I =]0 , t ] is kQ t ρ k = 1 − kP − t ρ k a monotonically nondecreasing quantity whichconverges to 1 as t → ∞ while kP − t ρ k tends monotonically to zero. As noticed by Misra andSudarshan [9], such quantity can not exist in the usual quantum mechanical treatment of thedecay processes. It should not be confused with the usual “survival probability of an unstablestate χ at time t ” defined by | < χ, e − itH χ > | where χ is an eigenstate of the free Hamiltonian.In fact, the last quantity is interpreted as the probability, at the instant t , for finding the systemundecayed when at time 0 it was prepared in the state χ . There is no general reason for thisquantity to be monotonically decreasing as should be any genuine probability distribution. Thisproblem does not appear in the time operator approach.Considered so, the time operator approach is non-standard. Actually, the key, non-standard,assumption that underlies the time super-operator formalism is that. In the Liouville space, given any initial state ρ , its survival probability in the unstable spaceis given by: p ρ ( t ) = kP e − i tL ρ k (2.14) This is the probability that, for a system initially in the state ρ , no decay is found during [0 , t ].Given any initial state ρ , its survival probability in the unstable space is given by [12] p ρ ( t ) = kP e − i tL ρ k = k U − t P U t ρ k = kP − t ρ k (2.15)Here we used the following relation: P − t = U − t P U t . Then, the survival probability is mono-tonically decreasing to as t → ∞ . This survival probability and the probability of finding thesystem to decay some time during the interval I =]0 , t ] , q ρ ( t ) = kQ ρ ( t ) k are related by: q ρ ( t ) = 1 − p ρ ( t ) (2.16)Therefore, q ρ ( t ) → t → + ∞ .The expression of the time operator is given in a spectral representation of H , that is, inthe representation in which H is diagonal. As shown in [13], H should have an unboundedabsolutely continuous spectrum. In the simplest case, we shall suppose that H is representedas the multiplication operator on H = L ( R + ) : Hψ ( ω ) = ωψ ( ω ) . (2.17)The Hilbert-Schmidt operators on L ( R + ) correspond to the square-integrable functions ρ ( ω, ω ′ ) ∈ L ( R + × R + ) and the Liouville-von Neumann operator L is given by : Lρ ( ω, ω ′ ) = ( ω − ω ′ ) ρ ( ω, ω ′ ) (2.18)Then we obtain a spectral representation of L via the change of variables: ν = ω − ω ′ (2.19)and E = min( ω, ω ′ ) (2.20)This gives a spectral representation of L : Lρ ( ν, E ) = νρ ( ν, E ) , (2.21)4here ρ ( ν, E ) ∈ L ( R × R + ). In this representation, T ρ ( ν, E ) = i ddν ρ ( ν, E ) so that the spectralrepresentation, of T is obtained by the inverse Fourier transform:ˆ ρ ( τ, E ) = 1 √ π Z + ∞−∞ e i τν ρ ( ν, E ) dν = ( F ∗ ρ )( τ, E ) (2.22)and T ˆ ρ ( τ, E ) = τ ˆ ρ ( τ, E ) . (2.23)The spectral projection operators P s of T are given in the ( τ, E )-representation by P s ˆ ρ ( τ, E ) = χ ] −∞ ,s ] ( τ )ˆ ρ ( τ, E ) (2.24)where χ ] −∞ ,s ] is the characteristic function of ] − ∞ , s ]. So, to obtain in the ( ν, E )-representationthe expression of these spectral projection operators, we use the Fourier transform: P s ρ ( ν, E ) = 1 √ π Z s −∞ e − i ντ ˆ ρ ( τ, E ) dτ = e − i νs Z −∞ e − iντ ˆ ρ ( τ + s, E ) dτ. (2.25)Let g ∈ L ( R ) and denote its Fourier transform by: F g ( ν ) = √ π R ∞−∞ e − i ντ g ( τ ) dτ . Using theHilbert transformation: H g ( x ) = 1 π P Z ∞−∞ g ( t ) t − x dt. (2.26)We have [14] the following formula:1 √ π Z −∞ e − i ντ g ( τ ) dτ = 12 ( F ( g ) − i H F ( g )) . (2.27)Finally, using the well-known property of the translated Fourier transform: σ s g ( τ ) = g ( τ + s ), F ( σ s g )( ν ) = e i νs F .g ( ν ) (2.28)(2.25) and (2.27) yield: P s ρ ( ν, E ) = 12 e − i νs [ e iνs ρ ( ν, E ) − i H ( e i νs ρ ( ν, E ))] . (2.29)Thus: P s ρ ( ν, E ) = 12 [ ρ ( ν, E ) − i e − i νs H ( e i νs ρ ( ν, E ))] . (2.30)It is to be noted that P s ρ ( ν, E ) is in the Hardy class H + (i.e. it is the limit as y → + of ananalytic function Φ( ν + i y ) such that: R ∞−∞ | Φ( ν + i y ) | dy < ∞ )[14]. The Friedrichs interaction Hamiltonian between the two discrete modes and the continuousdegree of freedom is given by the operator H on the Hilbert space of the wave functions of theform | ψ > = { f , f , g ( µ ) } , f , f ∈ C , g ∈ L ( R + ) H = H + λ V + λ V , (3.31)5here λ and λ are the complex coupling constants, and H | ψ > = { ω f , ω f , µg ( µ ) } , ( ω and ω > . (3.32)The operators V i ( i = 1 ,
2) are given by: V { f , f , g ( µ ) } = { < v ( µ ) , g ( µ ) >, , f .v ( µ ) } V { f , f , g ( µ ) } = { , < v ( µ ) , g ( µ ) >, f .v ( µ ) } (3.33)where < v ( µ ) , g ( µ ) > = Z dµv ∗ ( µ ) g ( µ ) , (3.34)is the inner product. Thus H can be represented as a matrix : H Friedrichs = ω λ ∗ v ∗ ( µ )0 ω λ ∗ v ∗ ( µ ) λ v ( µ ) λ v ( µ ) µ (3.35) ω , represent the energies of the discrete levels, and the factors λ i v ( µ ) ( i = 1 ,
2) represent thecouplings to the continuous degree of freedom. The energies µ of the different modes of thecontinuum range from −∞ to + ∞ when v ( µ ) = 1, but we are free to tune the coupling v ( µ ) inorder to introduce a selective cut off to extreme energy modes. Let us now solve the Schr¨odingerequation and trace out the continuum in order to derive the master equation for the two-levelsystem. The two-level Friedrichs model Schr¨odinger equation with ~ = 1 is formally written as ω λ ∗ v ∗ ( µ )0 ω λ ∗ v ∗ ( µ ) λ v ( µ ) λ v ( µ ) µ f f g ( µ ) = ω f f g ( µ ) . (3.36)That is to say: ω f ( ω ) + λ ∗ Z dµv ∗ ( µ ) g ( µ ) = ωf ( ω ) , (3.37) ω f ( ω ) + λ ∗ Z dµv ∗ ( µ ) g ( µ ) = ωf ( ω ) , (3.38)and λ v ( ω ) f ( ω ) + λ v ( ω ) f ( ω ) + µg ( ω ) = ωg ( ω ) . (3.39)The solution of (5.69), for “outgoing” wave, is: g ( µ ) = δ ( µ − ω ) − lim ǫ → λ v ( µ ) f + λ v ( µ ) f ω − µ − i ǫ . (3.40)inserting the above equation in the equations(3.37) yields f ( ω ) = λ ∗ v ∗ ( ω ) η +1 ( ω ) − (cid:18) λ ∗ λ lim ǫ → Z dµ | v ( µ ) | µ − ω − i ǫ (cid:19) f ( ω ) , (3.41)where η +1 ( ω ) = ω − ω + | λ | lim ǫ → Z dµ | v ( µ ) | µ − ( ω + i ǫ ) . (3.42)6e can also obtain the similar relations for f by changing the indices 1 with 2 and vis versaas: f ( ω ) = λ ∗ v ∗ ( ω ) η +2 ( ω ) − (cid:18) λ λ ∗ lim ǫ → Z dµ | v ( µ ) | µ − ω − i ǫ (cid:19) f ( ω ) . (3.43)By substituting f ( ω ) from the above equation in the equation (3.41) we obtain f ( ω ) = 11 − (cid:16) λ ∗ λ R dµ | v ( µ ) | µ − ω − i0 (cid:17) (cid:18) λ ∗ v ∗ ( ω ) η +1 ( ω ) − λ ∗ | λ | η +2 ( ω ) Z dµ | v ( µ ) | µ − ω − i0 (cid:19) = 11 − O ( | λ | ) (cid:18) λ ∗ v ∗ ( ω ) η +1 ( ω ) − O ( λ ∗ | λ | ) (cid:19) (3.44)Thus, to the order two approximation we have f ( ω ) ≃ λ ∗ v ∗ ( ω ) η +1 ( ω ) . (3.45)and the same formula for f as: f ( ω ) ≃ λ ∗ v ∗ ( ω ) η +2 ( ω ) . (3.46)Also denote η − i ( ω ) = η i ( ω − i ǫ ). η ± i ( ω ) , ( i = 1 ,
2) are complex conjugate of each other, we cansee that η ± i ( ω ) = ω − ω i + | λ i | P Z ∞ | v ( ω ′ ) | ω ′ − ω dω ′ ± i π | λ i | | v ( ω ) | , (3.47)where P indicates the “principal value” and we used the following identity in equation (3.47)lim ε → + x − x ± i ε = P x − x ∓ i πδ ( x − x ) . (3.48)Let | χ i = | ǫ f + ǫ f i where ǫ i , ( i = 1 ,
2) is a constant complex number. The physicalmeaning of such a state is that it corresponds to a coherent superposition of two exponentialdecay processes. In the following Section we shall compute the projection of | χ ih χ | on theunstable spaces of time operator and then the survival probability p ρ ( t ) introduced in theSection 2. We compute its expression for the density matrix | χ ih χ | in terms of the lifetimes andenergies of the (mesonic) resonances. It has been shown [15] that the average of time operator forthe state | χ ih χ | is equal to the lifetime 1 /γ in a first approximation (more precisely in the weakcoupling regime that is described in the next section (equation (3.52)). We shall characterizethe short time and long time behavior of this survival probability.Let us now identify the pure state χ with the element ρ = | χ >< χ | of the Liouville space,that is the kernel operator: ρ = X i =1 2 X j =1 ρ ij ( ω, ω ′ ) = X i =1 2 X j =1 ǫ i ǫ ∗ j f i ( ω ) f j ( ω ′ ) = X i =1 2 X j =1 ǫ i ǫ ∗ j F ij . (3.49)We shall compute the survival probability kP − s ρ k of the state ρ and show how it reaches thefollowing limit: lim s →∞ kP − s ρ k → . (3.50)7 .1 Weak coupling conditions As explained above the Liouville operator is given by equation (2.18) and the spectral represen-tation of L is given by the change of variables introduced in (2.19) and (2.20). Thus, we obtainfor F ij ( ν, E ) , ( i, j = 1 ,
2) : F ij ( ν, E ) = λ i λ ∗ j v ( E ) η − i ( E ) v ∗ ( E + ν ) η + j ( E + ν ) ν > λ ∗ i λ j v ∗ ( E ) η + j ( E ) v ( E − ν ) η − i ( E − ν ) ν < , (3.51)Admitting that η + i ( ω ) in (3.47) in the the O ( | λ | ) has one zero in the lower half-plane [16, 17]which approaches ω i for decreasing coupling, we can write: η + i ( ω ) = ω − z i . (3.52)where z i = e ω i − i γ i where γ i ∼ | λ i | is a real positive constant [17]. In this article we supposethat e ω < e ω . Easily, we can verify that η + i ( ω ) − η − i ( ω ) = i γ i . (3.53)From (3.47), we have i2 (cid:20) η + i ( ω ) − η − i ( ω ) (cid:21) = π | λ i | | v ( ω ) | | η + i ( ω ) | . (3.54)Consequently, the two above equations yield π | λ i | | v ( ω ) | | η + i ( ω ) | = γ i | η + i ( ω ) | . (3.55)Therefore, | f i ( ω ) | ∼ ω − e ω i ) + γ i which is a Breit-Wigner like distribution. This equation willbe used in the next sections. Kaons are bosons that were discovered in the forties during the study of cosmic rays. Theyare produced by collision processes in nuclear reactions during which the strong interactionsdominate. They appear in pairs K , K [3, 18].The K mesons are eigenstates of the parity operator P : P | K i = −| K i , and P | K i = −| K i .K and K are charge conjugate to each other C | K i = | K i , and C | K i = | K i . We get thus CP | K i = −| K i , CP | K i = −| K i . (4.56)Clearly | K i and | K i are not CP -eigenstates, but the following combinations | K i = 1 √ (cid:0) | K i + | K i (cid:1) , | K i = 1 √ (cid:0) | K i − | K i (cid:1) , (4.57)are CP -eigenstates. CP | K i = + | K i , CP | K i = −| K i . (4.58)8n the absence of matter, kaons disintegrate through weak interactions [18]. Actually, K andK are distinguished by their mode of production . K and K are the decay modes of kaons. Inabsence of CP -violation, the weak disintegration process distinguishes the K states which decayonly into “2 π ” while the K states decay into “3 π, πeν, ... ” [19]. The lifetime of the K kaon isshort ( τ S ≈ . × −
11 s ), while the lifetime of the K kaon is quite longer ( τ L ≈ . × − ). CP - violation was discovered by Christenson et al . [20]. CP -violation means that the long-lived kaon can also decay to “2 π ”. Then, the CP symmetry is slightly violated (by a factor ofthe order of 10 − ) by weak interactions so that the CP eigenstates K and K are not exacteigenstates of the decay interaction. Those exact states are characterized by lifetimes that arein a ratio of the order of 10 − , so that they are called the short-lived state (K S ) and long-livedstate (K L ). They can be expressed as coherent superpositions of the K and K eigenstatesthrough | K L i = 1 p | ǫ | (cid:2) ǫ | K i + | K i (cid:3) , | K S i = 1 p | ǫ | (cid:2) | K i + ǫ | K i (cid:3) , (4.59)where ǫ is a complex CP -violation parameter, | ǫ | ≪ ǫ does not have to be real. K L andK S are the eigenstates of the Hamiltonian for the mass-decay matrix [18, 19] which has thefollowing form in the basis | K i and | K i : H = M − i2 Γ ≡ (cid:18) M − i2 Γ M − i2 Γ M − i2 Γ M − i2 Γ (cid:19) (4.60)where M and Γ are individually hermitian since they correspond to observables (mass andlifetime). The corresponding eigenvalues of the mass-decay matrix are equal to m L − i2 Γ L , m S − i2 Γ S (4.61)The CP -violation was established by the observation that K L decays not only via three-pion,which has natural CP parity, but also via the two-pion (“2 π ”) mode with an experimentallyobserved violation amplitude | ǫ exp. | of the order of 10 − , which was truly unexpected at the time.Let us now reconsider how the simple model (4.59), (4.60) is related to the experimental data.A series of detections is performed at various distances from the source of a neutral kaon beamin order to estimate the variation of the populations of emitted pion π + , π − pairs in functionof the proper time. This is done for times of the order of τ S . The experiment shows that aninterference term is present in the expression of the excitation rates of detectors in function oftheir distance to the source. It follows from (4.59) that the transition amplitude of the K L beamis given by ψ ( t ) = A (cid:16) e − i( m S − i2 Γ S ) t + ǫ exp e − i( m L − i2 Γ L ) t (cid:17) (4.62)with A a global proportionality factor that remains constant in time. Then the intensity I ( t ) = | ψ ( t ) | is given by: I ( t ) = I (cid:16) e − Γ S t + | ǫ exp | e − Γ L t + | ǫ exp | e − ( Γ S +Γ L ) t ) cos( △ mt + arg( ǫ exp )) (cid:17) (4.63)where | ǫ exp | = Amplitude ( K L → π + , π − )Amplitude ( K S → π + , π − ) (4.64)By fitting the expressions (4.63) and (4.64) with the observed data one derives an estimation ofthe mass difference between the short and long lived state as well as the phase of ǫ exp and itsamplitude. 9ll this leads to an experimental estimation of ǫ exp [21] | ǫ exp | = (2 . ± . × − , arg( ǫ exp ) = (43 . . ◦ . (4.65) C P -violation in theHilbert space
Let us present the fundamental ideas of the theory of spontaneous emission of an atom inter-acting with the electromagnetic field, given by Wigner and Weisskopf . This treatment aims atobtaining an exponential time dependence for decaying states by integrating over the continuumenergy. That is, we assume that the modes of the fields are closely spaced. Then, we have toassume that the variation of v ( µ ) over µ is negligible with | µ | . “uncertainty of the originalstate energy”, i.e. v ( µ ) ≈ v independent of µ or in the simple case it is taken to obey v ( µ ) = 1.Also another assumption is that the lower limit of integration over ω is replaced by −∞ .The two-level Friedrichs model time-dependent Schr¨odinger equation, in the Wigner-Weisskopfregime becomes: ω λ ∗ ω λ ∗ λ λ µ f ( t ) f ( t ) g ( µ, t ) = i ∂∂t f ( t ) f ( t ) g ( µ, t ) . (5.66)which means: ω f ( t ) + λ ∗ Z ∞−∞ dµg ( µ, t ) = i ∂f ( t ) ∂t , (5.67) ω f ( t ) + λ ∗ Z ∞−∞ dµg ( µ, t ) = i ∂f ( t ) ∂t , (5.68)and λ f ( t ) + λ f ( t ) + µg ( µ, t ) = i ∂g ( µ, t ) ∂t . (5.69)Let us now solve the Schr¨odinger equation and trace out the continuum in order to derive themaster equation for the two-level system. From the equation (5.69) we can obtain g ( µ, t ), taking g ( µ,
0) = 0, as g ( µ, t ) = − i e − i ωt Z t dτ (cid:2) λ f ( τ ) + λ f ( τ ) (cid:3) e i ωτ , (5.70)where t >
0. Then, we substitute g ( µ, t ) in the equation (5.67) and we obtaini ∂f ( t ) ∂t = ω f ( t ) − i λ ∗ Z ∞−∞ dµe − i µt Z t dτ (cid:2) λ f ( τ ) + λ f ( τ ) (cid:3) e i µτ , (5.71)we also obtain the same relation for f ( t ) from equation(5.68):i ∂f ( t ) ∂t = ω f ( t ) − i λ ∗ Z ∞−∞ dµe − i µt Z t dτ (cid:2) λ f ( τ ) + λ f ( τ ) (cid:3) e i µτ . (5.72)Finally, one obtains the following Markovian form of the reduced Schr¨odinger equation [22]i ∂∂t (cid:18) f ( t ) f ( t ) (cid:19) = (cid:18) ω − i π | λ | − i πλ ∗ λ − i πλ λ ∗ ω − i π | λ | (cid:19) (cid:18) f ( t ) f ( t ) (cid:19) . (5.73)10hus, we obtain an effective non-Hermitian Hamiltonian evolution, H eff = M − i γ . The eigen-values of the above effective Hamiltonian under the weak coupling constant approximation are: ω + = ω − i π | λ | + O ( λ ) , ω − = ω − i π | λ | + O ( λ ) , (5.74)In a first and very rough approximation, the eigenvectors of the effective Hamiltonian are thesame as the postulated kaons states. | f + i = (cid:18) (cid:19) = | K i and | f − i = (cid:18) (cid:19) = | K i , (5.75)Phenomenology imposes that the complex Friedrichs energies ω ± coincide with the observedcomplex energies. The Friedrichs energies depend on the choice of the four parameters ω , ω , λ and λ and the observed complex energies are directly derived from the experimentaldetermination of four other parameters, the masses m S and m L and the lifetimes τ S and τ L .We must thus adjust the theoretical parameters in order that they fit the experimental data.This can be done by comparing the eigenvalue of the effective matrix with the eigenvalue of themass-decay matrix which is taken in the expression (4.61). Finally, we have ω = m S , π | λ | = Γ S ,ω = m L , π | λ | = Γ L . (5.76)The above identities yield λ = r Γ S π e i θ S , λ = r Γ L π e i θ L (5.77)where θ S and θ L are real constants. CP T invariance:
Let us now discuss the
CP T invariance in our model. As mentioned in thetexts books like [18, 19],
CP T invariance imposes some conditions on the mass-decay matrix,i.e. M = M , Γ = Γ , M = M ∗ and Γ = Γ ∗ (5.78)in the K and K bases. But, we note that our effective Hamiltonian is written in the K andK bases. Thus, we have to rewrite in the K and K bases. Thus, the transformation matrix T from the K and K bases to the K and K bases is obtained as T = 1 √ (cid:18) − (cid:19) = T − . (5.79)Then, the effective Hamiltonian in the K and K bases, H is obtained by H = T H eff T − = 12 (cid:18) − (cid:19) (cid:18) ω − i π | λ | − i πλ ∗ λ − i πλ λ ∗ ω − i π | λ | (cid:19) (cid:18) − (cid:19) . (5.80)we have, H = (cid:18) ( m S + m L ) − i2 (cid:0) Γ S + Γ L + 2 √ Γ S Γ L cos △ θ (cid:1) , ( m S − m L ) − i2 (cid:0) Γ S − Γ L + 2i √ Γ S Γ L sin △ θ (cid:1) ( m S − m L ) − i2 (cid:0) Γ S − Γ L − √ Γ S Γ L sin △ θ (cid:1) , ( m S + m L ) − i2 (cid:0) Γ S + Γ L − √ Γ S Γ L cos △ θ (cid:1) (cid:19) . (5.81)where △ θ = θ L − θ S . CP T invariance conditions in (5.78) impose that △ θ = kπ + π , ( k = · · · , − , , , · · · ) . (5.82)11ere we choose k = −
1, consequently, △ θ = − π . Then, we have M = M = ( m S + m L ) , Γ = Γ = Γ S + Γ L ,M = M ∗ = ( m S − m L ) , Γ = Γ ∗ = Γ S − Γ L − √ Γ S Γ L . (5.83) CP -violation: Let us study in this case the CP -violation. The Friedrichs model allows us toestimate the value of ǫ . For this purpose, the effective Hamiltonian (5.73) acts on the | K S i vectorstates (4.59) as an eigenstate corresponding to the eigenvalue ω + = ω − i π | λ | = m S − i Γ S ,so that we must impose that H eff | ( f + + ǫf − ) i = H eff (cid:0) ǫ (cid:1) = ω + (cid:0) ǫ (cid:1) , from which we obtain afterstraightforward calculations that ǫ = √ Γ L Γ S ( m L − m S ) − i2 (Γ L − Γ S ) . (5.84)By using the experimental ratio ( m L − m S ) − (Γ L − Γ S ) ≈ △ mτ S ≈ .
47 and the above experimental valuesof Γ L , Γ S , m L , m S , we obtain the following estimated value for ǫ : ǫ = r Γ L Γ S e i(46 . ◦ = r . × − e i(46 . ◦ = 14 ǫ exp . (5.85)We see that the ǫ argument is the same as the experimental value but the magnitude of the CP -violation parameter is quite larger than its experimental value.The reason is that, as we have shown in a previous work [11], we did not normalize correctlythe amplitudes associated to the two interfering decay processes (short and long). In that workwe solved the problem by developing an analogy between the temporal density of decay and thespatial density of presence (this constitutes the so-called wave function approach).Now we shall derive intensity formula for the meson decay [11] using the formalism of thetime operator ( T ′ ) sketched in Section 2. By considering the relations (3.52) and (3.55) andsupposing the v ( ω ) is a real function, we can write the f ( ω ) and f ( ω ), the equations (3.45)and (3.46), as: f i ( ω ) = q γ i e − i θ i ω − e ω i + i γ i , ( i = 1 , , (5.86)where θ i is the phase of the possibly complex coefficients λ i . By using the Fourier transforms,i.e. equation (2.3), for the above equation, (5.86), we obtain for ( i = 1 , f i ( τ ) = (cid:26) N √ π γ i e − (i e ω i + γi ) τ − i θ i , τ ≥ , τ < N is the normalization constant. For s = − τ <
0, we haveˆ f i ( s ) = √ πγ i e (i e ω i + γi ) s − i θ i , s ≤ , s > . (5.88)Finally, the normalization relation, i.e. Z + ∞−∞ ds | ˆ f i ( s ) | = 1 , ( i = 1 ,
2) (5.89)12ields: ˆ f i ( s ) = √ γ i e (i e ω i + γi ) s − i θ i , s ≤ , s > . (5.90)Here f i ( s ) , ( i = 1 ,
2) is the form of the density of the probability. Thus, the intensity is obtainedby I ( s ) = | C | | ǫ f ( s ) + ǫ f ( s ) | = I (cid:18) e γ s + | ǫ | γ γ e γ s + | ǫ | r γ γ e ( γ γ s cos(( e ω − e ω ) s + θ − θ + arg( ǫ )) (cid:19) (5.91)where ǫ = ǫ /ǫ and C and I = | C | ǫ γ are the constants. This corresponds to an effectivevalue for ǫ that is no longer 14 times too large as in expression (5.85) because it must berenormalized. Identifying equations (4.63) and (5.91) it is easy to show, as we have also donein [11], that, ǫ th , the theoretical prediction for the experimental CP -violation parameter, obeys ǫ th = ǫ r Γ L Γ S = Γ L Γ S △ m Γ S − i △ γ S . (5.92)Substituting in the expression (5.92) the physically observed masses and lifetimes of theshort and long kaon states we find that ǫ th ≈ . ǫ exp which constitutes an improvement incomparison to the non-renormalized estimation (5.85). We shall also reconsider similar resultsin the case of B and D particles in a next section.In the next coming section, we shall use the time super-operator ( T ) formalism as a nonWigner-Weisskopf approximation method to obtain the CP -violation parameter. This formalismalso predicts a CP -violation parameter comparable to the experimental value. T in a Friedrichs model In this section, we will compute the survival probability and we obtain the theoretical CP -violation parameters for the mesons K, B and D. Then, we compare our results to the ex-perimental CP -violation parameters. We shall see that our theoretical results provide a goodestimation of the experimentally measured quantities. Moreover, a fine structure appears in thecase of kaons, which brought us to conceive an experimental test aimed at falsifying the timesuper-operator approach, that we shall discuss in the conclusion.By considering v ( ω ) a real test function and using the equation (3.55) we obtain F ji ( ν, E )in the following form: F ji ( ν, E ) = λ j λ ∗ i ν ∗ j ( ν + ν i ) ν > λ ∗ j λ i ν i ( ν ∗ j − ν ) ν < . (6.93)where i, j = 1 , ν i := a i + i b i := ( E − e ω i ) + i γ i . (6.94)For obtaining P s F ij ( ν, E ) ( s < G ji ( ν, E ) = H ( e i sν F ji )( ν, E ) = 1 π P Z ∞−∞ e i sx F ji ( x, E ) x − ν dx (6.95)13ow, we substitute (6.93) in (6.95), so we have G ji ( ν, E ) = 1 π P " λ i λ ∗ j Z −∞ e i sx ν i ( x − ν )( ν ∗ j − x ) dx + λ ∗ i λ j Z + ∞ e i sx ν ∗ j ( x − ν )( ν i + x ) dx (6.96)which for the ν > G ji ( ν, E ) = 1 π " λ i λ ∗ j Z −∞ e i sx ν i ( x − ν )( ν ∗ j − x ) dx + λ ∗ i λ j P Z + ∞ e i sx ν ∗ j ( x − ν )( ν i + x ) dx . (6.97)A complete computation of the G ii ( ν, E ) is showed in [17]. Finally, P s F ij ( ν, E ) is obtained as:for i = j P s F ii ( ν, E ) = i | λ i | e − i sν (cid:20) − πν i ( ν ∗ i − ν ) (cid:18) Z −∞ e − sy y + i ν ∗ i dy − Z −∞ e − sy y + i ν dy (cid:19) + 12 πν ∗ i ( ν + ν i ) (cid:18) Z −∞ e − sy y − i ν i dy − Z −∞ e − sy y + i ν dy (cid:19)(cid:21) + | λ i | e − i sν [ e i sν ∗ i ν i ( ν ∗ i − ν ) − e − i sνi ν ∗ i ( ν i + ν ) ] , E < e ω , E > e ω . (6.98)and by considering e ω i < e ω j , F ij for i = j have the following form : P s F ji ( ν, E ) = i e − i sν (cid:20) − λ i λ ∗ j πν i ( ν ∗ j − ν ) (cid:18) Z −∞ e − sy y + i ν ∗ j dy − Z −∞ e − sy y + i ν dy (cid:19) + λ ∗ i λ j πν ∗ j ( ν + ν i ) (cid:18) Z −∞ e − sy y − i ν i dy − Z −∞ e − sy y + i ν dy (cid:19)(cid:21) + e − i sν [ λ i λ ∗ j e i sν ∗ j ν i ( ν ∗ j − ν ) − λ ∗ i λ j e − i sνi ν ∗ j ( ν i + ν ) ] , E < e ω i λ i λ ∗ j e − i sν e i sν ∗ i ν i ( ν ∗ j − ν ) , e ω i < E < e ω j , E > e ω j . (6.99)In the equations (6.98) and (6.99) the non-integral terms yield the poles and lead to the reso-nance, and the integral terms yield an algebraical term analog to the background in the Hamil-tonian theories [23]. We can also compute the same result for the case ν <
0. We will neglectthe the background (the integrals terms). Then, the above equation is rewritten as: P s F ii ( ν, E ) ≃ | λ i | e − i sν [ e i sν ∗ i ν i ( ν ∗ i − ν ) − e − i sνi ν ∗ i ( ν i + ν ) ] , E ≤ e ω , E > e ω . and for i = j P s F ij ( ν, E ) ≃ e − i sν [ λ i λ ∗ j e i sν ∗ j ν i ( ν ∗ j − ν ) − λ ∗ i λ j e − i sνi ν ∗ j ( ν i + ν ) ] , E ≤ e ω i λ i λ ∗ j e − i sν e i sν ∗ i ν i ( ν ∗ j − ν ) , e ω i < E ≤ e ω j , E > e ω j . p ρ ( s ) = k P ρ ( s ) k = k | ǫ | P s F ( ν, E ) + ǫ ǫ ∗ P s F ( ν, E ) + ǫ ǫ ∗ P s F ( ν, E ) + | ǫ | P s F ( ν, E ) k (6.100)where k · k = R ∞ dE R ∞−∞ dν | · | . We see that P ρ ( s ) can be written as : P ρ ( s ) ≃ e − i sν (cid:20) (cid:16) ǫ ∗ λ ν + ǫ ∗ λ ν (cid:17) (cid:18) ǫ λ ∗ e i sν ∗ ν ∗ − ν + ǫ λ ∗ e i sν ∗ ν ∗ − ν (cid:19) − (cid:16) ǫ λ ν ∗ + ǫ λ ν ∗ (cid:17) (cid:16) ǫ ∗ λ ∗ e − i sν ν + ν + ǫ ∗ λ ∗ e − i sν ν + ν (cid:17) (cid:21) E ≤ e ω ,e − i sν (cid:20) (cid:16) ǫ ∗ λ ν + ǫ ∗ λ ν (cid:17) ǫ λ ∗ e i sν ∗ ( ν ∗ − ν ) − (cid:16) ǫ λ ν ∗ + ǫ λ ν ∗ (cid:17) ǫ ∗ λ ∗ e − i sν ( ν + ν ) (cid:21) , e ω < E ≤ e ω , E > e ω (6.101)Now, by remembering that b i = | λ i | , ( i = 1 , P ρ ( s ) is obtained as: | P ρ ( s ) | ≃ (cid:12)(cid:12)(cid:12) ǫ λ ν + ǫ λ ν (cid:12)(cid:12)(cid:12) (cid:20) | ǫ | | λ | e b s | ν ∗ − ν | + | ǫ | | λ | e b s | ν ∗ − ν | + | ǫ | | λ | e b s | ν + ν | + | ǫ | | λ | e b s | ν + ν | + e ( b + b ) s (cid:16) ǫ ǫ ∗ λ ∗ λ e i( a − a s ( ν ∗ − ν )( ν − ν ) + ǫ ǫ ∗ λ ∗ λ e i( a − a s ( ν ∗ + ν )( ν + ν ) + C . C . (cid:17) (cid:21) , E ≤ e ω (cid:12)(cid:12)(cid:12) ǫ λ ν + ǫ λ ν (cid:12)(cid:12)(cid:12) (cid:20) | ǫ | | λ | e b s | ν ∗ − ν | + | ǫ | | λ | e b s | ν + ν | (cid:21) , e ω < E ≤ e ω , E > e ω (6.102)where the terms that oscillate with a frequency equal to the difference of the two masses, i.e.( e ω − e ω ) is kept, the other decay terms oscillating with the frequency of one of the masses onlyare neglected since we have in the weak-coupling regime and the high-mass.The integral over ν arrives at: Z ∞−∞ dν | P ρ ( s ) | ≃ π (cid:12)(cid:12)(cid:12) ǫ λ ν + ǫ λ ν (cid:12)(cid:12)(cid:12) (cid:20) | ǫ | e b s + | ǫ | e b s + (cid:16) ǫ ∗ ǫ λ ∗ λ e ( b b s e i( e ω − e ω s ( e ω − e ω )+i( b + b ) + C . C . (cid:17) (cid:21) , E ≤ e ω π (cid:12)(cid:12)(cid:12) ǫ λ ν + ǫ λ ν (cid:12)(cid:12)(cid:12) | ǫ | e b s , e ω < E ≤ e ω , E > e ω (6.103)Only the terms of the square norm are depended to E and we have (cid:12)(cid:12)(cid:12)(cid:12) ǫ λ ν + ǫ λ ν (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ǫ λ E − e ω + i b (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ǫ λ E − e ω + i b (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) ǫ ∗ λ ∗ ( E − e ω + i b ) ǫ λ ( E − e ω − i b ) + C . C . (cid:19) (6.104)The integral over E of the above expression is like the following integrals Z dE (cid:12)(cid:12)(cid:12)(cid:12) √ b i ( E − e ω i ) + i b i (cid:12)(cid:12)(cid:12)(cid:12) = arctan (cid:18) E − e ω i b i (cid:19) (6.105)15nd Z dE λ ∗ λ ( x − a + i b )( E − a − i b ) = − λ ∗ λ ( e ω − e ω ) + i( b + b ) (cid:18) i arctan b E − e ω +i arctan b E − e ω + log q ( E − e ω ) + b − log q ( E − e ω ) + b (cid:19) (6.106)Now, we integrate from equation(6.104) over E from 0 to ∞ . Firstly, for the interval E ∈ [0 , e ω ]we have I = Z e ω dE (cid:12)(cid:12)(cid:12)(cid:12) ǫ λ ν + ǫ λ ν (cid:12)(cid:12)(cid:12)(cid:12) = | ǫ | arctan e ω b + | ǫ | (cid:18) arctan e ω − e ω b + arctan e ω b (cid:19) − (cid:20) (cid:18) ǫ ∗ ǫ √ b b ( e ω − e ω ) + i( b + b ) (cid:19) (cid:18) i( π b e ω + arctan b e ω + arctan b e ω − e ω ) + 12 log b ( e ω + b )( e ω + b )(( e ω − e ω ) + b ) (cid:19) + C . C . (cid:21) . (6.107)For E ∈ ] e ω , e ω ] we have I = Z e ω e ω dE (cid:12)(cid:12)(cid:12)(cid:12) ǫ λ ν + ǫ λ ν (cid:12)(cid:12)(cid:12)(cid:12) = | ǫ | arctan e ω − e ω b + | ǫ | arctan e ω − e ω b − (cid:20) (cid:18) ǫ ∗ ǫ √ b b ( e ω − e ω ) + i( b + b ) (cid:19) (cid:18) i(arctan b e ω − e ω − arctan b e ω − e ω )+ 12 log b b (( e ω − e ω ) + b )(( e ω − e ω ) + b ) (cid:19) + C . C . (cid:21) . (6.108) For the weak-coupling constants we have b i ≪ e ω i , ( i = 1 ,
2) and also by supposing e ω ∼ e ω ,( e ω − e ω ) ∼ b and b b ≪
1, we have I ≃ π (cid:18) | ǫ | + 2 | ǫ | + (cid:18) ǫ ∗ ǫ λ ∗ λ ( e ω − e ω ) + i( b + b ) + C . C . (cid:19)(cid:19) ≈ π I ≃ π (cid:18) | ǫ | + 2 | ǫ | + (cid:18) ǫ ∗ ǫ λ ∗ λ ( e ω − e ω ) + i( b + b ) + C . C . (cid:19)(cid:19) ≈ π | ǫ | + | ǫ | ) = 1.Finally, we obtain p ρ ( s ) ≃ π (cid:20) | ǫ | e b s + 32 | ǫ | e b s + i ǫ ∗ ǫ λ ∗ λ e ( b + b ) s e i( e ω − e ω ) s ( e ω − e ω ) + i( b + b ) + C . C . ! (cid:21) ≃ π | ǫ | (cid:20) e b s + 32 | ǫ | e b s + i ǫλ ∗ λ e ( b + b ) s e i( e ω − e ω ) s ( e ω − e ω ) + i( b + b ) + C . C . ! (cid:21) . (6.111)where ǫ = | ǫ | e i φ := ǫ ǫ . (6.112)16he derivative of the equation (6.111) yields the time super-operator density of the probabilityor intensity: I ( s ) := dp ρ ( s ) ds = I (cid:20) e b s + | ǫ | b b e b s + 2 | ǫ | r b b e ( b + b ) s cos(( e ω − e ω ) s + φ + θ − θ ) (cid:21) . (6.113)where I = ( π | ǫ | b ) / λ i = √ b i e i θ i , ( i = 1 , term from theintensity derived previous by [11] from the integrated probability of decay of two exponentiallydecay process or relation (5.92) that we obtained in the Hilbert space which we call the timeoperator prediction.Let us now evaluate the predictions related to the above equation in the different timeintervals and let us compare them with the intensity introduced in the equation (4.63). Firstly,for t = − s ∼ × τ S or t ≫ τ S which that the term effective is: | ǫ | b b e b s and comparing forthe same time with the equation (4.63) yields the CP -violation parameter is | ǫ | b b . Thus, theequations (6.113) and (4.63) for t = − s ≫ τ S can be written approximately as I ( s ) ≈ I (cid:12)(cid:12)(cid:12) ǫ th (cid:12)(cid:12)(cid:12) e b s and I ( t ) ≈ I | ǫ exp | e − γ L t , ( − s = t ≫ τ S ) (6.114)where ǫ th = ǫ r b b (6.115)and the coefficient q b b in the above equation is the correction which is obtained by the timeoperator formalism and by using the condition ( e ω − e ω ) ∼ b = 0, then I = 0. Secondly, forthe time of the order oft τ S ( t < τ S ) we have I ( s ) ≈ I e b s and I ( t ) ≈ I e − γ S t , ( t < τ S ) (6.116)Finally, for intermediate times (5 τ S < t < τ S ) we have I ( s ) ≈ I | ǫ | b b e b s cos(( e ω − e ω ) s + φ + θ − θ ) and I ( t ) ≈ I | ǫ exp | e − “ Γ S +Γ L ” t cos(( m L − m L ) s + arg( ǫ exp )) (6.117)The equations (6.114), (6.116), (6.117) and (5.77) yield b = γ = Γ S = τ S , e ω = m S ,b = γ = Γ L = τ L , e ω = m L ,θ = θ S , θ = θ L (6.118)The ǫ is obtained in (5.85), thus, we have ǫ th = ǫ r
32 Γ L Γ S e − i π ! = r
32 Γ L Γ S △ m Γ S − i △ γ S (6.119)where △ m = ( m L − m S ) and △ γ = (Γ L − Γ S ). Then, by replacing the experimental data wehave ǫ th = 1 . × − e i(46 . ◦ ) = 0 . ǫ exp (6.120)17 .2 B-meson It easy to see that the integral I , for the B-mesons and D-mesons, is zero. So the intensity iswritten as: I ( s ) = I (cid:20) e b s + | ǫ th1 | e b s + 2 | ǫ th1 | e ( b + b ) s cos(( e ω − e ω ) s + φ ) (cid:21) (6.121)where ǫ th = ǫ r b b e − i π (6.122)This expression is the same, in the case of B and D particles in the time operator and in thesuper-operator approaches, and the theoretically estimated CP -violation parameter obeys thefollowing equation ǫ th = ǫ r Γ L Γ S e − i π = Γ L Γ S △ m Γ S − i △ γ S (6.123)which is not true in the case of K particles. Also for B and D particles the agreement withobservations is quite good as we shall now check.Another example is the CP -violation in the decay of B s and B s . The experimental valuesare [24] △ Γ s s = 0 . +0 . − . , s = 1 . +0 . − . ps , (6.124)or equivalently (Γ L,H = Γ s ± △ Γ s / L = 1 . +0 . − . ps , H = 1 . +0 . − . ps , (6.125)and the difference of masses is △ m = 17 . +6 . − . ps − (6.126)and the experimental CP -violation parameter of the B meson is [24, 25]: A exp SL ≃ R e ( ǫ exp B ) = ( − . ± . × − ⇒ (cid:12)(cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:12) exp = 1 . ± . . (6.127)where A exp SL ≈ − (cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12) exp . By replacing in the equation (6.123) we obtain: ǫ th B = Γ L Γ H △ m Γ s − i △ Γ s s = 0 .
018 + 0 . × − i (6.128)Thus, our theoretical (cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12) th prediction is: (cid:12)(cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:12) th = (cid:12)(cid:12)(cid:12)(cid:12) − ǫ th ǫ th (cid:12)(cid:12)(cid:12)(cid:12) = 0 .
96 (6.129)which is in fairly good agreement with the experimental value.18 .3 D-meson
The other example is the CP -violation in the decay of D meson. The experimental values for CP -violation of D → K S π + π − as reported by Belle [26] are as follows: △ Γ2Γ = (cid:0) . ± . +0 . . − . − . (cid:1) , (6.130) △ m Γ = (cid:0) . ± . +0 . . − . − . (cid:1) (6.131)where 1 / Γ = τ, ( ~ = 1) is the mean life time1Γ = τ = τ D + τ D . ± . × − ps (6.132)The CP -violation parameters are experimentally denoted by (cid:16) qp (cid:17) and given by: (cid:12)(cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:12) exp = (cid:12)(cid:12)(cid:12)(cid:12) − ǫ exp ǫ exp (cid:12)(cid:12)(cid:12)(cid:12) = (cid:0) . +0 . . − . − . (cid:1) (6.133)and φ exp = arg (cid:18) qp (cid:19) exp = arg (cid:18) − ǫ exp ǫ exp (cid:19) = (cid:0) − +16+5+2 − − − (cid:1) ◦ . (6.134)By replacing in the expression (6.123) we obtain ǫ th = (0 .
077 + 0 . . (6.135)Consequently, (cid:12)(cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:12) th = 0 . , φ th = − . ◦ . (6.136)which is once again in fairly good agreement with the experimental value. About the relevance and novelty of our results
As we can see, the accuracy of the prediction (6.119) is comparable to the one that we derivedwithin the Wigner-Weisskopf approach (5.92) (time operator instead of time super-operator).Now, as we said before, the present results were derived under the assumption that the spectrumof the continuous mode was not bounded by below (no cut-off). In a precedent publication[27], we considered the Friedrichs model with a Gaussian factor form and energy bounded bybelow (the spectrum of the continuous mode was assumed there to vary from 0 to + ∞ ). Weshowed that by introducing a cut-off in the coupling between discrete and continuous modes,the estimated value of ǫ slightly differs, depending on the shape that we impose to the cut-off. Therefore a fine tuning of the estimated CP -violation parameter is possible provided thatthe factor form is chosen conveniently. Considered so, the precision of the agreement withthe measured value of the CP -violation parameter is not very convincing by it self (3 timesthe experimental value of the kaon CP -violation parameter [27]). What is convincing in ourapproach is that we obtain the right order of magnitude for the K, B and D particles altogether.19 crucial experiment for testing the validity of the Time Super-Operator ( T )Formalism The most important novelty of the time operator approach is, in our eyes, that it predicts thatthe distribution in time of the measured populations of pions pairs significantly differs fromthe predictions that could be made in the standard approach and/or in the Wigner-Weisskopfapproach provided we make a fit over the full distribution (which means not only for timeslarger than the lifetime of the “Short” state but also for times comparable to it). Indeed,taking account of the three contributions of the distribution, which are the purely exponential,”Short” and ”Long” contributions, and the oscillating contribution, one sees that the expression(6.113) radically differs from the expressions (4.63) and (5.91). This is due to the presence of thecoefficient b b in the above equation which is the correction obtained by the time super-operatorformalism and by using the condition ( e ω − e ω ) ∼ b , that is, ( m L − m s ) ∼ Γ S for kaons. Sincein the case of the B and D mesons, no such relation exists, the formula obtained here coincidewith the one derived using the Wigner-Weisskopf time operator approach [11].So, one can conceive crucial experiments that would allow to falsify the time operator ap-proach and do not radically differ from the original Christenson experiment. These experimentsrequire to measure the population of pairs of pions over a large range of times (distances tothe source), and to check whether the best fit is provided by the expression (6.115) or by theexpressions (4.63) and (5.91).In principle these effects will be tested on the LHC at CERN in the coming months (years)so that the crucial experiment that we propose here is feasible in the future. Concluding remark
The formalism of the mass-decay matrix for the kaon decay was introduced by LOY [4]. Thenseveral other authors [8, 6, 22] improved this model. The LOY model requires the Wigner-Weisskopf approximation, i.e. it requires to assume that the energy interval varies from −∞ to+ ∞ and also that the coupling between discrete and continuous modes is not restricted by afactor form or cut-off.In [22], we used the 2-level Friedriche model and the Wigner-Weisskopf approach to obtaina mass-decay matrix. This approach was improved by using a new concept of probability decaydensity for mesons in [11]. Beyond the Wigner-Weisskopf approximation, we used the Friedrichsmodel with a cutoff that amounts to bound from below the energy spectrum of the Hamiltonian[27]. In the present paper, we derived the decay probability density in the formalism of the timesuper-operator, that also goes beyond the Wigner-Weisskopf approximation. References [1] Friedrichs K O 1948 On the perturbation of continuous spectra Communications on
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