On Experimental Confirmation of the Corrections to the Fermi's golden rule

Standards calculations by the Fermi's Golden rule involve approximations. These approximations could lead to deviations from the predictions of the standard model as discussed in another paper. In this paper we propose experimental searches for such deviations in the two photon spectra from the decay of the neutral pion in the process $\phi \rightarrow \pi^{+} \pi^{-} \pi^{0}$ and in the annihilation of the positron from nclear $\beta$ decay.


I. THE CORRECTION TO THE FERMI'S GOLDEN RULE
In interacting many-body system described by a Hamiltonian H 0 + H 1 , a state evolves with a Schrödinger equation. One particle state is specified by the momentum and noninteracting energy defined by H 0 . A transition by H 1 has been studied by the Fermi's golden rule. Although these transitions have been paid attention from researchers, those that do not conserve the energy often arise, when the approximations are taken into account. The Schrödinger equation includes these approximations, which affect transitions of any states. Surprisingly, a correction term beyond the Fermi's golden rule emerges. The correction becomes manifest in a transition of a finite time interval, in which that reveals different dependence on the time interval and on the energy difference. The correction terms would have been identified from experimental data. However it is not simple as was naively thought due to several reasons. One reason is that those signals that are caused by the corrections terms are similar to those of experimental background. In majority cases, they were considered as the background, and discarded. Another reason is on a difficulty to find the absolute value of the physical quantity in experiments, because the data is always modified by an efficiency of detector. The transition rate describes average behavior [1,2] of the process, and the correction terms give dominant contribution to rapidly changing processes.
Direct observation of these events might give signals of the correction terms, but has not been possible up to the present. In general it is difficult to separate these from the real background.
Accordingly the correction terms was not a major concern from researchers. Nevertheless, the correction is one part of the total probability and contributes to natural phenomena.
Fitting these experiments in approximate way without the correction term might be possible and viable for certain period. However, that should lead serious inconsistency or fatal outcome at later time, which must be avoided. It is urgent to confirm an existence of the correction term with simple and clean experiments.
Two photon processes of the neutral pion and the positron annihilation supply precise information on the transitions and can be candidates. The rates have theoretically been well-understood, and determined from the various experiments, in which the background have been subtracted. There are subtlety on the background subtraction, and a signal of the correction term has been insignificant. The correction terms are computed in a separate paper [3]and are found sizable. Due to their unusual properties, which will be presented later, it is not an easy task to disentangle them from the real background. Nevertheless, they give universal contributions to the phenomena. It will be shown that these are feasible in φ-factory for the pion and in nuclear beta decays for the positron.
The neutral pion, π 0 , is the lightest hadron composed of the quark and anti-quark and supplies many informations on particle physics [4]. The rate of π 0 decay to two photons [5,6] is proportional to the number of the color N c [7,8], and the measurement on life time τ = 10 −16 seconds determines N c = 3. Despite of this remarkable success, the average life-time obtained from various methods [9] has large uncertainty of about 10 per cent.
Accordingly, K-meson decays to two or three pions have also large uncertainties [4]. A large uncertainty arises also in the decay of para-positronium, which is a bound state of the electron and positron in Quantum Electrodynamics (QED). Its properties and transition rates are understood well, but the precision is not very good. The large uncertainty of the experimental values may suggest a fundamental problem on the transition probability.
We find the many body wave function |Ψ composed of normalized states from the Schrödinger equation , where H 0 and H int are the free and interaction parts, and compute the rigorous transition amplitude. Hereafter we employ the natural units = c = 1 unless otherwise stated. A transition probability from a state |i, 0 at t = 0 to a state |f, T at t = T is determined by the von Neumann's fundamental principle of the quantum mechanics (FQM) as , P (T ) = | f, T |i, 0 | 2 , for normalized states. For P (T ) ≪ 1, the average rate Γ = P (T )−P (T i ) between a small T i and a large T , is given from a ratio of fluxes of out-going waves over that of incident waves and is in agreement with that derived from the golden rule for the final state of continuous spectrum. In these standard calculations, the plane waves and the interaction that switches off adiabatically (ASI) are used. Although, this value has been used in the majority of the processes, experiments are made at the finite time intervals and the value is measured without average.
Theoretical values under these conditions are necessary.
Stueckelberg studied this problem sometime ago and found that the transition amplitudes of the plane waves for finite-time interval lead a divergence [10] even in the tree level. This is unconnected with the ultraviolet divergences due to the intermediate states but to non-normalized initial and final states. It is possible to avoid this difficulty by using the normalized states. Those computed in the previous paper [3] are applied to experiments. P (T ) at a large T is the sum, T i is determined by a time that the initial wave packets separates. This is determined by √ σ i , where σ i is the spatial size of the initial wave. At ΓT i < 1, and at T > T i , Γ, is computed with the standard S-matrix S[∞] under ASI [11][12][13], but P (d) is computed with the wavefunctions following FQM [14][15][16][17]. A rigorous probability will be obtained without facing the difficulty raised by Stueckelberg by using the wave packets instead of the plane waves.
Experimental proof of P (d) in the neutral pion decay, the positronium decay, and the positron annihilations are studied. Two photon decays of a para-positronium is almost equivalent to the neutral pion decay. Their systematic analyses are presented. It will be shown that the unique properties derived from the probability ΓT + P (d) can be confirmed experimentally.
The paper is organized as follows: In Section 2, the pion decay is analyzed and in Section 3, the positron annihilation is analyzed. In Section 4 the wave packets sizes and relevant parameters are estimated. In Section 5 the experiments are studied and summary and prospects are presented. Appendix A is devoted to various formula and Appendix B is devoted to a method for entanglement of the accidental background.

II. TWO PHOTON DECAY OF THE NEUTRAL PION
The interaction of a neutral pion or a para-positronium with two photons are derived from the triangle diagram of the quark or the electron as L int = −g ϕ ǫ µνρσ F µν F ρσ in which the coupling for the pion is g = α 4πfπ is almost constant from the confining mechanism and is related with the πγγ coupling [7,8]. For the positronium, the binding energy is small and the coupling varies with the momentum, which will be ignored for a while. Substituting this to Eq.(1), we have the transition amplitude for an initial state of a central momentum and position into two photons Gaussian wave packet [14] , [18], [19] , [20] satisfies the minimum uncertainty, which is idealistic for studying a transition of finite-time interval, and is used in majority of places. Non-Gaussian form is also physically relevant and studied later. Wave packets of the size σ i , the central momentum, and the central position are used throughout this paper, where i = 1, 2, is the group velocity of the momentum P A . Throughout this paper, the upper-case roman letters A, B, . . . run for π, 1, 2 so that e.g. A stands for A=π,1,2 , etc. An imaginary part is added to the energy of the unstable initial state according to Ref. [21,22]; see also e.g. Ref. [11] for a review is taken. Integration over the space position leads to a Gaussian function in the momentum difference, and that over the time leads to 1 where N 0 ( X i ) shows a dependence on the positions, , G(δω) is expressed with the error function erf(x+iy).
Their explicit forms are given in [3]. The transition probability is written as, As is shown in Ref. [3] in details , G(δω) depends on an intersection of the trajectories determined by the positions of X i ; i = 1, 2. If they intersect outside of the material, the interaction does not occur and the amplitude vanishes. If that is inside of the material, the interaction occurs. This is a bulk region. In the boundary region, the interaction occurs partly. This is the boundary region.
The integration in the bulk is proportional to the time interval due to the translational invariance along the initial momentum, and that in the boundary is proportional to the width of the boundary region, σ t , which depends on the wave packet size and the velocity variation, σ t = σs ∆V 2 . The derivation is given in [3] The momentum distribution is written as a sum of two terms, where for bulk, where T = T γ − T π , C is a constant of energy dimension E 1 and depends on the wave packet parameters. The squares of G(δω) in the asymptotic region is, where T R 0 is the time that the wave packets intersect. The bulk term decreases rapidly with δω and the boundary term decreases slowly with an inverse power of the energy difference.
In the decay of the high energy pion of p π = (E π , 0, 0, p π ), the momenta of the final states are almost parallel to the pion. In the boundary term, |G(δω)| 2 decreases slowly at K i → ∞, and leads a large contribution to the probability.
In the transition, the total energy is conserved but the kinetic energy is partly violated. The bulk contribution is narrow in the kinetic energy, and reveals the golden rule. The boundary contribution is broad in the kinetic energy, and reveals the correction term. The deviation of the kinetic energy from the total energy is the interaction energy V int = Ψ|H int |Ψ . The coupling strength g can be treated as constant for the golden rule, where k γ i · k γ i ′ ≪ m q . However, the boundary term is spread in wide kinetic region of [23]. Here m q is the composite quark mass of a magnitude around mp 3 , where m p is the proton's mass. Thus P (d) becomes maximum at around k γ i · k γ i ′ ≈ m 2 p 2 . Its magnitude is proportional to the proton's mass. This behavior shows that the average interaction energy |H int | is the order of the proton's rest energy, m p .
For a high energy pion, the initial and final waves overlap in wide area for photons propagating in the parallel direction to the pion. The boundary region becomes large in size , and gives large contribution to the probability.

III. POSITRON ANNIHILATION
Positron and electron are described by the field ψ(x), and photon is by A µ (x) in the Quantum Elecrodynamics, and the interaction is eψ(x)γ µ ψ(x)A µ (x). The para-positronium decay and the free positron annihilation are derived from this interaction. The former one is also expressed by an effective interaction equivalent to the pion-two photon interaction. The latter one is described by the 2nd order perturbative expansion with respect to the above interaction. P (T ) in these decays are studied. A.
Para-positronium decay Para-positronium is even in the charge conjugation and decays to two photons. The formula of decay probability Eq.(6) is applied after changing parameters with suitable ones.
The average lifetime of the Para-positronium is much longer than that of the pion and the wave packet size is also longer. The positronium decays and positron annihilation in porous material ,which are composed of small holes and many boundary regions, are analyzed. We will see that the boundary term is enhanced.

B. Free positron annihilation
The annihilation amplitude of the free positron and the free electron at rest for those of the central values of momentum and position, for the photons, the electron, and the positron, where T is the time interval that the positron crosses a grain of the target. The integrations over the coordinates x i , and over the momentum q for the intermediate state are made using Gaussian integrations.
The integration over times give the bulk and boundary terms, and lead the amplitude to be written as Eq.(4). Substituting these, we have the momentum distribution where Eqs.(A22) and (A12) are substituted, and where C is the constant [3]. In silica powder, this size is semi-microscopic of order few nano meter, and almost the same or slightly larger than √ σ γ . In the present situation, the target is composed of silica particles of L = 7 nano meter, and it is reasonable to assume the ratios √ 2σs L and √ 2σt T are 1 10 − 1 100 . The positron energy is E e + = m e with the energy uncertainty of 10 per cent. The spectrum of the boundary term is of the universal form but its magnitude has uncertainties due to the uncertainties on the wave packets. This ambiguity could be studied by a light scattering of the silica powder [24].

IV. INITIAL AND FINAL STATES
We apply the decay probability Eq. (6) to the neutral pion in the process , and that of the positron Eq.(12) in the process 22 Na → 22 Ne * + e + + ν, 22 Ne * → 22 Ne + γ.
The former experiment is made in a high energy laboratory and the latter experiment is made in a low-energy laboratory.

8
A. Wave packet shape and size The total transition rate Γ derived from Eqs. (6) and (12) is independent of the wave packet parameters. This is consistent with the general theorem given by Stodolsky [25] [26] [27] [28] on stationary physical quantities. This theorem, however, is not applied to a nonstationary quantity such as P (d) . In fact P (d) derived from Eqs. (6) and (12) depend on the forms and sizes of the wave packets. Up to here the Gaussian wave packet, which decreases exponentially in the position and the momentum and satisfies the minimum uncertainty δxδp = , and δp = 0 for δx = ∞ is used. This is idealistic for studying the transition for a finite time interval. Other wave packet satisfying δxδp ≥ is shown to lead almost equivalent results. σ π , σ γ , and σē stand for σ s of the pion, photon, and positron.
These particles interact with microscopic objects in matters and cause the final states to be produced, from which a number of the events and the probability are determined.
Accordingly the packet parameters in our formula are determined by these states in matter.
This method has been shown valid in [14][15][16][17], and in quantum transition of two atoms in an energy transfer process in photosynthesis [29] .
1. Sizes of wave functions :π 0 σ π 0 In order for the electron and the positron to produce a φ meson, they are accelerated from average electron momentum in matter, which is less than The detection process of the photon is governed by its reaction with the atoms and the following coherent transitions by which electronic signals emerge in the detector. They occur within finite spatial area occupied by the wavefunctions in solid. The transition amplitude of the photon is described by the wave packet of this size. Thus σ γ represents the spatial size of the electron wavefunction in the configuration space that the photon interacts with.
The initial process depends on the energy. In the energy 0.5 GeV, majority of the events are the pair production due to nucleus electric field. Accordingly, σ γ = sγ m 2 π , where s γ ≤ 1 , and s γ = 0.5 is used for a following estimation.  An idealistic detector that detects and gives an energy of a particle or a wave directly does not exist. For its measurement, signals caused by its reactions with matter are read first and is converted to the energy using a conversion rule justified by other processes.
The energy is measured within finite uncertainty. This is the energy resolution, and all the detector have the finite energy resolution. This causes an experimental uncertainty. The energy resolution, σ(E), has various origins such as a statistical one and an intrinsic one.
That is written as where σ statitics (E) is determined normally from Poisson statistics and other is written as σ intrinsic (E), in which an effect due to the finite size of wavefunctions, Eq. (5), is included.
The former depends on the detector's type, and the latter does not and has universal properties regardless of detector type. In scintillation detector, an electric signal of a γ-ray is obtained according to the number of the scintillation photons N, and the energy resolution, σ statitics (E), is given by where N is a number of the sample and F is a correction factor, the Fano-factor. For NaI(TI), F = 1, and σ statistics (E)/ E is around 5 − 10 per cent , and the energy resolution is 25 − 50 keV for the energy 500 keV. Ge detector is of different mechanism of much smaller statistical uncertainty, due to the small F and large N. The distribution around the central value decreases exponentially with E.
The wave-packet size determined by the size of the atom is π(10 −10 ) 2 M 2 and should be almost the same in NaI(TI) and Ge detectors, and leads to the energy uncertainty, σ intrinsic (E) = 1 keV. Accordingly in the NaI, σ statitics (E) is the dominant one and σ intrinsic (E) is negligible, but in Ge detector, σ intrinsic (E) shares the substantial part.

D. Energy distribution
The energy distributions of the bulk term and the boundary term are very different. That from Γ for the plane waves under ASI is proportional to δ(E i − E f ), but for the wave packets that behaves as e −( δω σ(E) ) 2 , where the width is of universal nature and behaves differently from those of statistical one. That of P (d) decreases in E −n , where n ≥ 0 depends on the decay dynamics. P (d) can be identified easily in the energy region E ≫ σ(E) if the relative fraction over ΓT is of substantial magnitude of the order 10 −3 or larger, even with the detector of large energy resolution. Despite of large energy resolution, NaI(Tl) scintillator is useful for the confirmation of P (d) . The detector of much smaller resolution such as the Ge detector is also useful.

V. EXPERIMENTAL CONFIRMATIONS
As P (d) possesses many unusual properties, phenomena originated from P (d) reveal intriguing properties. By detecting these events, P (d) can be confirmed. Γ has been well established, and phenomena of Γ origin have been understood precisely with a help of numerical methods. They are compared with the data from the natural phenomena and observations.
If clear disagreements are found, and if it is resolved by P (d) , this may confirm P (d) .

A. Magnitude of P (d)
A magnitude of P (d) for para-positronium decay, P (d) (pp), and direct annihilation, , is estimated and given in Figure. They depend on the size and shape of the wave packets. We use the value σ γ > 10 −20 m 2 , and the Gaussian wave function and power law wave function, and find At the moment we are not aware of the precise shape and size of the wave function. Light scattering may be useful for a study of the wave function [24].
The photon distribution is modified by P in the high energy regions, clear signals may be obtained. Although accidental coincident events may contribute, the separation of them can be made and and events of P (d) origin in the data is estimated. It is our expectation that with 10 8 events of the positron annihilation a confirmation of P d could be in scope.
GEANT4 [33]is a simulation program that includes the transition probability and the detector performance. The probabilities derived from the golden rule are employed . Hence this is quite useful for analyzing the natural phenomena including the detector's response and backgrounds. Comparing the events derived from the golden rule of the standard theory with the observations, we are able to see if a non-standard component is included.

B. Backgrounds from decay ( annihilation ) in flight
The signals from the decay or the annihilation in flight are in energy regions different from those at rest and give background. Positron loses its energy in insulator in pico seconds [34], and stops. A photon produced before the stop has an energy higher than m e and its contribution is estimated in two steps.
The average positron lifetime, due to the annihilation or the decay is, 100-500 pico seconds, which depends on various conditions. Hereafter we use 200 pico seconds for the average life time and 2 pico second for the thermalization time. The annihilation events of the positron in flight over that at rest is less than the ratio , 2 200 = 10 −2 . The experimental value seems to be less than 10 −3 or 10 −4 [35]. Among the events of energy E 1 + E 2 > 2m e c 2 , a fraction in the energy region E 1 + E 2 − 2m e c 2 ≥ 3σE, where σE is the width of NaI(Tl) detector, is obtained as 1 230 from Bethe's formula [36]. A further suppression factor 1 10 is multiplied due to a specific configuration of the detector setup of the present experiment.
Combining these numbers, the fraction is 0.43 × 10 −6 or 0.43 × 10 −7 . This gives the magnitude of the background from the inflight annihilation, is less than 10 −6 .

C. Uncertainties
Possible sources of uncertainties and ambiguities are matter effects, accidental coincident events (double hits) , and environmental gammas.
The photon spectrum in the high energy region is not modified by Moeller scattering, photo-electric effect, the Compton effect, and the pair production. Accordingly the matter effects are irrelevant. The environmental gammas or those of cosmic ray origins are avoided by selecting coincedent events of multiple gammas. In two gamma's case, the coincedence between one gamma from Ne radiative decay and another from the positron annihilation are taken. In three gammas case, the coincedence between one gamma from 21 Ne * radiative decay and two photons from the positron annihilation are taken. In these multiple coincident events, there remain accidental coincident events (double hits). Because their strength depends upon the initial positron flux and the spectrum has different momentum dependence than the signal from P (d) , it is possible to disentangle them following Appendix B.

D. Related processes
Para-positronium decays is included in the text. (1) The energies of the photons in the positron annihilation at rest from the golden rule satisfy E γ 1 +E γ 2 = 2m e , whereas those from P (d) satisfy E γ 1 +E γ 2 < 2m e or E γ 1 +E γ 2 > 2m e .
The photon loses its energy by the Compton scattering, and that produced by the golden rule can be detected in the former region, but not in the latter region. The events of the energies E γ 1 + E γ 2 > m e are generated only by P (d) , and may be worthwhile for its confirmation.
(2) For the neutral pion, our finding P (d) ≈ O(0.8) suggest that for the analysis P (d) must be implemented. The previous large uncertainty of about 10 per cent in the life time would be due to P (d) , and will be reduced in an analysis that includes P (d) .
(4) Many-body wave functions of δE = E initial −(E γ 1 +E γ 2 ) = 0 have interaction energies, which are independent of the frequency of each wave. This leads an extra component to the energy momentum tensor in addition to those proportional to the frequencies. Normal detection processes measure the wave's frequencies, but these interaction energies. Accordingly, this corresponds to an invisible energy. This state may be considered as a kind of halo.
(5) Once the confirmation of P d is made, (a) methods to reduce current uncertainties in the experiments and (b) mechanisms to solve current puzzling phenomena will be found.
where θ( X γ , volume), θ( X γ , b t ), and θ( X γ , b (t,s) ) show that the intersection of trajectories are in the inside of the volume L 3 , in the boundary in time, and in the boundary in space and time.
is different. The momentum dependence of the bulk term and that of the time-boundary are spherically symmetric as before but that of the space-boundary is asymmetric.
The integration over the positions X γ l , and over the position X e + in the region of L 3 and the time interval T , and for the boundary of the width √ 2σ t and √ 2σ s are, . (A15) Substituting these , we have the momentum distribution where Eqs.(A22) and (A12) are substituted, and where C is the constant. In the present situation, the target is composed of silica particles of L = 7 nano meter, and it is reasonable to assume E e + = m e (1± 1 10 ), √ 2σs L , √ 2σt T ≈ 1 10 − 1 100 . The spectrum of the boundary term is of the universal form but its magnitude has uncertainties due to the uncertainties on the wave packets. This ambiguity could be studied by a light scattering of the silica powder.