1. INFLUENCE OF ELECTRONIC SHELL ON NEUTRINOLESS DOUBLE BETA DECAY
M. Ya. Amusia , E. G. Drukarev, and
L. V. Chernysheva

2. 1. Introduction There are 35 nuclei, for which the single beta decay
(A,Z)->(A,Z-1)+e+v
is forbidden, while the double beta decay
(A, Z)->(A,Z-2)+2e+2v
is allowed.
The energy of the decay is shared between two outgoing electrons and two electron antineutrino.

3. 2. Diagrammatic representation I
The single beta-decay: e
n p ν
The double beta-decay: e1
n p1 ν1 ν2
n p2 e2

4. 2. Diagrammatic representation II
Neutrinoless double beta-decay: e1
n p1 ν
n p2 e2
It is possible only if
ν = ν
Detectable by measuring the energy loss

5. 2. Diagrammatic representation III
Energy loss in the electron shell:
Atomic
excitation e1
n p1 ν
n p2 e2

6. 3. Double beta decay of an atom I
If the electron neutrino is a Majorana particle,
i.e. if it coincides with corresponding antineutrino, there exist a neutrinoless channel of the double beta decay
(A,Z)->(A,Z-2)+2e.
Denote as Q=M(Z)-M(Z-2) the mass difference of the “parent” and “daughter” nuclei ; T - the energy carried away by the beta electrons.
The relation E=Q is the “fingerprint” of the neutrinoless decay
Note: Until now the neutrinoless decay have not been observed.

7. 3. Double beta decay of an atom II
In the neutrinoless beta decay its energy is shared between the beta electrons and the atomic electrons, which can be excited or knocked into continuum.
If the “daughter” atom remains in the ground state, energy available for the beta electrons is
Q=M(Z)-M(Z+2)+ ε(Z)-ε(Z+2),
Here ε (Z) and ε (Z+2) are the energies of the bound electrons in the ground states of the “parent” and “daughter” atoms, respectively.

8. 3. Double beta decay of an atom III
Neglecting the possible excitations of the “daughter” atom, we find that the largest energy available for the beta electrons is
E=Q
If we include the possible excitations, it reads
E=Q-δQ
Our aim is to calculate δQ.

9. 4. Derivation of formulas
δQ=Σnεn0(dWn/dΓ)/ (dW0/dΓ),
εn0= εn(Z+2)- ε0(Z+2)
Index n labels the excited state of the “daughter atom, 0 stands for the ground state.
Let us present δQ as
δQ=δ1 + δ2,
where
δ1= [ε0(Z)- ε0(Z+2)] Σn(dWn/dΓ)/(dW0/dΓ)
δ2= Σn[εn(Z+2)- ε0(Z)] (dWn/dΓ)/ (dW0/dΓ).

10. 5. Calculation details I Since the beta electrons carry the energy of
several MeV the probabilities dWn /dΓ
can be calculated in the shake-off approximation
dWn/dΓ=dWn/dΓ|<Φn|Ψ>|2,
Here Ψ and Φn are the wave functions of Z electrons in the ground state of the field of the nucleus with the charge Z and in the state n of the field of the nucleus with the charge Z+2.

11. 5. Calculation details II
This leads to the following expressions
δ1= [ε0(Z)- ε0(Z+2)] Σn Wn,
δ2= Σn[εn(Z+2)- ε0(Z)]Wn .
where Wn=|< Φn|Ψ>|2.
If the atom is treated nonrelativistic, the
states Φn compose a closed system. Thus,
δ1= ε0(Z)- ε0(Z+2),
δ2= <Ψ|H(Z+2)-H(Z)|Ψ>=<Ψ|Σk(-2α/rk)|Ψ>.

12. 5. Calculation details III
Both δ1 and δ2 can be calculated with reasonably high accuracy. We did it employing our Hartree-Fock computer codes.
We focused on the double beta decay of germanium Ge (Z=32) and xenon Xe (Z=54),
which are studied in nowadays experiments.
We have found
δQGe = 352 eV
δQXe = 414 eV
Corrections to the shake-off approximation are estimated to be of the order 1 eV.

13. 6. Summary I
We carried out nonrelativistic calculation for the shift of the limiting energy available for the ejected electrons in double beta decay caused by inelastic processes in electronic shell. The energy diminishes by about 400 eV.
We estimated the accuracy of our calculations.
Our result support the earlier conclusion that the neutrinoless mode is not yet observed

14. Thank you very much for attention!
6. Summary II
We calculated an integrated characteristic of the process. It is reasonable to investigate also the differential characteristics such as the spectrum of the shake-off electrons or the probabilities of the shake-up processes.
However, such data for neutrinoless double beta decay is beyond the scope of our contemporary experiment.
Thank you very much for attention!

15. 7. References I
E. G. Drukarev, M. Ya. Amusia, and L. V. Chernysheva, Phys. Rev. A, In print, 2016
B. Schwingenheuer, Ann. Phys. (Berlin) 525, 269 (2013).
I. V. Kirpichnikov, arXiv: [hep-ex].
I. V. Kirpichnikov, arXiv: ,v2 [hep-ex].
I. V. Kirpichnikov, arXiv: ,v2 [nucl-ex].