New model of the kaon regeneration

It is shown that in the standard model of $K^0_{S}$ regeneration a system of noncoupled equations of motion is used instead of the coupled ones. A model alternative to the standard one is proposed. The calculation performed by means of diagram technique agrees with that based on exact solution of equations of motion.


Introduction
The effect of kaon regeneration has been known since the 1950s. However, in the previous calculations [1][2][3] a system of noncoupled equations of motion was considered (see Eq. (1)) instead of the coupled ones. This is a fundamental defect because it leads to a qualitative disagreement in the results. This means that the regeneration has not been described at all.
The result obtained in [2] was adduced in [4,5] and subsequent papers. In this paper we consider the model based on the exact solution of the coupled equations of motion with the potentials taken in general form. The comparison with the previous model and our calculation performed by means of the diagram technique is given as well.
Let K 0 L fall onto the plate at t = 0. The probability of finding K 0 S is of our particular interest. Our approach is as follows. Since K 0 N-andK 0 N-interections are known, we go to K 0 ,K 0 representation. The problem is described by coupled equations of motion for K 0 (t) and K 0 (t). We find the corresponding solutions and revert to K 0 L , K 0 S representation.
We consider all the possibilities. If α and α ′ correspond to K 0 S and K 0 L , α 1 and α 2 describe K 0 andK 0 , whereas our interest is with K 0 S and K 0 L . Besides, the indexes of refraction for K 0 L and K 0 S are unknown. Let α = K 0 and α ′ =K 0 . Then α 1 = K 0 S and α 2 = K 0 L . This variant follows from the initial conditions (see [2]): α 2 (0) = 1, α 1 (0) = 0. There is no off-diagonal mass ǫ = (m L − m S )/2. 2 Let K 0 L fall onto the plate at t = 0. We use the model described in paragraph 2 of introduction. In Ref. [6] the exact wave function K S (t) of K 0 S has been calculated. The probability of finding K 0 S or, what is the same, the probability of K 0 L K 0 S transition is given by Eq. (12) of Ref. [6]: This expression is exact; | K S (t = 0) | 2 = 0.
If ImV = 0, then Imp = 0 as well. In this case This is a pure oscillation regime. Here regeneration by scattering takes place. The regeneration by absorption is described by ImV .
As in [6] we put p and Γ d are the mass and width of decay of K 0 , respectively). Let's denote Γ a K 0 and Γ aK 0 as widths of absorption (not decay) of K 0 andK 0 , respectively. Then where Equation (2) gives: where Γ(K L → K S ) is the width of K L K S transition (regeneration). The value ∆m is involved in Γ(K L → K S ) and cos(Re(pt)).
In this case The t-dependence is given by the exponential decay law. It is significant that ReV = 0 in contrast to [6]. (Note that (14) is valid if ∆Γ > 2ǫ.)

Connection between the models based on the diagram technique and the exact solution
The calculation presented above is cumbersome and formal. So verification is required. In [11] an approach based on the perturbation theory was proposed. Regeneration followed by the decay K 0 L → K 0 S → ππ was considered. The similar approach is used for the nn transition in a medium followed by annihilation (see Refs. [12][13][14][15]). The process amplitude M(K 0 (See the second term of Eq. (23) of Ref. [11].) Here M d (K 0 S → ππ) is the in-medium amplitude of the decay K 0 S → ππ. The corresponding process width is where Γ d (K 0 S → ππ) is the width of the decay K 0 S → ππ. Consider now the connection between the models based on the diagram technique and the exact solution. In this case we write (16) in the form where W is the probability of the K 0 S decay in the channel K 0 S → ππ. The physical sense of (17) is obvious: the multistep process K 0 L → K 0 S → ππ involves the subprocess of K L K S transition (regeneration). Equation (17) is verification of the models considered above.
Due to a strong absorption ofK 0 and zero momentum transfer in the K 0K 0 transition vertex the description of competition between scattering and absorption is of particular importance.
In this regard the diagram technique has some advantage over the model based on the equations of motion (see Refs. [14,15]).
The opposite case when ReV = 0 is more interesting. Then and Here regeneration by absorption takes place. Comparing (19) and (21) we see that ReV = 0 violates t 2 -dependence. Let us revert to Eqs. (8)-(10). Γ a K 0 and Γ aK 0 are given by standard expressions which follow from the optical theorem: and where N n and N p are the number of netrons and protons in a unit of volume, respectively; σ(K 0 N) and σ(K 0 N) are the total cross sections of K 0 N-andK 0 N-interactions, v is the velocity of the K 0 meson. By way of illustration we take σ(K 0 n) = σ(K 0 p) = 15 mb [16]. As in Ref. [2], we use σ(K 0 N) = 1 3 σ(K 0 N). Instead of cross sections one can use the forward scattering amplitudes of kaons by the molecules of the medium. In this case f 21 = f −f . Here N m is the number of molecules in a unit of volume, f andf are the forward scattering amplitudes of K 0 andK 0 , respectively.
For the copper absorber the probability of finding K 0 S is shown in Figs. 1 and 2. ImV is determined by Eqs. (6), (7), (23) and (24), ReV is the parameter. Solid and dot-dashed curves correspond to | ReV |= ∆Γ/2 and ReV = 0, respectively. Dashed curve corresponds to the copper plate and V defined from (25). The amplitudes f andf are taken from [17]. In the case ReV = 0 only the regeneration by absorption takes place. It is seen that ReV leads to the suppression of regeneration. Compared to [2], | K S (t) | 2 is about 10 times smaller. (Although the comparison with [2] is meaningless for the reasons given above.)

Conclusion
The main results of this paper are given in the abstract. The most distinctive feature of the model presented above is the inverse ∆Γ-and ∆m-dependences of the amplitude of regenerated K 0 S , or parameter regeneration r (see Eqs. (19)-(24) of Ref. [6]). The main uncertainty in the numerical results is conditioned by the uncertainty in the cross sections σ(K 0 N) and σ(K 0 N).
The same is also true for the previous results [1][2][3][4][5] since they have been obtained by means of above-mentioned cross sections as well. In this connection we would like to recall that ∆m is extracted from free-space oscillations without recourse to potentials of K 0 andK 0 .
Nevertheless, in any case the regeneration should be described correctly.
This paper accepted for publication in Chinese Physics C. The author is grateful to Michael Bayev for help in numerical calculations.