The Exposure-Background Duality in the Searches of Neutrinoless Double Beta Decay

Tremendous efforts are required to scale the summit of observing neutrinoless double beta decay ($0 \nu \beta \beta$). This article quantitatively explores the interplay between exposure (target mass X data taking time) and background levels in $0 \nu \beta \beta$ experiments. In particular, background reduction can substantially alleviate the necessity of unrealistic large exposure as the normal mass hierarchy (NH) is probed. The non-degenerate (ND)-NH can be covered with an exposure of O(100) ton-year, which is only an order of magnitude larger than those planned for next generation projects - provided that the background could be reduced by 0($10^{-6}$) relative to the current best levels. It follows that background suppression will be playing increasingly important and investment-effective, if not determining, roles in future $0 \nu \beta \beta$ experiments with sensitivity goals of approaching and covering ND-NH.


I. INTRODUCTION
The nature of the neutrinos [1], and in particular whether they are Majorana or Dirac particles, is an important problem in particle physics, the answer to which will have profound implications to the searches and formulation of physics beyond Standard Model and the Grand Unified Theories. Neutrinoless double beta decay (0νββ) is the most sensitive experimental probe to address this question [2]. Observation of 0νββ implies: (i) that neutrinos are Majorana particles, and (ii) lepton number violation. Since several decades, there are intense activities world-wide committed to the experimental searches of 0νββ.
Neutrino oscillation experiments [1,3] are producing increasingly precise information on the mass differences and mixings among the three neutrino mass eigenstates. The latest data imply slight preferences of the "Normal Hierarchy" (NH) over the "Inverted Hierarchy" (IH) in the structures of the neutrino mass eigenstates [4]. In parallel, cosmology data [5] provide stringent upper bounds on the total mass of the neutrinos, with good prospects on an actual measurement in the future. Together, a picture emerges providing a glimpse on the parameter space where positive observations of 0νββ may reside. Experimental studies are expected to require significant efforts and resources − especially so if NH is confirmed. Detailed quantitative studies on the optimal strategies "to scale this summit" with finite resources would be highly necessary.
The current work addresses one aspect of this issue. We studied the required exposures of 0νββ-projects ver- * Corresponding Author: htwong@phys.sinica.edu.tw sus the expected background B 0 before the experiments are performed. The notations and formulation are described in Section II. The effects on the "discovery potentials" with varying B 0 , and the implied experimental strategies, are discussed in Section II B. The connections with the current landscape in neutrino physics are made in Section II E via the choice of a particular model on the evaluation of nuclear matrix elements. Various aspects on the interplay between exposure and background in 0νββ experiments are discussed in Section III A Background typically includes two generic components each having different energy dependence − the ambient background and the irreducible intrinsic background from cosmogenic radioactivity and two-neutrino double beta decay (2νββ). Only the combined background is considered in this work, while on-going research efforts are attending the different roles of the two components. In particular, the constraints imposed by the 2νββ background to detector resolution are discussed in Section III B.

A. Double Beta Decay
The process 0νββ in candidate nucleus A ββ refers to the decay The experimental signature is distinctive. The summed kinetic energy of the two emitted electrons corresponds to a peak at the transition Q-value (Q ββ ), which is known and unique for each A ββ . The width of the 0νββ-peak (denoted by ∆ in %) characterizes the energy resolution of the detector, and is de-fined − a natural choice and also following convention in the literature for clarity − as the ratio of full-width-halfmaximum (FWHM, denoted by w 1/2 ) to the total measureable energy Q ββ , such that w 1/2 =(∆·Q ββ ).
Various beyond-standard-model processes invoking lepton-number-violation can give rise to 0νββ [6]. In the case of the "mass mechanism" where 0νββ is driven by the Majorana neutrino mass, the 0νββ half-life (T 0ν 1/2 ) can be expressed by [2,7] 1 T 0ν where m e is the electron mass, g A is the effective axial vector coupling [8], G 0ν is a known phase space factor [9] due to kinematics, |M 0ν | is the nuclear physics matrix element [10], while m ββ is the effective Majorana neutrino mass term which depends on neutrino masses (m i for eigenstate ν i ) and mixings (U ei for the component of where α and β are the Majorana phases. The measureable half-life T 0ν 1/2 from an experiment which observes N 0ν obs -counts of 0νββ-events in time t DAQ in a "Region-of-Interest" (RoI) at an efficiency of ε RoI can be expressed as where N (A ββ ) is the number of A ββ atoms being probed. For simplicity in discussions and to allow the results be easily convertible to different configurations − while capturing the essence of the physics, results in this article are derived in the special "ideal" case where the target is made up of completely enriched A ββ isotopes. That is, the isotopic abundance (IA) is 100%. In additional, the various experimental efficiency factors are all unity (ε expt =100%). Accordingly, Eq. 4 becomes T 0ν where N A is the Avogadro Number, M (A ββ ) is the molar mass of A ββ , and Σ denotes the combined exposure (mass×t DAQ ) expressed in units of ton-year (ton-yr) at A ββ at IA=100% and ε expt =100%. Effects due to these parameter choices and other assumptions will be discussed in Section II D where conversion relations to those for realistic experiments are given. The expression of Eq. 5 applies to experiments with counting analysis. More sophisticated statistical methods are usually adopted to extract full information from a given data set. These typically exploit the energy spectral shapes, which are known for the signal and are predictable with uncertainties for the background. However, in the conceptual-design and sensitivity-projection stage of experiments, the simplified and intuitive approach of Eq. 5 will suffice, especially so in the low count rate Poisson statistics regime which is of particular interest in this article.

B. Discovery Potential
In our context, B 0 is expected background counts within the RoI around Q ββ . This can, in principle, be predicted with good accuracies prior to the experiments. The sensitivity goals of experiments are typically expressed in the literature [11] as: "Discovery Potential at 3σ with 50% probability" (P 3σ 50 ) and "upper limits at 90% confidence level" which characterize possible positive and negative outcomes, respectively. We focus on P 3σ 50 in this work, for the reason that next-generation 0νββ experiments should be designed to have the maximum reach of discovery, rather than setting limits.
Poisson statistics is necessary to handle low background and rare signal processes. The dependence of the required average signal (S 0 ) versus B 0 under P 3σ 50 and other discovery potential criteria are depicted in Figure 1a. For a given real and positive B 0 as input and using P 3σ 50 as illustration, the Poisson distribution P (i; µ) is constructed with mean µ=B 0 . The observed count N 3σ obs is evaluated as the smallest integer which satisfies where 0.00135 is the fraction of a Gaussian distribution in the interval [+3σ, ∞]. This is the minimal observed event integer number with ≥3σ significance over a predicted average background B 0 . The output S 0 is the minimal signal strength corresponding to the case where the average total event (B 0 +S 0 )=N 3σ obs with ≥50% probability. This is evaluated as the minimum value which satisfies another Poisson distribution under the condition: It can be inferred from Figure 1a   The variations of (a) S0 and (b) ratios of S0 to S ref versus B0 in the discovery potential of ≥ 3σ, 5σ with ≥ 50%, 90%. The specific case of P 3σ 50 at 3σ and 50% is adopted as the criteria in this work. The level of S ref at the background-free condition for P 3σ 50 is represented with a black dot in (a) 6.7(16.8) stronger signals to establish positive results at P 3σ 50 when B 0 increases from < 10 −3 to 1(10).
While the predicted average background B 0 can be continuous and real numbers, only integer counts can be observed in an experiment. This gives rise to the relations being inequalities in Eqs. 7&8 and consequently the steps in Figures 1a&b. In addition, signal and background events are indistinguishable experimentally. The P 3σ 50 criteria is applied to (B 0 +S 0 ) versus B 0 , while the S 0 dependence on B 0 is shown in Figures 1a&b. This is the origin of the negative slopes in various segments.

C. Background Index
The theme of this work is to study the interplay between required exposure and background in 0νββ experiments to meet certain m ββ target sensitivities.
In realistic experiments, it is more instructive to characterize background with respect to exposure and the RoI energy range, such that the relevant parameter is the "Background Index" (BI) defined as: which is the background within the RoI (chosen to be ≡ w 1/2 , following convention) per 1 ton-year of exposure, with dimension [counts/(w 1/2 -ton-yr)]. Background levels expressed in BI are universally applicable to compare sensitivities of varying A ββ in different experiments.

D. Conversion to Realistic Configurations
As explained in Section II A, the (BI, Σ) results presented in this article correspond to the ideal case where IA=100% and ε expt =100%. In addition, while the range of g A ∈[0.6, 1.27] is generally considered possible [7,8], the "unquenched" free nucleon value of g A =1.27 is adopted.
The required exposure (Σ ) in realistic experiments would be larger and can be readily converted from the Σ-values via where W Σ (g A ) is the weight factor for Σ due to the g Adependence [10,12] of T 0ν 1/2 in Eq. 2, relative to the values at g A =1.27. It is depicted in Figure 3 for the case of 76 Ge. The finite band width as a function of g A is the consequence of the spread in |M 0ν | 2 predictions [10,12]. The specific case where |M 0ν | 2 is independent of g A implies Σ∝[g A ] -4 and is denoted by the dotted line.
The background index defined relative to Σ for realistic configurations can accordingly be expressed as such that Σ >Σ and BI <BI. Realistic experiments naturally imply larger exposure and more stringent background requirements.

E. Neutrino Physics Connections
Results from neutrino oscillation experiments [1,3] indicate that the m i of the three active ν i have structures corresponding to either IH or NH. The values of m ββ are constrained and depend on the absolute neutrino mass scale, and are typically expressed in terms of the Variations of "specific phase space" g 4 A H 0ν , as defined in Eq. 6, versus |M 0ν | 2 for various A ββ . The geometric mean of the range of |M 0ν | 2 is presented as the data points. The best-fit and other diagonal lines correspond to the matching m ββ values at gA=1.27 that give rise to a 0νββ rate of 1 event per ton-year at full efficiency. This formulation is adopted from Figures 2&3 of Ref. [7].
TABLE I: Summary of the key parameters used in this work. Inputs are the IH and NH bands at ±3σ of the ND scenario at mmin<10 −4 eV from existing measurements [1], such that m ββ −< m ββ < m ββ +. The posterior m ββ 95% denotes the 95% lower bound for the m ββ -distribution, taking an uncorrelated (α, β) and the uncertainty range in |M 0ν | 2 as prior [11]. The corresponding minimal-exposures at background-free levels under criteria P 3σ 50 are given as Σmin. The reduction fraction in Σ from m ββ − to m ββ 95% is denoted by f 95% . The values of ( m ββ −, m ββ +) and m ββ 95% define the IH/NH band width and dotted lines, respectively, in Figures 4,5&8 in this article.  Table I.
There are no experimental constraints on the Majorana phases (α, β). It is in principle possible to have accidental cancellation which leads to very small m ββ at m min ∈[1, 10]×10 −3 eV. However, under the reasonable assumption that they are uncorrelated and have uniform probabilities within [0, 2π], a posterior probability distributions of m ββ can be assigned [11]. The 95% lower limit, denoted as m ββ 95% and listed in Table I, shows that the vanishing values of m ββ are disfavored.
The current generation of oscillation experiments may reveal Nature's choice between the two hierarchy options. In particular, there is an emerging preference of NH over IH [1,4]. Moreover, the combined cosmology data may provide a measurement on the sum of m i [5]. Therefore, it can be expected that the ranges of parameter space of m ββ in 0νββ searches will be further constrained.
Extracting neutrino mass information via Eq. 2 from the experimentally measured T 0ν 1/2 requires knowledge of |M 0ν | 2 and g A . There are different schemes to calculate |M 0ν | 2 for different A ββ [10]. Deviations among their results are the main contributors to the theoretical uncertainties. Another source of uncertainties is the values of g A , which may differ between a free nucleon and complex nuclei [8].
Studies of Ref. [7] suggest that, in the case where 0νββ is driven by the neutrino mass mechanism, there exists an inverse correlation between G 0ν and |M 0ν | 2 in Eq. 2, the consequence of which is that the decay rates per unit mass for different A ββ are similar at given m ββ and constant g A . That is, there is no favored 0νββ-isotope from the nuclear physics point of view.
This empirical observation originates partially to the large uncertainties in |M 0ν | 2 and g A . To derive numerical results which would shed qualitative insights without involving excessive discussions on the choice of |M 0ν | 2 , we assume that this correlation is quantitatively valid.
We follow Ref. [7] in adopting the geometric means of the realistic ranges for the various |M 0ν | 2 in different isotopes. The data points can be parametrized by straight lines at given m ββ , as depicted in Figure 2. That is, is a constant at fixed m ββ independent of A ββ . The displayed m ββ values in Figure 2 correspond to 0νββ decay rates of [N 0ν obs /Σ]=1/ton-yr at g A =1.27 and full efficiency. The best-fit at this decay rate corresponds to m ββ =(35×10 −3 ) eV.
Following Eq. 6, this model leads to a simplifying consequence that at IA=100% and ε expt =100%, which is universally applicable to all A ββ . The proportional constant can be derived via the best-fit values of Figure 2. Given a background B 0 as input, the required S 0 to establish signal under P 3σ 50 can be derived via Figure 1a. This is related to the mean of N 0ν obs at known ε RoI . Neutrino physics provides constraints on m ββ with several scales-of-interest given in Table I of the nominal values) to match our current understanding of |M 0ν | 2 .

III. SENSITIVITY DEPENDENCE
It is well-known, following Eqs. 2&5, that the sensitivity to [1/ m ββ ] is proportional to Σ 1 2 as B 0 →0 and to Σ 1 4 at large B 0 . We further investigate the B 0 -dependence quantitatively and in the context of the preferred IH and NH ranges with the model of Ref. [7]. The specific m ββvalues of Table I − ( m ββ − , m ββ + , m ββ 95% ) for both IH and NH − serve to provide reference scales.

A. Required Exposure and Background
The variations of m ββ versus B 0 with different Σ at RoI=w 1/2 (such that ε RoI 76%) under the criteria of P 3σ 50 are depicted in Figure 4, with the IH and NH bands superimposed. The matching T 0ν 1/2 for 76 Ge is illustrated. The equivalent half-life sensitivities for other isotopes A ββ can be derived via The figure depicts how the same exposure can be used to probe longer T 0ν 1/2 and smaller m ββ with decreasing background.
The dependence of m ββ sensitivities to BI is depicted in Figure 5. Taking RoI=w 1/2 is obviously not the optimal choice when the expected background B 0 →0. An alternative choice for low B 0 is RoI≡w 3σ covering ±3σ of Q ββ , such that ε RoI ∼ =100%. Both schemes are illustrated in Figure 5. The choice of RoI=w 3σ at B 0 →0 would expectedly give better sensitivity by a factor of ε RoI (w 1/2 )=0.76, such that the covered T 0ν 1/2 is 32% longer, or the required Σ is 24% less.
The required exposure to probe m ββ 95% and m ββ − with both RoI selections in both IH and NH are depicted in Figure 6a. Superimposed as a blue contour is the  The range of (S0, B0) to qualify a positive signal to cover m ββ − for both IH and NH under P 3σ 50 , given the observed number of events in RoI − 0νββ-signals and background are combined but indistinguishable at event-by-event level. The smaller Σ values among the alternatives of RoI=w1/2 or w3σ are selected. The sixth column shows the required BI which is universal to all A ββ . Last column lists the required background specifically for 76 Ge normalized to "/(keV-ton-yr)", and the conversion to other isotopes is referred to Eq. 14. The N 0ν obs =1 row correpsonds to the background-free conditions. The BI-values follow from Eq. 9.

Counts Within RoI
Optimal Covering m ββ − for NH: "benchmark" background level at 1 count/(w 1/2 -Σ) where the first background event would occur at a given exposure. The benchmark level also represents the transition in the effectiveness of probing m ββ with increasing exposure. The shaded regions correspond to the preferred hardware specification space for future 0νββ experiments − where the exposure should be sufficient to cover at least m ββ

IH(NH) 95%
, and there would be less than one background event per w 1/2 over the full exposure.
This background would correspond to Σ cantly reduced to allow the quest to advance. The target exposure is Σ=10 ton-yr for the next generation 0νββ projects to cover IH with ton-scale detector target [14].

IH(NH) ref
Following Figure 5, this exposure would require BI<(0.21, 0.033) counts/(w 1/2 -ton-yr) to cover ( m ββ 95% , m ββ  Table II. event can establish the signal at the P 3σ 50 -criteria. Their values at the benchmark m ββ 's are given in Table I.
The choice of m ββ − to define Σ is a conservative one. Since m ββ 95% > m ββ − from Table I, the minimum exposure Σ min corresponding to m ββ 95% is reduced relative to that for m ββ − by a fraction given as f 95% .
The variations of (BI min , Σ min ) with m ββ are depicted in Figure 7.
The interplay between fractional reduction of BI and Σ relative to BI 0 and Σ

IH(NH) ref
to cover m ββ 95% and m ββ − in IH(NH) is depicted in Figure 6b. Backgroundfree conditions require additional BI-suppression by factors of 3.1 × 10 −4 (0.96 × 10 −6 ), to cover m ββ IH(NH) − in which cases Σ can be reduced by factors of 0.016(5 × 10 −5 ). The shaded regions match those of Figure 6a in displaying the preferred hardware specification space.
In realistic experiments, signals and background are indistinguishable at the event-by-event level. The expected average background B 0 and the observed event counts (an integer) in the RoI are the known quantities. They can be used to assess whether a signal is "established" under certain criteria like P 3σ 50 . Listed in Table III  are   corresponds to the background-free condition, in which one single event is sufficient to establish a signal. Accordingly, the (BI, Σ) values match the entries in the last rows of Table II, and are displayed in Figure 7.
Results of Table III, apply generically to all A ββ except those for the last column when background is expressed in "/(keV-ton-yr)" unit. The values are specific for 76 Ge, where the best published ∆( 76 Ge)=0.12% of the MJDexperiment [15] is adopted as input. The background requirements for other A ββ can be derived via: [Background/(keV-ton-yr)](A ββ ) [Background/(keV-ton-yr)]( 76 Ge)

B. Limiting Irreducible Background
It is instructive and important to quantify the interplay between various irreducible background channels to the required exposure. In particular, one such irreducible background is the Standard Model-allowed 2νββ The contamination levels to 0νββ at the Q ββ -associated RoI depend on its half-life (T 2ν 1/2 ) and the detector resolution. A worse resolution (larger ∆) implies a larger RoI range to search for 0νββ signals, and therefore a higher probability of having background events from the 2νββ spectral tail.  [2,13], such that the 2νββ background within RoI=w3σ would contribute less than the levels specified by the benchmark and background-free conditions to cover m ββ  Depicted in Figure 9 are variations of the required ∆ with m ββ such that 2νββ background within RoI=w 3σ would contribute less than the BI-values specified by the benchmark and background-free conditions. The finite width of the band is a consequence of the spread of measured T 2ν 1/2 [2,13]. Faster 2νββ rates typically require better detector resolution to define smaller RoI. The relative locations for different A ββ within the bands are depicted in the inset.
Listed in Table IV are the required ranges of ∆ to cover m ββ
This provides a comfortable margin relative to that which satisfies the background-free conditions for m ββ NH − at BI≤0.96×10 −6 counts/(w 1/2 -ton-yr).

IV. SUMMARY AND PROSPECTS
As current neutrino oscillation experiments reveal a preference of NH, the strategy of scaling the summit of 0νββ should take this genuine possibility into account.
This work studies the relation between the two main factors in improving experimental sensitivities: (BI, Σ). We recall that the presented results are derived with certain input parameter choice: IA=100%, ε expt =100% and g A =1.27, and that 0νββ is driven by the Majorana neutrino mass terms via the mass mechanism while the Signal-to-Background analysis is based on counting experiments without exploiting the spectral shape information at this stage.
Advancing towards ND-NH to cover m ββ NH − will require large and costly exposure.
An unrealistic O(10) Mton-yr enriched target mass is necessary at the current best achieved background level BI 0 ∼1 count/(w 1/2 -ton-yr). Reduction of BI will be playing increasingly significant, if not determining, roles in shaping future 0νββ projects.
For instance, following Table II, backgroundfree conditions for m ββ NH − correspond to additional background suppression from the current best BI 0 ∼1 count/(w 1/2 -ton-yr) and benchmark [1 count/(w 1/2 -Σ)] levels by factors of (0.96 × 10 −6 ) and (4.4 × 10 −3 ), respectively. This would reduce the required Σ from 11 Mton-yr and 4600 ton-yr, respectively, to 550 ton-yr. The corresponding minimal-exposure to cover m ββ NH 95% is Σ min ∼37 ton-yr, which is only a modest factor beyond the goals of next-generation experiments [14]. The pursuit of background towards BI∼O(10 −6 ) counts/(w 1/2 -ton-yr) to probe ND-NH, while challenging, is highly investment-effective, as it is equivalent to reduction of Σ by O(10) Mton-yr and O(1) kton-yr relative to those required for the current best and benchmark background levels, respectively.
This article serves to quantify the merits of background reduction in 0νββ experiments, but does not attempt to address the experimental issues on how to realize the feat and how to demonstrate that the suppression factors are achieved when experiments are constructed. We project that the continuous intense efforts and ingenuities from the experimentalists world-wide, with motivations reinforced by the increasing equivalent "market" values, will be able to meet the challenges.
Boosting Σ involves mostly in the accumulation of enriched A ββ isotopes and turning these into operating detectors. These processes are confined to relatively few locations and small communities of expertise. The room of development which may overcome the known hurdles is limited. Suppression of the 0νββ experimental background, on the other hand, would be the tasks of mobilizing and coordinating the efforts of a large pool of expertise. It is related to the advances in diverse disciplines from novel materials to chemistry processing to trace measurement techniques. Research programs on many subjects requiring low-background techniques may contribute to − and benefit from − the advances. There would be strong potentials of technological breakthroughs and innovative ideas as the sensitivity goals are pursued.
Signal efficiencies are also increasingly costly as sensitivities advance towards ND-NH. For instance, at Σ min =550 ton-yr to cover m ββ NH − , a high 90% efficiency to certain selection criterion corresponds to discarding data of O(10) ton-yr strength − already an order of magnitude larger than the combined exposure of all 0νββ experiments. It follows that background suppression would preferably be attended at the root level − that radioactive contaminations are suppressed to start with, rather than relying on special signatures and software selection algorithms to identify them.
The next generation of 0νββ experiments would cover m ββ IH − . In addition, they should be able to explore the strategies and demonstrate sufficient margins to advance towards m ββ NH − . A significant merit would be to have no irreducible background before reaching the BI∼O(10 −6 ) counts/(w 1/2 -ton-yr) background-free configuration. The detector requirements to achieve this for 2νββ are summarized in Figure 9 and Table IV. Detailed studies of this background as well as other channels like those due to residual cosmogenic radioactivity and longlived radioactive isotopes are themes of our on-going research efforts.