$^{100}\rm Mo$ decay to excited states

Alternative approach to gamma search was used in this analysis. The basic idea is to use only time information, to construct gamma cluster, instead of geometrical positions.

Algorithm. Let it be $p_1,p_2,..,p_n$ - list of fired PMTs, not associated to any track. One can make two ordered groups $p^1_1,..,p^1_{n1}$ and $p^2_1,..,p^2_{n2}$ ($n_1+n_2=n$). The hypothesis is that each group is a gamma, emmited from the foil and rescattered inside the detector. Using TOF information, one can calculate $\chi^2$ criteria for this hypothesis:

$$\chi^2_1=\sum^{n_1}_{i=1} [((t_{mes}(p^1_i)-t_{foil})-(L(vtx,p^1_1)+..+L(p^1_{i-1},p^1_i))/c)/\sigma]^2$$

$$\chi^2_2=\sum^{n_2}_{i=1} [((t_{mes}(p^2_i)-t_{foil})-(L(vtx,p^2_1)+..+L(p^2_{i-1},p^2_i))/c)/\sigma]^2$$

$$\sigma^2=\sigma(t_{mes}(p_i))^2+\sigma(t_{foil})^2+\sigma(L)^2$$

$$\chi^2=(\chi^2_1+\chi^2_2)/n$$

,where $t_{mes}(p_i)$ - mesuared time of PM $i$; $t_{foil}$ decay time in the foil (known from 2 electrons), $L(a,b)$ - distance between objects $a$ and $b$.
Foil time is known from two electrons detected. Let it be $t_1$ and $t_2$ measured PM times for electron 1 and, and $tof_1$, $tof_2$ - there calculated times of flight. $\sigma (t_1)$, $\sigma (t_2)$, $\sigma (tof_1)$, $\sigma(tof_2)$ - errors in this quantities. Then the foil time and its error can be calculated as weighed mean value:

$$\sigma(t_{foil})^2=\sigma (t_1)^2+\sigma (t_2)^2+\sigma (tof_1)^2+\sigma(tof_2)^2\\$$

$$t_{foil}=\sigma(t_{foil})^2 * (\frac{t_1}{\sigma (t_1)^2+\sigma (tof_1)^2} +\frac{t_2}{\sigma (t_2)^2+\sigma (tof_2)^2} )$$

Looping other all possible combinations of group 1 and 2, select one with the best $\chi^2_{min}$. If its value less then cutoff (<3), then accept this event as two-gamma event. Gamma energy is the sum of all PM's energy in the group.
So my basic cuts (cut1) are:

• Two internal electrons in the events (common cuts)
• Distance between electron vertecies $\Delta XY<6$cm, $\Delta Z<8$cm
• Electron energy $0.6 > E_\beta > 0.175$ MeV
• Two internal gamma found in the event, $\chi^2<3.$
• Gamma energy $0.55 > E_\gamma > 0.175$ MeV

I used analysis developed by Jenny and me in UCL. So we have the same amount of data (8022h) and MC background (Bi214,Tl208,K40 in PMTs, Bi214 in the air outside, Bi214 on the wires, Bi214 on the surf., Mo100 decay to ground state). Obtained result is:

• number of events in Mo 95 (MC background 39.6)
• efficiency $\varepsilon=0.00096$
• $T_{1/2}=4.6^{+0.9}_{-0.7}\cdot 10^{20}$ y
• $T^{metal}_{1/2}=4.4^{+1.6}_{-0.9}\cdot 10^{20}$
• $T^{comp}_{1/2}=4.7^{+1.4}_{-0.8}\cdot 10^{20}$
• Signal/ background = 1.4
• events with $\alpha$ in Mo 14 (MC estimation 8.8)
• events in Cu+Te+Te-nat 20 (MC estimation 15.7)

Figure: Events after cut1. Electrons and gammas energy for Mo and non-Mo foils. Green - MC background, red - $2\beta$ signal.

Figure: Events with detected $\alpha$ after cut1. Electrons and gammas energy for Mo and non-Mo foils. Green - MC background.

Figure: Events after cut1. Minimal electron and gamma energy, totla energy for Mo and non-Mo foils. Green - MC background, red - $2\beta$ signal.

In order to improve signal/background ratio one can use additional cuts. Looking at the gamma energy distribution I noticed that for $^{100}\rm Mo$ decay summed gamma energy is rather high. So additioanl cut3 is $E_{\gamma 1}+E_{\gamma 2}>0.6$ MeV. Results of cut2 are following:

• number of events in Mo 62 (MC background 23)
• efficiency $\varepsilon=0.000688$
• $T_{1/2}=4.7^{+1.2}_{-0.8}\cdot 10^{20}$ y
• $T^{metal}_{1/2}=4.2\cdot 10^{20}$, $T^{comp}_{1/2}=5.8\cdot 10^{20}$
• Signal/ background = 1.7
• events with $\alpha$ in Mo 10 (MC estimation 6.5)
• events in Cu+Te+Te-nat 10 (MC estimation 9.2)

This cut is significantly improves S/B ratio. Also the number of events in test sectors is small, so it really supress the background.