^{3}

Standard calculations by 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

In an interacting many-body system described by a Hamiltonian

Accordingly, the correction terms were not a major concern for researchers. Nevertheless, the correction is one part of the total probability and contributes to natural phenomena. Fitting these experiments in an approximate way without the correction term might be possible and viable for a certain period; however, that could lead to serious inconsistency or an incorrect outcome at a later time, which must be avoided. It is thus important to confirm the existence of the correction term with simple and clean experiments.

Two-photon processes of the neutral pion and positron annihilation supply precise information on the transitions, and could be candidates. The rates are well understood theoretically, and have been determined from various experiments, in which the background has been subtracted. There are subtleties in the background subtraction, and the signal of the correction term has been insignificant. The correction terms are computed in a separate paper [

The neutral pion,

We find the many-body wave function

Theoretical values under these conditions are necessary. Stueckelberg studied this problem some time ago and found that the transition amplitudes of the plane waves for a finite time interval lead to a divergence [

Experimental proof of

The paper is organized as follows: In Sect. 2 pion decay is analyzed, and in Sect. 3 positron annihilation is analyzed. In Sect. 4, the wave packet sizes and relevant parameters are estimated. In Sect. 5, the experiments are studied and a summary and prospects for the future are presented.

The interactions of a neutral pion or para-positronium with two photons are derived from the triangle diagram of the quark or the electron as in which the coupling for the pion is

A Gaussian wave packet [^{1}

As shown in Ref. [

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,

The momentum distribution is written as a sum of two terms,

In the decay of a high-energy pion of

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

Here,

For a high-energy pion, the initial and final waves overlap in a wide area for photons propagating in the direction parallel to the pion. The boundary region becomes large in size, and gives a large contribution to the probability.

Positrons and electrons are described by the field

Para-positronium is even in charge conjugation and decays to two photons. The formula of the decay probability in Eq. (

The annihilation amplitude of a free positron and free electron at rest for the central values of momentum and position

Integration over time gives the bulk and boundary terms, and leads to the amplitude being written as in Eq. (^{2}

We apply the decay probability of Eq. (

The former experiment is made in a high-energy laboratory, and the latter experiment is made in a low-energy laboratory.

The total transition rate ^{3}

These particles interact with microscopic objects in matter 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 formulae are determined by these states in matter. This method has been shown to be valid in Refs. [

The wave packet size for a detected positron is estimated based on the detector used. When a plastic scintillator containing benzene is used, the spatial size of the benzene molecule, around 1 nm, shows the positron wave packet size.

The wave functions of the electron and positron overlap at the boundary region of the matter, and their annihilation takes place. The area is large, and the events increase in porous material. The porous material size determines the effective size of the area and the time interval of the transition. The transition amplitudes and probabilities depend on these sizes. This is used in positron experiments.

For experiments that use small powders, electrons are inside the small region, and the interaction takes place in the inside or at the boundary region. The transition amplitudes and probabilities depend on these sizes.

An ideal detector that detects and gives the 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 converted to the energy using a conversion rule justified by other processes. The energy is measured with finite uncertainty. This is the energy resolution, and all detectors have finite energy resolution. This causes an experimental uncertainty. The energy resolution,

The wave packet size determined by the size of the atom is

The energy distributions of the bulk term and the boundary term are very different. That from

As

The magnitude of

The variation of the expected ratio of events per stopping positron in silica with

At the moment we are not aware of the precise shape and size of the wave function. Light scattering may be useful to study the wave function.

The photon distribution is modified by

GEANT4 [

The signals from decay or annihilation in flight are in energy regions different from those at rest, and give background. A positron loses its energy in an insulator in picoseconds [

The average positron life time due to annihilation or decay is 100–500 ps, and depends on various conditions. Hereafter we use 200 ps for the average life time and 2 ps for the thermalization time. The ratio of annihilation events of positrons in flight over those at rest is less than

Among the events of energy

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, the photo-electric effect, the Compton effect, or pair production. Accordingly, matter effects are irrelevant. Environmental gammas or those of cosmic ray origin are avoided by selecting coincident events of multiple gammas. In the two-gamma case, coincidences between one gamma from Ne radiative decay and another from positron annihilation are taken. In the three-gamma case, coincidences between one gamma from

Para-positronium decay is included in the text. Another spin component, orth-positronium, may be used for a

(1) The energies of the photons in positron annihilation at rest from the golden rule satisfy

(2) For the neutral pion, our finding of

(3) Tagging

(4) Many-body wave functions of

(5) Once the confirmation of

^{1} We have put the central momentum

^{2} A raman scattering experiment has seen anomalous signals. Private communications from Professor M. Takesada

^{3} Reference [

This work was partially supported by a Grant-in-Aid for Scientific Research (Grant No. 24340043). The authors thank Dr. K. Hayasaka, Dr. K. Oda, and Mr. H. Nakatsuka for useful discussions.

Open Access funding: SCOAP

The amplitude for a free positron annihilation is

Applying Wick’s theorem,

Note that this is slightly different from that for positronium decays.

In scatterings in the laboratory frame where the target is composed of small particles of volume

The momentum-dependent term in the bulk, Eq. (21), and the boundary term in time, Eq. (22), lead to a probability of the same form as before,

The integrations over the positions

Substituting these, we have the momentum distribution

The function

For non-Gaussian wave packets, the momentum-dependent amplitudes in the bulk and boundary terms are

The integration over the positions

Suppose the probability is a sum of duplicate (accidental coincident and pile-up) events and

Plot

For