QValue $2\nu\beta\beta$

Author: Toby Dixon ([email protected]) A tool to fit the $2\nu\beta\beta$ decay spectrum treating $Q_{\beta\beta}$ as a free parameter.

Mathematical formulation

The idea of this analysis is to extract the $Q_{\beta\beta}$ value for $2\nu\beta\beta$ decay directly from the $2\nu\beta\beta$ spectrum. This is described in https://arxiv.org/pdf/2210.08722. The $2\nu\beta\beta$ spectrum can be described as:

$$ \frac{d\Gamma}{dE}=A(E)\times (Q-E)^5 $$

Where $A(E)$ is the 'shape' factor, fairly complicated to compute depending on the atomic and nuclear physics of the decay. The basic principle of our analysis is to perform a fit to extract an analytic approximation / interpolation of A(E), the $2\nu\beta\beta$ decay distribution can then be included in a background model fit with $Q_{\beta\beta}$ as a free parameter. Within the improved description (https://arxiv.org/pdf/2307.14086) of $2\nu\beta\beta$ decay:

$$ \frac{d\Gamma}{dE}=g_{A,eff}^4|M_{GT-1}|^2\Big[\frac{dG_0}{dE}+\xi_{3,1}\frac{dG_2}{dE}+...] $$

Or alternatively to allow the parameterisation above:

$$ \frac{d\Gamma}{dE}=g_{A,eff}^4|M_{GT-1}|^2(Q-E)^5\Big[A_0(E)+\xi_{3,1}A_2(E)+...] $$

Thus we can parameterise independently each of the additive contributions to the spectrum, for inclusion in a fit considering both the spectral shape uncertainty and $Q_{\beta\beta}$ uncertainities.

BetaSpectrumHandler

The first part of the analysis is contained in a script called BetaSpectrumHandler.cxx, which performs this parameterisation. This code takes in a MC simulation of the $2\nu\beta\beta$ decay obtained with the $Q=3034.4$ keV. It then fits this spectrum to $A(E)\times (Q-E)^5$, whether $A(E)$ is parameterised as a 6th order polynomial. The code to run this can be compiled with:

g++ -std=c++11 -g -o BetaSpectrumHandler BetaSpectrumHandler.cxx $(root-config --cflags --libs)  

It can then be run with:

./BetaSpectrumHandler -l [plot_path] -i [input_path] -p [polynomial order] -q [Qbb] -o [out_path] 

add the option -I to use the improved model and -h for this help

• [input_path] should be the path to the output file of the background model
• [plot_path] is a path to save some plots of the interpolation
• [out_path] is the path to the output files used for QbbFit this is a ROOT file with the functions parameterising A(E).

QbbFit

The next part of the analysis chain consists of a BAT code called QbbFit which performs a background model fit on the spectrum using the analytic approximation of the $2\nu\beta\beta$ decay spectrum shape. Currently the implemented fit is very simple and just models the data as a simple sum of signal and background.