TThe Exposure-Background Duality in the Searches of Neutrinoless Double Beta Decay
M.K. Singh,
1, 2
H.T. Wong, ∗ L. Singh,
1, 3
V. Sharma,
1, 2
V. Singh,
1, 2, 3 and Q. Yue Institute of Physics, Academia Sinica, Taipei 11529 Department of Physics, Institute of Science, Banaras Hindu University, Varanasi 221005 Department of Physics, School of Physical and Chemical Sciences, Central University of South Bihar, Gaya 824236 Department of Engineering Physics, Tsinghua University, Beijing 100084 (Dated: January 28, 2020)Tremendous efforts are required to scale the summit of observing neutrinoless double beta decay(0 νββ ). This article quantitatively explores the interplay between exposure (target mass × datataking time ) and background levels in 0 νββ experiments. In particular, background reduction cansubstantially 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, whichis only an order of magnitude larger than those planned for next generation projects − providedthat the background could be reduced by O (10 − ) relative to the current best levels. It followsthat background suppression will be playing increasingly important and investment-effective, if notdetermining, roles in future 0 νββ experiments with sensitivity goals of approaching and coveringND-NH. PACS numbers: 14.60.Pq, 23.40.-s, 02.50.-r.Keywords: Neutrino Mass and Mixing, Double Beta Decay, Statistics.
I. INTRODUCTION
The nature of the neutrinos [1], and in particularwhether they are Majorana or Dirac particles, is an im-portant problem in particle physics, the answer to whichwill have profound implications to the searches and for-mulation of physics beyond Standard Model and theGrand Unified Theories. Neutrinoless double beta de-cay (0 νββ ) is the most sensitive experimental probe toaddress this question [2]. Observation of 0 νββ implies:(i) that neutrinos are Majorana particles, and (ii) lep-ton number violation. Since several decades, there areintense activities world-wide committed to the experi-mental searches of 0 νββ .Neutrino oscillation experiments [1, 3] are producingincreasingly precise information on the mass differencesand mixings among the three neutrino mass eigenstates.The latest data imply slight preferences of the “Nor-mal Hierarchy” (NH) over the “Inverted Hierarchy” (IH)in the structures of the neutrino mass eigenstates [4].In parallel, cosmology data [5] provide stringent upperbounds on the total mass of the neutrinos, with goodprospects on an actual measurement in the future. To-gether, a picture emerges providing a glimpse on the pa-rameter space where positive observations of 0 νββ mayreside. Experimental studies are expected to require sig-nificant efforts and resources − especially so if NH isconfirmed. Detailed quantitative studies on the optimalstrategies “to scale this summit” with finite resourceswould be highly necessary.The current work addresses one aspect of this issue.We studied the required exposures of 0 νββ -projects ver- ∗ Corresponding Author: [email protected] sus the expected background B before the experimentsare performed. The notations and formulation are de-scribed in Section II. The effects on the “discovery po-tentials” with varying B , and the implied experimentalstrategies, are discussed in Section II B. The connectionswith the current landscape in neutrino physics are madein Section II E via the choice of a particular model onthe evaluation of nuclear matrix elements. Various as-pects on the interplay between exposure and backgroundin 0 νββ experiments are discussed in Section III ABackground typically includes two generic componentseach having different energy dependence − the ambi-ent background and the irreducible intrinsic backgroundfrom cosmogenic radioactivity and two-neutrino doublebeta decay (2 νββ ). Only the combined backgroundis considered in this work, while on-going research ef-forts are attending the different roles of the two com-ponents. In particular, the constraints imposed by the2 νββ background to detector resolution are discussed inSection III B. II. FORMULATIONS AND NOTATIONSA. Double Beta Decay
The process 0 νββ in candidate nucleus A ββ refers tothe decay NZ A ββ → N − Z +2 A + 2 e − . (1)The experimental signature is distinctive. The summedkinetic energy of the two emitted electrons correspondsto a peak at the transition Q-value ( Q ββ ), which is knownand unique for each A ββ .The width of the 0 νββ -peak (denoted by ∆ in %) char-acterizes the energy resolution of the detector, and is de- a r X i v : . [ h e p - e x ] J a n fined − a natural choice and also following convention inthe literature for clarity − as the ratio of full-width-half-maximum (FWHM, denoted by w / ) to the total mea-sureable energy Q ββ , such that w / =(∆ · Q ββ ).Various beyond-standard-model processes invokinglepton-number-violation can give rise to 0 νββ [6]. Inthe case of the “mass mechanism” where 0 νββ is drivenby the Majorana neutrino mass, the 0 νββ half-life ( T ν / )can be expressed by [2, 7] (cid:20) T ν / (cid:21) = G ν g A | M ν | (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) m ββ (cid:105) m e (cid:12)(cid:12)(cid:12)(cid:12) (2)where m e is the electron mass, g A is the effective axialvector coupling [8], G ν is a known phase space factor [9]due to kinematics, | M ν | is the nuclear physics matrixelement [10], while (cid:104) m ββ (cid:105) is the effective Majorana neu-trino mass term which depends on neutrino masses ( m i for eigenstate ν i ) and mixings ( U ei for the component of ν i in ν e ): (cid:104) m ββ (cid:105) = | U e m + U e m e iα + U e m e iβ | (3)where α and β are the Majorana phases.The measureable half-life T ν / from an experimentwhich observes N νobs -counts of 0 νββ -events in time t DAQ in a “Region-of-Interest” (RoI) at an efficiency of ε RoI can be expressed as T ν / = ln 2 · N ( A ββ ) · t DAQ · (cid:20) ε RoI N νobs (cid:21) (4)where N ( A ββ ) is the number of A ββ atoms being probed.For simplicity in discussions and to allow the resultsbe easily convertible to different configurations − whilecapturing the essence of the physics, results in this articleare derived in the special “ideal” case where the targetis made up of completely enriched A ββ isotopes. Thatis, the isotopic abundance (IA) is 100%. In additional,the various experimental efficiency factors are all unity( ε expt =100%). Accordingly, Eq. 4 becomes T ν / = ln 2 · (cid:20) N A M ( A ββ ) (cid:21) · Σ · (cid:20) ε RoI N νobs (cid:21) (5)where N A is the Avogadro Number, M ( A ββ ) is the mo-lar 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 theseparameter choices and other assumptions will be dis-cussed in Section II D where conversion relations to thosefor realistic experiments are given.The expression of Eq. 5 applies to experiments withcounting analysis. More sophisticated statistical meth-ods are usually adopted to extract full information froma given data set. These typically exploit the energy spec-tral shapes, which are known for the signal and are pre-dictable with uncertainties for the background. However,in the conceptual-design and sensitivity-projection stage of experiments, the simplified and intuitive approach ofEq. 5 will suffice, especially so in the low count rate Pois-son statistics regime which is of particular interest in thisarticle.Combining the theoretical and experimental descrip-tions of T ν / from, respectively, Eqs. 2&5 gives: | M ν | (cid:2) g A · H ν (cid:3) = 1 (cid:104) m ββ (cid:105) (cid:20) · N νobs ε RoI (cid:21) , where H ν ≡ ln 2 (cid:20) N A M ( A ββ ) · m e (cid:21) G ν (6)is called “specific phase space” in the literature [7]. B. Discovery Potential
In our context, B is expected background countswithin the RoI around Q ββ . This can, in principle, bepredicted with good accuracies prior to the experiments.The sensitivity goals of experiments are typically ex-pressed in the literature [11] as: “Discovery Potential at3 σ with 50% probability” (P σ ) and “upper limits at 90%confidence level” which characterize possible positive andnegative outcomes, respectively. We focus on P σ in thiswork, for the reason that next-generation 0 νββ experi-ments should be designed to have the maximum reach ofdiscovery, rather than setting limits.Poisson statistics is necessary to handle low back-ground and rare signal processes. The dependence ofthe required average signal (S ) versus B under P σ and other discovery potential criteria are depicted in Fig-ure 1a. For a given real and positive B as input and us-ing P σ as illustration, the Poisson distribution P ( i ; µ ) isconstructed with mean µ =B . The observed count N σobs is evaluated as the smallest integer which satisfies N σobs (cid:88) i =0 P ( i ; B ) ≥ (1 − . σ, ∞ ]. This is the minimal observed eventinteger number with ≥ σ significance over a predictedaverage background B . The output S is the minimalsignal strength corresponding to the case where the av-erage total event (B +S )= N σobs with ≥
50% probability.This is evaluated as the minimum value which satisfiesanother Poisson distribution under the condition: N σobs (cid:88) i =0 P ( i ; [B +S ]) ≥ . . (8)It can be inferred from Figure 1a that the“background-free” level with P σ criteria corresponds toa background of B < − and a reference-point signal ofS ≡ S ref =0 . relative to S ref are depicted in Fig-ure 1b. It can be seen that one would require factors of (a) (counts) B - - - -
10 1 10 ( c o un t s ) S - · s P s P s P s P (counts) B - - - - - ( c o un t s ) S - · (b) (counts) B - - - -
10 1 10 - · s P s P s P s P ref S S (counts) B - - - - - - · ref S S FIG. 1: The variations of (a) S and (b) ratios of S toS ref versus B in the discovery potential of ≥ σ, σ with ≥ , σ at 3 σ and 50% isadopted as the criteria in this work. The level of S ref at thebackground-free condition for P σ is represented with a blackdot in (a) σ when B increases from < − to 1(10).While the predicted average background B can becontinuous and real numbers, only integer counts can beobserved in an experiment. This gives rise to the re-lations being inequalities in Eqs. 7&8 and consequentlythe steps in Figures 1a&b. In addition, signal and back-ground events are indistinguishable experimentally. TheP σ criteria is applied to (B +S ) versus B , while theS dependence on B is shown in Figures 1a&b. This isthe origin of the negative slopes in various segments. C. Background Index
The theme of this work is to study the interplay be-tween required exposure and background in 0 νββ exper-iments to meet certain (cid:104) m ββ (cid:105) target sensitivities.In realistic experiments, it is more instructive to char-acterize background with respect to exposure and theRoI energy range, such that the relevant parameter isthe “Background Index” (BI) defined as:BI ≡ B (RoI)Σ (9)which is the background within the RoI (chosen to be ≡ w / , following convention) per 1 ton-year of exposure,with dimension [counts / ( w / -ton-yr)]. Background lev-els expressed in BI are universally applicable to comparesensitivities of varying A ββ in different experiments. D. Conversion to Realistic Configurations
As explained in Section II A, the (BI , Σ) results pre-sented in this article correspond to the ideal case whereIA=100% and ε expt =100%. In addition, while the rangeof g A ∈ [0 . , .
27] is generally considered possible [7, 8], the“unquenched” free nucleon value of g A =1 .
27 is adopted.The required exposure (Σ (cid:48) ) in realistic experimentswould be larger and can be readily converted from theΣ-values viaΣ (cid:48) (cid:39) Σ · · ε expt · W Σ ( g A ) , (10)where W Σ ( g A ) is the weight factor for Σ due to the g A -dependence [10, 12] of T ν / in Eq. 2, relative to the valuesat g A =1.27. It is depicted in Figure 3 for the case of Ge.The finite band width as a function of g A is the conse-quence of the spread in | M ν | predictions [10, 12]. Thespecific case where | M ν | is independent of g A impliesΣ ∝ [ g A ]- and is denoted by the dotted line.The background index defined relative to Σ (cid:48) for realis-tic configurations can accordingly be expressed asBI (cid:48) (Σ (cid:48) ) (cid:39) BI · (cid:20) ΣΣ (cid:48) (cid:21) , (11)such that Σ (cid:48) > Σ and BI (cid:48) < BI. Realistic experiments nat-urally imply larger exposure and more stringent back-ground requirements.
E. Neutrino Physics Connections
Results from neutrino oscillation experiments [1, 3] in-dicate that the m i of the three active ν i have structurescorresponding to either IH or NH. The values of (cid:104) m ββ (cid:105) are constrained and depend on the absolute neutrinomass scale, and are typically expressed in terms of the | n |M -
10 1 10 ) ( / t o n - y r - e V n H A g e V m e V m e V . m e V Ca Nd Xe Zr Cd Sn Te Se Ge Mo Pd FIG. 2: Variations of “specific phase space” g A H ν , as de-fined in Eq. 6, versus | M ν | for various A ββ . The geometricmean of the range of | M ν | is presented as the data points.The best-fit and other diagonal lines correspond to the match-ing (cid:104) m ββ (cid:105) values at g A =1.27 that give rise to a 0 νββ rate of1 event per ton-year at full efficiency. This formulation isadopted from Figures 2&3 of Ref. [7].TABLE I: Summary of the key parameters used in thiswork. Inputs are the IH and NH bands at ± σ of the NDscenario at m min < − eV from existing measurements [1],such that (cid:104) m ββ (cid:105) − < (cid:104) m ββ (cid:105) < (cid:104) m ββ (cid:105) + . The posterior (cid:104) m ββ (cid:105) denotes the 95% lower bound for the (cid:104) m ββ (cid:105) -distribution,taking an uncorrelated ( α, β ) and the uncertainty range in | M ν | as prior [11]. The corresponding minimal-exposuresat background-free levels under criteria P σ are given asΣ min . The reduction fraction in Σ from (cid:104) m ββ (cid:105) − to (cid:104) m ββ (cid:105) is denoted by f . The values of ( (cid:104) m ββ (cid:105) − , (cid:104) m ββ (cid:105) + ) and (cid:104) m ββ (cid:105) define the IH/NH band width and dotted lines, re-spectively, in Figures 4,5&8 in this article. (cid:104) m ββ (cid:105) − (cid:104) m ββ (cid:105) + (cid:104) m ββ (cid:105) f IH: (cid:104) m ββ (cid:105) ( × eV) 14 51 20 − Σ min (ton-yr) 1.7 0.13 0.83 0.49 NH: (cid:104) m ββ (cid:105) ( × eV) 0.78 4.3 3.0 − Σ min (ton-yr) 550 18 37 0.068 lowest mass eigenstates m min . The ± σ ranges of (cid:104) m ββ (cid:105) with the non-degenerate (ND) mass eigenstate scenariosat m min < − eV, denoted by (cid:104) m ββ (cid:105) − and (cid:104) m ββ (cid:105) + , areconstant and listed in Table I.There are no experimental constraints on the Majo-rana phases ( α, β ). It is in principle possible to have ac-cidental cancellation which leads to very small (cid:104) m ββ (cid:105) at m min ∈ [1 , × − eV. However, under the reasonableassumption that they are uncorrelated and have uniform probabilities within [0 , π ], a posterior probability distri-butions of (cid:104) m ββ (cid:105) can be assigned [11]. The 95% lowerlimit, denoted as (cid:104) m ββ (cid:105) and listed in Table I, showsthat the vanishing values of (cid:104) m ββ (cid:105) are disfavored.The current generation of oscillation experiments mayreveal Nature’s choice between the two hierarchy options.In particular, there is an emerging preference of NH overIH [1, 4]. Moreover, the combined cosmology data mayprovide a measurement on the sum of m i [5]. Therefore,it can be expected that the ranges of parameter space of (cid:104) m ββ (cid:105) in 0 νββ searches will be further constrained.Extracting neutrino mass information via Eq. 2 fromthe experimentally measured T ν / requires knowledge of | M ν | and g A . There are different schemes to calculate | M ν | for different A ββ [10]. Deviations among theirresults are the main contributors to the theoretical un-certainties. Another source of uncertainties is the valuesof g A , which may differ between a free nucleon and com-plex nuclei [8].Studies of Ref. [7] suggest that, in the case where 0 νββ is driven by the neutrino mass mechanism, there existsan inverse correlation between G ν and | M ν | in Eq. 2,the consequence of which is that the decay rates per unitmass for different A ββ are similar at given (cid:104) m ββ (cid:105) andconstant g A . That is, there is no favored 0 νββ -isotopefrom the nuclear physics point of view.This empirical observation originates partially to thelarge uncertainties in | M ν | and g A . To derive numeri-cal results which would shed qualitative insights withoutinvolving excessive discussions on the choice of | M ν | ,we assume that this correlation is quantitatively valid.We follow Ref. [7] in adopting the geometric means ofthe realistic ranges for the various | M ν | in different iso-topes. The data points can be parametrized by straightlines at given (cid:104) m ββ (cid:105) , as depicted in Figure 2. That is,[ | M ν | ( g A H ν )] is a constant at fixed (cid:104) m ββ (cid:105) indepen-dent of A ββ . The displayed (cid:104) m ββ (cid:105) values in Figure 2correspond to 0 νββ decay rates of [ N νobs / Σ]=1 / ton-yr at g A =1.27 and full efficiency. The best-fit at this decayrate corresponds to (cid:104) m ββ (cid:105) =(35 × − ) eV.Following Eq. 6, this model leads to a simplifying con-sequence thatΣ(ton-yr) · (cid:20) ε RoI N νobs (cid:21) ∝ (cid:104) m ββ (cid:105) (12)at IA=100% and ε expt =100%, which is universally ap-plicable to all A ββ . The proportional constant can bederived via the best-fit values of Figure 2.Given a background B as input, the required S toestablish signal under P σ can be derived via Figure 1a.This is related to the mean of N νobs at known ε RoI . Neu-trino physics provides constraints on (cid:104) m ββ (cid:105) with severalscales-of-interest given in Table I. The output values ofΣ and BI can be derived with Eqs. 12&9, respectively.The Σ-values thus inferred in what follows could beinterpreted with the typical uncertainties of a “factor oftwo, both directions” (that is, within a factor of [0 . , . A g S W -dependent + uncertainty) A (g | n |M Ge -independent) A (g | n |M FIG. 3: Variation of W Σ as defined in Eq. 10 due to changesin g A relative to that of g A =1.27 in the case of Ge. Thefinite band width is the consequence of the spread in | M ν | predictions [10, 12]. The specific case where | M ν | is inde-pendent of g A such that Σ ∝ [ g A ]- is denoted by the dottedline. of the nominal values) to match our current understand-ing of | M ν | . III. SENSITIVITY DEPENDENCE
It is well-known, following Eqs. 2&5, that the sensitiv-ity to [1 / (cid:104) m ββ (cid:105) ] is proportional to Σ as B → at large B . We further investigate the B -dependencequantitatively and in the context of the preferred IH andNH ranges with the model of Ref. [7]. The specific (cid:104) m ββ (cid:105) -values of Table I − ( (cid:104) m ββ (cid:105) − , (cid:104) m ββ (cid:105) + , (cid:104) m ββ (cid:105) ) for bothIH and NH − serve to provide reference scales. A. Required Exposure and Background
The variations of (cid:104) m ββ (cid:105) versus B with different Σ atRoI= w / (such that ε RoI (cid:39) σ are depicted in Figure 4, with the IH and NH bands su-perimposed. The matching T ν / for Ge is illustrated.The equivalent half-life sensitivities for other isotopes A ββ can be derived via (cid:2) T ν / (cid:3) A ββ = (cid:2) T ν / (cid:3) Ge (cid:18) A ββ (cid:19) . (13)The figure depicts how the same exposure can be usedto probe longer T ν / and smaller (cid:104) m ββ (cid:105) with decreasingbackground.The dependence of (cid:104) m ββ (cid:105) sensitivities to BI is depictedin Figure 5. Taking RoI= w / is obviously not the opti-mal choice when the expected background B →
0. Analternative choice for low B is RoI ≡ w σ covering ± σ (counts) B - - - -
10 1 10 ( e V ) æbb m Æ - - - ( y r ) G e ] / n T [ = 1.0 ton-yr S = 10.0 ton-yr S ton-yr = 10 S ton-yr = 10 S IHNH
FIG. 4: The variation of (cid:104) m ββ (cid:105) with B following P σ ,with RoI= w / for Ge at Σ=(1; 10; 100; 1000) ton-yr follow-ing Eq. 6 at g A =1.27. The IH(NH) bands are defined by( (cid:104) m ββ (cid:105) − , (cid:104) m ββ (cid:105) + ), while their (cid:104) m ββ (cid:105) -values are denotedas dotted lines. The corresponding variations of (cid:2) T ν / (cid:3) Ge ver-sus B are displayed as the right vertical axis. This is specificto Ge, while the equivalent values for other A ββ can bederived via Eq. 13. -ton-yr)] w BI [counts/( - - - - - -
10 1 10 ( e V ) æbb m Æ - - - ton-yr = 10 S ton-yr = 10 S = 10.0 ton-yr S = 1.0 ton-yr S RoI = w s w IH NH
FIG. 5: Sensitivities of (cid:104) m ββ (cid:105) versus BI, as defined in Eq. 9which is universally applicable to all A ββ , following P σ un-der different exposures at Σ=(1; 10; 100; 1000) ton-yr. TheIH(NH) bands are defined by ( (cid:104) m ββ (cid:105) − , (cid:104) m ββ (cid:105) + ), while their (cid:104) m ββ (cid:105) -values are denoted as dotted lines. Both optionsof RoI= w / and w σ are displayed. Black dots correspond tothe benchmark [1 count / ( w / -Σ)] background levels. of Q ββ , such that ε RoI ∼ =100%. Both schemes are illus-trated in Figure 5. The choice of RoI= w σ at B → ε RoI ( w / )=0.76, such that the covered T ν / is 32% longer,or the required Σ is 24% less.The required exposure to probe (cid:104) m ββ (cid:105) and (cid:104) m ββ (cid:105) − with both RoI selections in both IH and NH are depictedin Figure 6a. Superimposed as a blue contour is the TABLE II: Required exposure (Σ) to cover (cid:104) m ββ (cid:105) IH(NH)95% and (cid:104) m ββ (cid:105) IH(NH) − at different background scenarios in descending orderof intensity. The BI-values follow from Eq. 9.Background Required Σ (ton-yr) To CoverScenario BI (cid:104) m ββ (cid:105) IH (cid:104) m ββ (cid:105) IH − (cid:104) m ββ (cid:105) NH (cid:104) m ββ (cid:105) NH − [counts / ( w / -ton-yr)]Best Published [13] 1 27 110 4 . × × Next Generation } . × . × Projected [14]Benchmark IH : { / ( w / -Σ)] 0.067 – 15 – –NH : { . × − – – 330 –2 . × − – – – 4 . × “Background IH : { ≤ (6 . × − ) 0.83 – – –-Free” ≤ (3 . × − ) 0.83 1.7 – –NH : { ≤ (1 . × − ) 0.83 1.7 37 – ≤ (0 . × − ) 0.83 1.7 37 550TABLE III: The range of (S , B ) to qualify a positive signal to cover (cid:104) m ββ (cid:105) − for both IH and NH under P σ , given theobserved number of events in RoI − νββ -signals and background are combined but indistinguishable at event-by-event level.The smaller Σ values among the alternatives of RoI= w / or w σ are selected. The sixth column shows the required BI whichis universal to all A ββ . Last column lists the required background specifically for Ge normalized to “ / (keV-ton-yr)”, andthe conversion to other isotopes is referred to Eq. 14. The N νobs =1 row correpsonds to the background-free conditions. TheBI-values follow from Eq. 9.Counts Within RoI Optimal Required Universal Background / (keV-ton-yr) N νobs S B RoI Exposure BI for Ge atΣ (ton-yr) [counts / ( w / -ton-yr)] ∆ = 0.12%Covering (cid:104) m ββ (cid:105) − for IH:1 ≥ ≤ × − w σ ≤ × − ≤ × − ≥ ≤ × − w σ ≤ × − ≤ × − ≥ ≤ w σ ≤ × − ≤ × − ≥ ≤ w / ≤ × − ≤ × − ≥ ≤ w / ≤ × − ≤ × − ≥ ≤ w / ≤ ≤ × − Covering (cid:104) m ββ (cid:105) − for NH:1 ≥ ≤ × − w σ × ≤ × − ≤ × − ≥ ≤ × − w σ × ≤ × − ≤ × − ≥ ≤ w σ × ≤ × − ≤ × − ≥ ≤ w / × ≤ × − ≤ × − ≥ ≤ w / × ≤ × − ≤ × − ≥ ≤ w / × ≤ × − ≤ × − “benchmark” background level at 1 count / ( w / -Σ) wherethe first background event would occur at a given expo-sure. The benchmark level also represents the transitionin the effectiveness of probing (cid:104) m ββ (cid:105) with increasing ex-posure. The shaded regions correspond to the preferredhardware specification space for future 0 νββ experiments − where the exposure should be sufficient to cover atleast (cid:104) m ββ (cid:105) IH(NH)95% , and there would be less than one back-ground event per w / over the full exposure.The required exposures under various background con- ditions are summarized in Table II. The best pub-lished background level is 1 . +0 . − . counts / (keV-ton-yr)or BI ∼ / ( w / -ton-yr) from the GERDA experi-ment on Ge [13]. For simplicity, the “best” currentbackground is taken to be BI ≡ BI =1 count / ( w / -ton-yr)in what follows. This background would corre-spond to Σ IH(NH)ref =110 ton-yr(11 Mton-yr) to cover (cid:104) m ββ (cid:105) IH − ( (cid:104) m ββ (cid:105) NH − ).Such a large required exposure is inefficient and un-realistic, so that the background should be signifi- (a) -ton-yr)] w BI [counts/( - - - - - - -
10 1 10 (t o n - y r ) S -
10 110 - NH æ bb m Æ æ bb m Æ - IH æ bb m Æ æ bb m Æ I H N H s w RoI = w RoI =
Benchmark )] S - w [1 count/( { (b) BI/BI - - - - - - -
10 1 - - - - - -
10 1 - NH æ bb m Æ æ bb m Æ - IH æ bb m Æ æ bb m Æ IHNH ref
S S
FIG. 6: (a) Required exposure universally applicable to all A ββ to cover (cid:104) m ββ (cid:105) and (cid:104) m ββ (cid:105) − under P σ at IH andNH versus BI. The benchmark background [1 count / ( w / -Σ)]condition is superimposed as the blue contour. The shadedregions correspond to the preferred hardware specificationspace for future 0 νββ experiments. (b) The relation betweenexposure and background reduction relative to the currentBI =1 count / ( w / -ton-yr) and its corresponding required ex-posures Σ IH(NH)ref =110 ton-yr(11 Mton-yr) for IH(NH). Thereference points (BI , Σ IH(NH)ref ) are represented by black dotsin (a). Sensitivities with both RoI= w / and w σ are displayedin (a), while the more sensitive of the two schemes are shownin (b). The shaded regions match those of (a). cantly reduced to allow the quest to advance. Thetarget exposure is Σ=10 ton-yr for the next gener-ation 0 νββ projects to cover IH with ton-scale de-tector target [14]. Following Figure 5, this expo-sure would require BI < (0 . , . / ( w / -ton-yr)to cover ( (cid:104) m ββ (cid:105) , (cid:104) m ββ (cid:105) IH − ). This matches the back-ground specifications of BI= O (0 .
1) counts / ( w / -ton-yr).The background-free (BI min ) − equivalently, minimal-exposure (Σ min ) − condition is where one single observed (eV) æ bb m Æ - - - -t o n - y r )] / w [ c o un t s / ( m i n B I - - - - - - - (t o n - y r ) m i n S - - NH IH
FIG. 7: The variations of the background-free and minimal-exposure conditions (BI min , Σ min ) at P σ -criteria with (cid:104) m ββ (cid:105) .The IH(NH) bands are defined by ( (cid:104) m ββ (cid:105) − , (cid:104) m ββ (cid:105) + ), whiletheir (cid:104) m ββ (cid:105) -values are denoted as dotted lines. The blackdots correspond to the values listed in the last rows of Table II. event can establish the signal at the P σ -criteria. Theirvalues at the benchmark (cid:104) m ββ (cid:105) ’s are given in Table I.The choice of (cid:104) m ββ (cid:105) − to define Σ is a conservative one.Since (cid:104) m ββ (cid:105) > (cid:104) m ββ (cid:105) − from Table I, the minimum ex-posure Σ min corresponding to (cid:104) m ββ (cid:105) is reduced rela-tive to that for (cid:104) m ββ (cid:105) − by a fraction given as f .The variations of (BI min , Σ min ) with (cid:104) m ββ (cid:105) are de-picted in Figure 7. As shown by the black dotsand also listed in Table II, Σ min =(0.83,1.7) ton-yr atBI min ≤ (6 . × − , . × − ) counts / ( w / -ton-yr) are re-quired to cover ( (cid:104) m ββ (cid:105) , (cid:104) m ββ (cid:105) − ) IH . The correspond-ing requirements for NH are Σ min =(37,550) ton-yr atBI min ≤ (1 . × − , . × − ) counts / ( w / -ton-yr). Therequired Σ min from (cid:104) m ββ (cid:105) − to (cid:104) m ββ (cid:105) is reduced by f =0.49(0.068) for IH(NH).Alternatively, the Σ=10 ton-yr targetexposure of next-generation projects canprobe (cid:104) m ββ (cid:105) > (5 . × − ) eV, approaching (cid:104) m ββ (cid:105) NH+ =(4 . × − ) eV, when the background-free condition BI min < . × − counts / ( w / -ton-yr) isachieved.The interplay between fractional reduction of BI andΣ relative to BI and Σ IH(NH)ref to cover (cid:104) m ββ (cid:105) and (cid:104) m ββ (cid:105) − in IH(NH) is depicted in Figure 6b. Background-free conditions require additional BI-suppression by fac-tors of 3 . × − (0 . × − ), to cover (cid:104) m ββ (cid:105) IH(NH) − inwhich cases Σ can be reduced by factors of 0.016(5 × − ). The shaded regions match those of Figure 6a indisplaying the preferred hardware specification space.The impact of background suppression to the requiredexposure is increasingly enhanced as smaller values of (cid:104) m ββ (cid:105) are probed. This is illustrated in Figures 8a(b)which display the reduction fraction in Σ relative toΣ IH(NH)ref at different background levels. For instance, the (a) - - · - · - - - -
10 1 IH B I = . = . - = - = - = refIH S S (eV) æ bb m Æ (b) - · - - · - - - - - - -
10 110 NH B I = . = . - = - = - = refNH S S (eV) æ bb m Æ FIG. 8: Fractional reduction of required exposure to probe(a) IH and (b) NH as a function of (cid:104) m ββ (cid:105) under differ-ent BI background levels in unit of [counts / ( w / -ton-yr)].The IH(NH) bands are defined by ( (cid:104) m ββ (cid:105) − , (cid:104) m ββ (cid:105) + ),while their (cid:104) m ββ (cid:105) -values are denoted as dotted lines.The reference exposures Σ IH(NH)ref =110 ton-yr(11 Mton-yr)to cover (cid:104) m ββ (cid:105) IH(NH) − , denoted as black dots, correspondto those required under the current background level ofBI =1 count / ( w / -ton-yr). suppression of BI from 1 to 10 − counts / ( w / -ton-yr) willcontribute to the reduction of Σ from Σ=(27 , , (cid:104) m ββ (cid:105) , (cid:104) m ββ (cid:105) − ) IH and( (cid:104) m ββ (cid:105) , (cid:104) m ββ (cid:105) − ) NH , respectively.In realistic experiments, signals and background areindistinguishable at the event-by-event level. The ex-pected average background B and the observed eventcounts (an integer) in the RoI are the known quantities.They can be used to assess whether a signal is “estab-lished” under certain criteria like P σ . Listed in Table IIIare the required ranges on (S , B ) to qualify positive sig-nals given the number of observed events. The first row (eV) æ bb m Æ - - - ( % ) D - - · - · IHNH B e n c h m a r k m i n B I Xe Ge Te Se Ca Cd Zr Nd Mo (eV) æ bb m Æ - · ( % ) D Xe Ge Te Se Ca Cd Zr Nd Mo FIG. 9: Variations of the required ∆ with (cid:104) m ββ (cid:105) such that2 νββ background within RoI= w σ would contribute less thanthose specified by the benchmark [1 count / ( w / -Σ)] level andthe background-free (BI min ) condition of Figure 7 as a func-tion of (cid:104) m ββ (cid:105) . The relative locations of A ββ within bothbands are depicted in the inset, using the BI min -band as il-lustration. The best achieved ∆’s in past and ongoing exper-iments are displayed at the right vertical axis. corresponds to the background-free condition, in whichone single event is sufficient to establish a signal. Ac-cordingly, the (BI , Σ) values match the entries in the lastrows of Table II, and are displayed in Figure 7.Results of Table III, apply generically to all A ββ exceptthose for the last column when background is expressedin “ / (keV-ton-yr)” unit. The values are specific for Ge,where the best published ∆( Ge)=0.12% of the MJD-experiment [15] is adopted as input. The backgroundrequirements for other A ββ can be derived via:[Background / (keV-ton-yr)]( A ββ )[Background / (keV-ton-yr)]( Ge)= (cid:20) ∆( Ge)∆( A ββ ) (cid:21) (cid:20) Q ββ ( Ge) Q ββ ( A ββ ) (cid:21) . (14) B. Limiting Irreducible Background
It is instructive and important to quantify the interplaybetween various irreducible background channels to therequired exposure. In particular, one such irreduciblebackground is the Standard Model-allowed 2 νββ NZ A ββ → N − Z +2 A + 2 e − + 2¯ ν e . (15)The contamination levels to 0 νββ at the Q ββ -associatedRoI depend on its half-life ( T ν / ) and the detector resolu-tion. A worse resolution (larger ∆) implies a larger RoIrange to search for 0 νββ signals, and therefore a higherprobability of having background events from the 2 νββ spectral tail. TABLE IV: The required ∆ for selected A ββ , listed in descending order of their measured T ν / [2, 13], such that the 2 νββ background within RoI= w σ would contribute less than the levels specified by the benchmark and background-free conditionsto cover (cid:104) m ββ (cid:105) IH(NH)95% and (cid:104) m ββ (cid:105) IH(NH) − . The BI-values follow from Eq. 9.Inverted Hierarachy Normal HierarachyBI [counts / ( w / -ton-yr)] BI [counts / ( w / -ton-yr)]Sensitivity: Benchmark Background-Free Benchmark Background-Free ≤ (cid:104) m ββ (cid:105) ≤ . − ≤ . × − − ≤ . × − − ≤ . × − −≤ (cid:104) m ββ (cid:105) − − ≤ . × − − ≤ . × − − ≤ . × − − ≤ . × − A ββ Q ββ T ν / Best † (MeV) (yr) ∆(%) Required ∆ (%) Xe 2.458 2 . × ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Ge 2.039 1 . × ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Te 2.528 8 . × ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Se 2.998 9 . × ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Ca 4.268 6 . × ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Cd 2.814 2 . × ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Zr 3.350 2 . × ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Nd 3.371 9 . × ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ Mo 3.034 6 . × ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ † Best achieved w / resolution at Q ββ from past and ongoing 0 νββ experiments [2, 15, 16],not including detector R&D programs and future projects. Depicted in Figure 9 are variations of the required ∆with (cid:104) m ββ (cid:105) such that 2 νββ background within RoI= w σ would contribute less than the BI-values specified by thebenchmark and background-free conditions. The finitewidth of the band is a consequence of the spread of mea-sured T ν / [2, 13]. Faster 2 νββ rates typically require bet-ter detector resolution to define smaller RoI. The relativelocations for different A ββ within the bands are depictedin the inset.Listed in Table IV are the required ranges of ∆ tocover (cid:104) m ββ (cid:105) IH(NH)95% and (cid:104) m ββ (cid:105) IH(NH) − . In particular, therequired resolutions to cover (cid:104) m ββ (cid:105) − for IH and NHunder background-free conditions are ∆ ≤ (0 . − . ≤ (0 . − . νββ experiments [2, 15, 16]are included in Table IV and depicted in the rightvertical axis of Figure 9 for comparison. In partic-ular, the best published ∆( Ge)=0.12% [15] corre-sponds to an irreducible 2 νββ background contribu-tion of BI < × − counts / ( w / -ton-yr). This pro-vides a comfortable margin relative to that which sat-isfies the background-free conditions for (cid:104) m ββ (cid:105) NH − atBI ≤ . × − counts / ( w / -ton-yr). IV. SUMMARY AND PROSPECTS
As current neutrino oscillation experiments reveal apreference of NH, the strategy of scaling the summit of 0 νββ should take this genuine possibility into account.This work studies the relation between the two mainfactors in improving experimental sensitivities: (BI , Σ).We recall that the presented results are derived withcertain input parameter choice: IA=100%, ε expt =100%and g A =1.27, and that 0 νββ is driven by the Majorananeutrino mass terms via the mass mechanism while theSignal-to-Background analysis is based on counting ex-periments without exploiting the spectral shape infor-mation at this stage.Advancing towards ND-NH to cover (cid:104) m ββ (cid:105) NH − willrequire large and costly exposure. An unrealis-tic O (10) Mton-yr enriched target mass is neces-sary at the current best achieved background levelBI ∼ / ( w / -ton-yr). Reduction of BI will be play-ing increasingly significant, if not determining, roles inshaping future 0 νββ projects.For instance, following Table II, background-free conditions for (cid:104) m ββ (cid:105) NH − correspond to addi-tional background suppression from the currentbest BI ∼ / ( w / -ton-yr) and benchmark[1 count / ( w / -Σ)] levels by factors of (0 . × − )and (4 . × − ), respectively. This would reduce the re-quired Σ from 11 Mton-yr and 4600 ton-yr, respectively,to 550 ton-yr. The corresponding minimal-exposureto cover (cid:104) m ββ (cid:105) NH95% is Σ min ∼
37 ton-yr, which is onlya modest factor beyond the goals of next-generationexperiments [14]. The pursuit of background towardsBI ∼O (10 − ) counts / ( w / -ton-yr) to probe ND-NH,while challenging, is highly investment-effective, as it0is equivalent to reduction of Σ by O (10) Mton-yr and O (1) kton-yr relative to those required for the currentbest and benchmark background levels, respectively.This article serves to quantify the merits of backgroundreduction in 0 νββ experiments, but does not attempt toaddress the experimental issues on how to realize the feat and how to demonstrate that the suppression factors areachieved when experiments are constructed. We projectthat the continuous intense efforts and ingenuities fromthe experimentalists world-wide, with motivations rein-forced by the increasing equivalent “market” values, willbe able to meet the challenges.Boosting Σ involves mostly in the accumulation of en-riched A ββ isotopes and turning these into operating de-tectors. These processes are confined to relatively fewlocations and small communities of expertise. The roomof development which may overcome the known hur-dles is limited. Suppression of the 0 νββ experimentalbackground, on the other hand, would be the tasks ofmobilizing and coordinating the efforts of a large poolof expertise. It is related to the advances in diversedisciplines from novel materials to chemistry processingto trace measurement techniques. Research programson many subjects requiring low-background techniquesmay contribute to − and benefit from − the advances.There would be strong potentials of technological break-throughs and innovative ideas as the sensitivity goals arepursued.Signal efficiencies are also increasingly costly as sen-sitivities advance towards ND-NH. For instance, atΣ min =550 ton-yr to cover (cid:104) m ββ (cid:105) NH − , a high 90% efficiencyto certain selection criterion corresponds to discarding data of O (10) ton-yr strength − already an order of mag-nitude larger than the combined exposure of all 0 νββ ex-periments. It follows that background suppression wouldpreferably be attended at the root level − that radioac-tive contaminations are suppressed to start with, ratherthan relying on special signatures and software selectionalgorithms to identify them.The next generation of 0 νββ experiments would cover (cid:104) m ββ (cid:105) IH − . In addition, they should be able to explorethe strategies and demonstrate sufficient margins to ad-vance towards (cid:104) m ββ (cid:105) NH − . A significant merit would beto have no irreducible background before reaching theBI ∼O (10 − ) counts / ( w / -ton-yr) background-free con-figuration. The detector requirements to achieve this for2 νββ are summarized in Figure 9 and Table IV. Detailedstudies of this background as well as other channels likethose due to residual cosmogenic radioactivity and long-lived radioactive isotopes are themes of our on-going re-search efforts. V. ACKNOWLEDGEMENT
This work is supported by the Academia SinicaPrincipal Investigator Award AS-IA-106-M02, contracts104-2112-M-259-004-MY3 and 107-2119-M-001-028-MY3from the Ministry of Science and Technology, Taiwan,and 2017-ECP2 from the National Center of TheoreticalSciences, Taiwan. [1] M. Tanabashi et al., Particle Data Group, Phys. Rev.
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