Absorption effects in particle oscillations

Particle oscillations in absorbing matter are considered. The approach based on the optical potential is shown to be inapplicable in the strong absorption region. Models with Hermitian Hamiltonian are analyzed.


Introduction
In particle oscillations in the medium absorption can play an important role, for example, in the K 0K 0 [1][2][3][4] and nn [5][6][7][8] oscillations. In this paper we consider nn transitions in the medium followed by annihilation n →n → M. (1) Here M are the annihilation mesons. The reason for considering this process is that the absorption (annihilation) ofn is extremely strong.
In the standard approach (later on referred to as a potential model) then-medium interaction is described by antineutron optical potential Un. We have objections to this model (Sect. 2). The alternative models with a Hermitian Hamiltonian are considered in Sect. 3. This fact should be emphasized particularly.
In Sect. 5 the results are summarized. The problems of the model with a Hermitian Hamiltonian are pointed out as well. The restriction on the free-space nn oscillation time τ critically depends on the description of absorption. In this regard, the main goal of this paper is to consider the absorption model itself.

Potential model
We consider process (1). In the standard approach [5][6][7] the nn transitions in the medium are described by Schrodinger equations ImUn = −Γ/2,n(0, x) = 0. Here U n and Un are the potential of n and the optical potential of n, respectively; ǫ nn is a small parameter with ǫ nn = 1/τ , where τ is the free-space nn oscillation time, Γ being the annihilation width ofn.
In the lowest order in ǫ nn the process width is [5][6][7] Γ pot = ǫ 2 Un is the basic element of the model. In this connection the following problems arise: 1. The optical model was developed for the Schrodinger type equations. The physical meaning of ImUn follows from the corresponding continuity equation. Coupled Eqs. (2) give rise to the following equation: The continuity equation cannot be derived from (4).

2.
To get Γ pot , the optical theorem or condition of probability conservation are used. However, the S-matrix is essentially non-unitary.
3. The structure and Γ-dependence of (3) provoke some objections. Due to this an alternative model should be considered.

Models with a Hermitian Hamiltonian
The interaction Hamiltonian of process (1) is where H nn and H are the Hamiltonians of nn conversion [5] and then-medium interaction, respectively. The background neutron potential is included in the neutron wave function:

Model with a bare propagator
The nn conversion comes from the exchange of Higs bosons with m H > 10 5 GeV. Then annihilates in a time τ a ∼ 1/Γ. We deal with a two-step process with a characteristic time τ a .
The general definition of the antineutron annihilation amplitude M a is Here | 0n p > is the state of the medium containing then with the 4-momentum p = (ǫ, p); < M | denotes the annihilation mesons, N includes the normalization factors of the wave functions.
The antineutron annihilation width Γ is expressed through M a : where N 1 is the normalization factor.
The amplitude of process (1) M 1 is given by In the lowest order in H nn one obtains where G 0 is the antineutron propagator. Since pn = p, ǫn = ǫ, then G 0 ∼ 1/0. M a contains all then-medium interactions followed by annihilation including antineutron rescattering in the initial state. So in this case the antineutron propagator is bare.
We deal with infrared singularity. For solving the problem a field theoretical approach with a finite time interval has been proposed [9]. The process (1) probability was found to be [10] where W f is the free-space nn transition probability. Equation (12) leads to a very strong restriction on the free-space nn oscillation time: τ = 10 16 yr.

Auxiliary process
Starting from (5) and (6) we have drawn the singular amplitude M 1 . To gain a better understanding of the problem, we consider the nn transitions in the medium followed by β + -decay: The neutron wave function is given by (6). The interaction Hamiltonian is where V is defined by (2), H W is the Hamiltonian of the decayn →pe + ν. In the lowest order in H nn the amplitude M 2 is where M d is the amplitude of the β + -decay, G is the antineutron propagator.
The process width Γ 2 is The propagator is dressed due to the additional field V .

Model with a dressed propagator
We return to process (1). Let us try to compose a model with a dressed propagator. By analogy with (14) in the Hamiltonian H (see (5)) we separate out the scalar field V 1 : where H a is the annihilation Hamiltonian. Now the antineutron annihilation amplitude M an is defined through H a : The interaction Hamiltonian is given by In the lowest order in H nn the amplitude of process (1) is The antineutron propagator G d is dressed. V 1 plays the role of antineutron self-energy Σ. M 3 corresponds to the first order in H nn and all the orders in V 1 and H a . Compared to (7), M an is calculated through the reduced Hamiltonian H a instead of H.
The process width Γ 3 is The amplitude M 3 is non-singular because the propagator is dressed. The antineutron selfenergy Σ = V 1 appears due to separation of the field V 1 . This procedure seems to be artificial and unjustified. There are no similar problems for process (13) since the self-energy and decay of n are generated by different fields H W and V 1 . This point should be given particular emphasis.
In any case Γ an ∼ Γ, and so

Discussion
First of all we compare the potential model with the model with a dressed propagator. In (21) we have to take the same parameters as in the potential model: V 1 = V and Γ an = Γ. Then we get Equation (23)  The same conclusion has been drawn in [8]. It was shown that double counting leads to full cancellation of the leading terms. However, in [8] the model with a bare propagator has been considered. The approach with a finite time interval was used, but it can provoke additional questions. The above-given consideration is transparent.
If we want to remove double counting, we have to make direct calculations of the off-diagonal matrix element (see Sect. 3).
Let us compare the Γ-dependence of the results. In (21) one should use realistic parameters.
We take V 1 = ReV , then we obtain Therefore, Γ 3 ∼ Γ. For the K 0K 0 transitions in the medium followed by decay and regeneration of the K 0 S -component an identical Γ-dependence takes plays [11,12]. In the potential model Γ pot ∼ Γ only at light absorption. Indeed, if Γ/2 ≪| ReV |, then (25) In the first approximation (25) coincides with (24). This agreement was expected since the dominant role was played by ReUn.
We consider the difference in the results in the region of strong absorption. It is seen from the ratio For nuclear matter we take Γ = 100 MeV. If | ReV |= 50 MeV, then r = 1. If | ReV |= 10 MeV, then r = 25. When | ReV | decreases, Γ 3 and r increase.
In the oscillation of other particles (for example, K 0K 0 ) the difference between Γ 3 and Γ pot is less, however this difference can be essential for the problem under study.
For the realistic parameters Γ = 100 MeV and | ReV |= 10 MeV, the lower limit on the free-space nn oscillations time is τ = 1.2 · 10 9 s. When V 1 = 0, the model with a dressed propagator converts to the model with a bare propagator. It gives τ = 10 16 yr. On the basis of this one can accept that the lower limit on the free-space nn oscillations time is in the range 10 16 yr > τ > 1.2 · 10 9 s.
Thus we conclude the following: (1) The smaller | ReV | (antineutron self-energy), the greater the difference in the results (see (27)). It is a maximum for the model with a bare propagator.
Finally, in the strong absorption region the model with an optical potential is inapplicable.
In the models with a Hermitian Hamiltonian the optical potential is not used. The conclusions made above do not depend on the specific models of the blocks M a and M an .

Conclusion
The potential model is applicable only in the case of slight absorption.
If absorption is strong, the potential model is inapplicable: (1) It contains double counting.
This is a main statement of this paper. The chief drawback in the model with a dressed propagator is that the procedure of separation of V 1 is artificial and unjustified. In our opinion the model with a bare propagator is preferable. There are a lot of arguments in favor of the model with a bare propagator [10]. The only objection to this model is that it gives the result which essentially differs from the result of the potential model. The potential model has been considered above.
Since the problem is of a great nicety, further investigations in the framework of the approach with a Hermitian Hamiltonian are needed.