Summary of the model





lnL =Pvariables v

Pslices

s lnLv,s(Nisoϵv,s) P

sϵv,s = 1 ∀v

(3.30)

where ϵ_v,s is the fraction of isotope in slice s of variable v. In the following results, we shall focus on using four different variables, namely prompt energy (E_p), delay energy (E_d), prompt-delay coincidence time (dt) and prompt-delay distance (dist) with uniform slicing over the range of selection criteria. Also, we shall vary the number of slices to see the robustness of our estimator and as an estimation of the systematic uncertainty with our estimation. Also, we shall use the predicted prompt energy spectrum of⁹Li/⁸He to estimate potential variations of our measurements as another source of systematic uncertainties.

3.4 Summary of the model

In section 3.2 and 3.3 we have discussed the underlying mechanics of how to measure cosmogenic isotopes and how we can use spectral information to increase our resolution of the measurements. The parameters used in the model is summarized in table 3.1.

Remember that the combined model is just the sum of individual log likelihood function for each IBD subsample, that is

lnL =X

r,v,s

lnLr,v,s (3.31)

Name Number of parameters

R_µ N_µ-regions

N⁹_Li/⁸_He N_µ-regions

N¹²_B N_µ-regions

N¹²_N N_µ-regions

N_uncorr 1

ϵ_iso,v,s N_componentsN_variables(N_slices− 1)

Table 3.1: Summary of free parameters in the combined model. Here Nµ-regions = 3 (low, mid, high) denotes the number of muon regions used and N_componentsdenotes the number of components (⁹Li,⁸He...) which may differ depend on the analysis performed, N_variables = 4 is the number of variables used in IBD slicing (Ep, Ed, dt, dist) and N_slicesis the number of slices used to slice the variables. Note that we only need N_slices−1 parameters to model the ϵ’s since they should sum to 1. So the total number of parameters is 4N_µ-regions + N_componentsN_variables(N_slices− 1) + 1.

Name Value

R_µ R_µ,r

N⁹_Li/⁸_He N⁹_Li/⁸_He,rϵ⁹_Li/⁸_He,v,s N¹²_B N¹²_B,rϵ¹²_B,v,s N¹²_N N¹²_N,rϵ¹²_N,v,s N_uncorr N_uncorr+P

iso

P

r̸=r^′N_iso,r′ϵ_iso,v,s

Table 3.2: Summary of free parameters in a single likelihood function of muon range r, slice variable v and slice s.

where r is the muon region, v is the variable of slicing and s is the slice of IBD. And for each individual lnLr,v,s, the fitting function is of the form shown in equation 3.14 with α being multiplied by the total number of events. In other words, we have to put the corresponding numbers of isotope, muon rates into the fitting function to get the likelihood value and these values are summarized in table 3.2.³

3In practice⁹Li and⁸He are combined into a single component due to the similarity of the reciprocal of

−1 −1 −1 −1

Chapter 4

Signal selection

The following analyses presented in this thesis are based on the dataset used in [15] which constitutes of 1958 days of data collected starting from December 2011 to January 2017.

For event reconstruction, we use the reconstruction method B as described in [17] which was also discussed in the event reconstruction section 2.4.

4.1 Signal selection

Remember that our goal is to measure the contaminations of ⁹Li and ⁸He in our IBD candidates. Therefore, the selection criteria for these signals shall follow the IBD selection criteria as described in [17]. For study purposes, here we adapt a more general selection criteria, namely the unified IBD selection criteria, where both nH and nGd are included in the sample. This can be achieved with slight modifications to the original IBD selection criteria where we enlarge our delayed energy window and the capture time window (which in turn affects our choices of multiplicity cuts).

In fact, we can retain the nGd sample from the unified sample by tightening the delayed energy, capture time window and discarding the distance cut. Similarly, we can also retain the nH sample from the unified sample by modifiying the variables correspondingly. The selection criteria for these samples are summarized in table

For the purpose of this study, we shall remove shower muon veto in the selection criteria to increase the statistics of⁹Li and ⁸He (shower muon contribute ∼ 70% of the statistics). The efficiency of shower muon veto on⁹Li and⁸He can be calculated, assuming

Analysis Unified nGd nH

Flasher cut Yes

µ_WS(NHIT>12) (-2, 600)µs

µ_AD(p.e.>3000) (0, 1.4)ms

µ_Shower(p.e.> 3× 10⁵) (0, 0.4)s

E_p(MeV) (0.7, 12) (0.7, 12) (1.5, 12)

E_d(MeV) (1.5, 12) (6, 12) (1.9, 2.7)

Capture time(µs) (1, 400) (1, 200) (1, 400)

P-D distance <500 mm N/A <500 mm

Multiplicity cut(p) Only one prompt candidate 800µs before delayed Multiplicity cut(d) No other delayed candidate 400µs after delayed

Table 4.1: Summary of IBD selection criteria for unified, nGd and nH analyses.

exponential decays, by

ϵ = P (t_decay > T_veto) = Z _∞

Tveto

1

τe^−τtdt = e^−Tvetoτ . (4.1) where T_veto = 0.4s and τ ∼ 257ms for ⁹Li and 172ms for ⁸He for our case. Since the contamination of ⁹Li is much higher ⁸He, we shall approximate this efficiency by e^{−Tveto/257ms} ≈ 0.211. In other words, ∼79% of the⁹Li and ⁸He are removed by shower muon veto.

For the record, previous estimations in both [17, 15] used a more stringent selection criteria to increase the significance of⁹Li and⁸He by removing uncorrelated backgrounds.

For example, IBDs and⁹Li/⁸He have capture time distributions following the exponential distributions while uncorrelated backgrounds (accidentals) have a flat distribution. This suggests that using shorter capture time windows may actually increase the significance of⁹Li/⁸He thus lower the uncertainties in the estimations. For our study, we will not be using these augmented cuts for we could simply include these spectra into the model and benefit from them without using these cuts.

4.1.1 Signal distributions

Below we shall examine some of distributions for the selected signals used in the following analyses. Aside from the three selection criteria (nGd, nH and unified) shown above, we also include another the unified sample without using distance cut, which is dominated by accidental backgrounds, to demonstrate the major differences between IBD signals and uncorrelated backgrounds (accidentals).

In figure 4.1, 4.2 and 4.3, we show the distributions of prompt energy, delayed energy, capture time and prompt-delayed distance under different selection criteria and different sites. First of all, one can think of the differences between the black lines (unified sample without distance cut) and blue lines (unified sample with distance cut) as the distributions for uncorrelated backgrounds which have the following characteristics

• Prompt energy and delayed energy distributions are similar to those from isolated signals because they are mostly. originated from isolated signals accidentally being paired up and passing the selection criteria.

• Capture time distributions are mostly flat because they are temporally uncorrelated.

• Increasing prompt-delayed distance distributions because they are also spatially uncorrelated.

These features will show up again later in our estimations of⁹Li and⁸He because we found that certain cosmogenic isotopes tend to pair up with other isolated signals and form pairs of accidentals backgrounds which shall be discussed in the next chapter.

In general, nGd samples from all three sites agree very well with each other because they are less contaminated by accidentals. nH and unified samples, on the other hand, are vastly different between the near sites and the far site. This is because the signal-to-background ratio is much lower for the far site (less IBD, similar accidentals). Aside from that, the distributions agree very well with each other and our expectations (∼ 2 Mev and

∼ 8 MeV neutron capture peaks for the delayed energies, exponential decay capture times and decreasing prompt-delay distances).