Electron Capture (EC) is a type of nuclear beta decay. An atomic nucleus captures one of its orbital electrons, whereby one proton changes into a neutron and an electron neutrino is emitted. A hole in the atomic shell is filled in a subsequent process by another electron which is accompanied by emission of a characteristic X-ray. This radiation tells us from which orbit the electron was captured. Normally, most probable is the capture from the K-shell (1s capture) which is heralded by KX radiation. In this story we consider only K-captures.  

The REC spectra were measured for many cases of so called allowed beta transitions (nuclear spin changes by 0 or 1 units, parity does not change) and they were found in good agreement with theory assuming the IB mechanism. These experiments and the IB theory are summarized in:

The IB theory was extended to all nuclear transitions, including so called forbidden ones, by two Russian physicists Zon and Rapoport (ZR) in 1968, but it was not tested experimentally. Initially, our motivation was to provide such a test. In the following we will discuss two classes of decays where nuclear spin changes by two units:

The first case we've measured was the decay 41Ca -> 41K which is in the 1u class (nuclear transition from 7/2- to 3/2+). The upper panel on the right shows the REC spectrum normalized to non-radiative capture rate. In three different experiments the K-capture component was selected by measuring gamma spectrum in coincidence with the potassium KX-rays. The red line shows the simplest model (Morrison and Schiff) which serves as a reference. To see better the shape details, the ratio of the REC spectrum to this reference is determined. Such a shape factor for 41Ca is shown in the lower panel. In the following, all REC spectra will be presented in the form of the corresponding shape factor. In the lower panel the shape factor for the ZR theory is also shown (solid line). 

As can be seen, the measured intensity and the shape strongly deviate from the theory. The REC intesity is found 6 times larger than the prediction of the ZR theory!      

Important: in the 1u case, the ratio of REC to normal capture does not depend on nuclear details (nuclear matrix elements cancel out)! The well known theory of weak interactions and the quantum electrodynamics should suffice to describe the data! 

The second measured case was the 2nu decay 59Ni -> 59Co. The corresponding shape factor, from two different experiments, is shown on the right. Here, the difference with respect to the 1u decays is that the REC spectrum depends on a combination of nuclear matrix elements, denoted by Lambda. Since they are very difficult to calculate from nuclear theory, we treat Lambda as an adjustable parameter. We found, however, that no value of Lambda can fit the shape and the intensity of 59Ni spectrum simultaneously! The prediction of ZR theory for the value of Lambda=3.5, which fits best the shape of the experimental spectrum, is shown by the solid line. Then, the measured intensity is 2 times larger than predicted! The results for 59Ni were published in:

In fact, there is another way photons can be emitted in the EC decay. The radiation can come from the nucleus! This mechanism, represented by diagrams shown on the right, was considered to be negligible in comparison with IB. It was noted, however, that in case of forbidden EC the situation can be different. The direct 1u transition, for example, can be realized by a combination of allowed GT decay (much more probable than 1u) and  E1 nuclear gamma transition!

Ł. Kalinowski working in the framework of ZR theory, calculated contributions of the detour diagrams to the REC spectra of 41Ca and 59Ni. Here, an additional parameter, effective charge, had to be introduced. On the plots below the predictions of the new model (DT: ZR theory plus detour transitions) are shown by the solid magenta line.The agreement is good when the value of the effective charge is 0.78 in both cases! Although the physical meaning of this effective charge was not entirely clear, it looked like the puzzle was solved.
The model of Kalinowski is descibed in:
Kalinowski et al., Nucl. Phys. A537 (1992) 1  and  Kalinowski, Nucl. Phys. A551 (1993) 473
 

The problem reappeared when we measured REC in 204Tl which is of 1u type, see on the right. Here, in contrast to previous cases, the ZR theory predicts 4 times more radiation than is observed! Adding detour transitions makes it even worse, because for both signs of the effective charge the detour contribution is positive.
The 204Tl case is described in:
Kurcewicz et al., Nucl. Phys. A728 (2003) 3

5. The breaktrough

The puzzle was finally solved by Krzysztof Pachucki (Faculty of Physics, University of Warsaw). He found an error in the ZR theory. Zon and Rapoport made a wrong approximation leading to some diverging terms. Pachucki noticed that when the electromagnetic field is described in the length gauge instead of previoiusly used Coulomb gauge, the diverging terms vanish! In addition, he showed that in the length gauge the contribution from the detour diagrams is negligible "unless there are close-lying states in the daughter or in the parent nucleus". The new theory (KP) was applied to the 1u REC and the very good agreement with the data was found (see below)! Afterwards, we measured another case of 1u transition, 81Kr, which fully confirmed the REC theory of Pachucki (below, on the right). Note that in the new theory there are no free parameters! As expected, the standard theory of weak interactions and QED do explain the REC in 1u transitions!

The new theory is published in: Pachucki et al., Phys. Rev. C75 (2007) 055502,
while the experiment on 81Kr in: Mianowski et al., Phys. Rev. C82 (2010) 044308

It remains to be seen how the new theory works in the 2nu case of 59Ni. It happens that the REC spectrum depends on the same nuclear matrix combination Lambda, as did the ZR theory. By setting its value to 1.5 the predicted spectrum almost agrees with the measured one (see on the right). In addition, there are good arguments that the value of Lambda equal to 1.5 is much more reasonable than 3.5, considered previously. We see, however, an excess of radiation in the high-energy end.

This excess, we believe, is a manifestation of a process which was predicted in 1960s and sought for ever since. The radiative decay of 59Ni can proceed via an excited state in 59Co, located about 30 keV above the initial state (see below on the left). This is an example of a virtual (detour) beta-gamma transition via a well defined nuclear state. The influence of such a close-lying state has to be explicitely added to the REC spectrum in the Pachucki's theory. The contribution of this particular virtual transition was calculated and the result is shown below on the right by the solid green line. Finally, the gamma radiation emitted in the EC decay of 59Ni is a coherent sum of both electronic (IB) and nuclear (detour) components. The latter is responsibe for about 4% of the total intensity.

Now it became clear that previous efforts to observe the virtual transition in 59Ni failed because the IB part was described by the wrong ZR theory. The evidence for the detour beta-gamma decay path poped-up once the contribution from the Internal Bremsstrahlung was correctly calculated. The radiative decay of 59Ni nicely illustrates the quantum mechanical principle that the evolution of a physical system proceeds simultaneously through all possible paths, including those that may temporarily violate the law of energy conservation.

The detailed analysis of the 59Ni case is described in : Pfützner, Pachucki, Żylicz, Phys. Rev. C92 (2015) 044305.