A Precise Calculation of Delayed Coincidence Selection Efficiency and Accidental Coincidence Rate
aa r X i v : . [ phy s i c s . i n s - d e t ] N ov Submitted to ’Chinese Physics C’
A Precise Calculation of Delayed Coincidence Selection Efficiency and Accidental Coincidence Rate
Jingyi Yu, Zhe Wang ∗ , and Shaomin Chen Center for High Energy Physics,Department of Engineering Physics,Tsinghua University, Beijing, China (Dated: June 5, 2018)A model is proposed to address issues on the precise background evaluation due to the complex data structuredefined by the delayed coincidence method, which is widely used in reactor electron-antineutrino oscillationexperiments. In this model, the effects from the muon veto, uncorrelated random background, coincident signaland background are all studied with the analytical solutions, simplifying the estimation of the systematic uncer-tainties of signal efficiency and accidental background rate determined by the unstable single rate. The result ofcalculation is validated numerically with a number of simulation studies and is also applied and validated in therecent Daya Bay hydrogen-capture based oscillation measurement.
PACS numbers: 29.85.Fj, 14.60.Pq, 29.40.McKeywords: delayed coincidence; accidental coincidence; inverse beta decay; analytical model
I. INTRODUCTION
The delayed coincidence method is used very broadly innuclear and high energy physics experiments. The existenceof a delayed signal greatly relaxes the critical requirement onthe random background level. Recently three reactor electron-antineutrino experiments Double CHOOZ [1–3], RENO [4]and Daya Bay [5–8] adopted this established technique toprecisely measure the neutrino mixing angle θ [9, 10].Electron-antineutrinos from reactors were distinguished bydetecting the coincidence of the prompt positron and delayedneutron capture signals of inverse-beta-decay IBD interac-tions, ¯ ν e + p → e + + n . The expected high precision ofthe sin θ measurements required a better understandingof the acceptance of delayed-coincidence signals and theirbackground contamination, especially for the situation whenthe backgrounds are much higher if using the neutron cap-ture signals on hydrogen (nH) [3, 8], instead of the capture ongadolinium (nGd).This article describes a complete mathematical model withanalytical solutions for these neutrino experiments, which ad-dresses realistic muon veto, high accidental background andvarying single rate. They were not completely discussed bythe formula with the neutron-like signal rate, Eq. 5 of [2] orthe off-window method [3]. It improves the understanding ofdelayed-coincidence signals and backgrounds, and was par-ticularly applied to one recent Daya Bay study [8]. It is alsouseful to other on-going and future reactor neutrino experi-ments to study the neutrino mass hierarchy [11–14].These three experiments have a very similar design. Theantineutrino detectors all have a three-layer structure, wherethe central region is filled with gadolinium-doped liquid scin-tillator GdLS, the middle region with pure liquid scintillatorLS, and the outmost layer with mineral oil. The IBD sig- ∗ Correspondence: [email protected] nals are categorized according to neutron capture nuclei. Inthe GdLS region, ∼
84% of neutrons capture on Gd and re-lease several gammas with a total energy of ∼ ∼ µs . Inthe LS region, most neutrons capture on hydrogen and re-lease a 2.2 MeV gamma, but the average capture time is muchlonger and is ∼ µs . Two types of dedicated analyses arein progress. The nGd analyses have the best precision whilethe nH analyses make a non-trivial contribution.In these analyses, the full data-taking time is chopped intofragments by cosmic muons. Take the Daya Bay nH analysisas an example for explanation. The total muon rate at the nearsite was ∼
200 Hz. Different veto windows were applied aftereach muon, i.e. µs or 400 µs depending on the energydeposition of the muon track and its distance to the detector.A longer veto window is needed when the energy depositionof a muon is larger, because more long-lived spallation back-grounds may be found immediately after it [15]. The totalvetoed time took ∼
20% at the near site. It is useful to under-stand the distribution for the time fragment durations betweenthese muon vetoes.The background situation is complicated and particularlyvery high for the nH analyses. For the Daya Bay case, fourmajor backgrounds were identified, which included acciden-tal coincidence background, spallation Li/ He and fast neu-tron backgrounds induced by cosmic-ray muons, and
Am- C calibration source background. The goal of the precisionof sin θ is less than one percent, however, the acciden-tal background statistics is comparable to the IBD statisticsfor both the Double Chooz and Daya Bay far-site detectors inthe nH analyses. Can these backgrounds be described by acomplete mathematical model along the time axis? Can thesystematic uncertainty of the accidental background be con-trolled to sub-percent?Besides these concerns, other issues from the random back-ground rate also deserve attention in the nH analyses. The ran-dom background was decreasing slowly in the Daya Bay case,since it is caused by residual radioactivity, which decreasedafter the antineutrino detectors were immersed in water. Aftermuons, the random rate also increase quickly, because muonsmay introduce a lot of spallation backgrounds. II. DATA MODEL AND SIMPLIFICATIONA. Data model
A model was built up to address issues on the precise back-ground evaluation due to the complex data structure. In thefollowing we use e and n to represent the positron and neutronfrom an IBD reaction respectively, s a random background (orsingle) and µ for a muon. The features of each type of eventare described below.1) The event of central importance is the delayed-coincidence IBD event, including a prompt signal e and a de-layed signal n . The event rate of IBD’s is assumed to be R IBD .The time between the prompt and delay signals can be com-plicated. For the three neutrino experiments mentioned above,the neutron capture cross section varies with its momentum,so that an exponential distribution is a poor approximation forthe neutron capture time. An abstract form P delay ( t ) is usedto represent the delayed signals’ time distribution.All Li/ He, fast neutron and
Am- C backgrounds havea correlated delayed signal caused by a real neutron. Forthe Li/ He background, neutrons are emitted from the beta-delayed neutron decay of Li or He. the fast neutron back-ground’s prompt signal is the recoil of the neutron. Similarly,the
Am- C background’s prompt signal is also from therecoil of the neutron from the source. In terms of delayed-coincidence selection, they are not distinguishable from theIBD events, so that they are all categorized as coincidentevents and presented as IBDs in later discussion. Details onhow to identify these coincident backgrounds with energy in-formation, etc. can be found elsewhere [1–8, 15] and is be-yond the scope of this paper.2) Random signals, or singles, include decays from residualradioactive nuclei in the detector and from the environment,or other non-correlated detector noise. Random signals oc-cur with a uniform distribution, i.e. the time interval betweentwo random signals follows an exponential distribution withan average value of /R s , where R s is the single rate.3) With the inclusion of cosmic ray muons, all elements arecovered with the model.These different types of signals in the full data-taking timeaxis are shown in Fig. 1, as well as one type of delayed co-incidence searching method (multiplicity two selection). Firstcome two muon events, A and B, on the full-time axis. MuonA is supposed to be closer to the sensitive region of the detec-tor than muon B, and so, a longer veto window is applied tomuon A. One pair of delayed coincidence signals (a positronand a neutron) occurs between muon A and B. A fixed-lengthcoincidence window T c is opened after the positron which isa possible prompt signal candidate, since it is not vetoed byany muons or occupied by any other coincidence-searchingwindows. Window T c is usually comparable to the average arrival time of the delayed signal, for example 400 µs for nHanalyses. For a delayed signal to not fall into a muon vetowindow, the time between a prompt signal and the next muonevent cannot be smaller than T c . The total dead time intro-duced by a muon is the veto time plus T c for prompt candi-dates. After muon B, a single event occurs, since no delayedsignal is found within T c . Finally two muons, C and D, arevery close to each other in time, so that their veto windowsoverlap. In this example all coincident signals and singles arewell separated. The situations with overlaps are discussed inthe following sections. B. Simplification
The first attempt of simplification is to project all signalsfrom the full-time axis to the live-time axis. As illustrated inFig. 1, a muon with its dead time is contracted as a point onthe live-time axis and counted as one net muon, like muon Aand B. Muons C and D have overlapping dead times, so theyare contracted together as one point on the live-time axis andcounted as only one net muon. All other signals are simplymoved to the live-time axis. Since every muon’s dead time isalready removed from the live-time axis, the full length of thelive-time axis is the total time of the prompt signal searching.The important assumption making this analytic calculationpossible is to postulate that the net muons are uniformly dis-tributed on the live-time axis and the time interval follows anexponential distribution with an average value of /R µ where R µ is the net muon rate on the live-time axis.In the calculation below, we always have R IBD ≪ R s . Forthe three reactor neutrino experiments mentioned, R s is aboutfour orders of magnitude higher than R IBD . III. CALCULATION METHODA. Delayed coincidence events and other combinations
Besides the necessary predictions of the event rates ofdelayed-coincidence signals and the accidental background,other types of combinations may also be of interest. They aregrouped according their multiplicity.One-fold coincident events: s , e , n .Two-fold coincident events: ss , se , sn , en , es , ns .Three-fold coincident events: sss , sse , ses , sns , ssn , sen , ens , esn , ess , nss . B. Two-step calculation
The calculation of these event rates is divided into twosteps: a) determine the probability of a type of signal to starta coincidence searching window; b) determine the probabilitythat there is a second or third signal in the searching windowfor two-fold or three-fold coincidences or that there is no othersignal for one-fold events. In the following sections, the start-ing probabilities of a single background ( P s − start ), positron FIG. 1: (Color online) On the top is a full-time axis, and from the left to right are muon A, a pair of delayed-coincidence events (positron andneutron), muon B, random single, and muon C and D. A fixed-length coincidence window T c is opened for each possible candidate, i.e. thepositron and the single. The total dead time for prompt signals is T c plus the muon veto window (the shaded area). In this example, muon Ahas a longer veto window than B, and muon C and D’s veto windows overlap. On the bottom is a live-time axis where the dead time of eachmuon is contracted as a single point and all other events are unchanged. More details can be found in Section II. ( P e − start ) and neutron ( P n − start ) will be calculated first, fol-lowed by the rates of all kinds of combinations. C. Starting probability
On the live-time axis, each signal except a muon can start acoincident searching window, as long as it is not in the previ-ous coincidence searching window. Note that on the live-timeaxis, R IBD and R s are exactly the same as on the full-timeaxis. A single event may start a searching window in differentsituations. In the formulas below, t µ is the time to its previousnet muon event.Case a) As shown in Fig. 2 a, when t µ < T c , if there is noother signal between the single event and the muon, a search-ing window will be started by the single event. The probabilityof this situation is P a = Z T c P (0 | R s t µ ) · P ( t µ ) · dt µ = Z T c ( R s t µ ) k k ! e − R s t µ | k =0 · R µ e − R µ t µ · dt µ = R µ R s + R µ [1 − e − ( R s + R µ ) T c ] , (1)where P (0 | R s t µ ) is the probability that there is no other sin-gle event in between, which is calculated as a Poisson distri-bution with a mean of R s t µ and count k = 0 , and the sec-ond term P ( t µ ) gives the probability of finding a muon at t µ before the target single event being considered, which is cal-culated according to an exponential distribution with a rate of R µ .Case b) When t µ > T c , if there is no other signal within T c before the target single event, a searching window will bestarted as the panel b of Fig. 2 shows. The starting probability FIG. 2: (Color online) Three situations of how a target ‘Single’ eventstarts a searching window. For case a, that the time to previous muon t µ is less than the coincidence window T c , and no event occur within t µ ensures that the target single is not within any muon veto or pre-vious coincidence windows. In case b, t µ is longer than T c , and asearching window can start as long as it is not occupied by a pre-vious coincidence window, i.e. nothing occurs within T c . Case cshows an extension based on b, the prior single ‘S1’ is very close,but is within its previous searching window opened by ‘S2’. See thetext in Sec. III for details. of this situation is P b = P (0 | R s T c ) · Z ∞ T c P ( t µ ) · dt µ = ( R s T c ) k k ! e − R s T c | k =0 · Z ∞ T c R µ e − R µ t µ · dt µ = e − ( R s + R µ ) T c , (2)where the first term corresponds to no signals in T c and thesecond term is to ensure t µ > T c .Case c) is an extension of b). When t µ > T c , there is an-other scenario that the target single can start a searching win-dow as depicted in panel c of Fig. 2. There is a random signal, s , before the target single event, however it occurs within aprevious searching window started by s . The probability ofthis situation is P c = Z T c dtR s e − R s t · [1 − P (0 | R s t )] · Z ∞ T c + t P ( t µ ) · dt µ = R s R s + R µ e − R µ T c [1 − e − ( R s + R µ ) T c ] − R s R s + R µ e − R µ T c [1 − e − (2 R s + R µ ) T c ] , (3)where the first integral gives the probability to find the firstrandom background signal s at time t before the target singleevent, the second term calculates the probability of having atleast one random background s in the early t window, that is T c away from the target single event, and the last integral justgives the probability of finding a muon at some time largerthan T c + t .Finally, the starting probability is P s − start = P a + P b + P c . (4)There should be higher order corrections after P c , but, withthe example parameters used in the simulation presented later,they are estimated to be five orders of magnitude smaller thanthese leading terms.The starting rate is R s − start = R s · P s − start . (5)For the prompt signals of IBD events, the starting proba-bility is different from P s − start by only a positron detectionefficiency ε e : P e − start = P s − start · ε e . (6)Then the rate of searching windows started with positrons is R e − start = R IBD · P s − start · ε e . (7)The situation of a neutron starting a searching window is abit complicated, because by nature it is always possible thata prompt positron signal is ahead of it. It only happens whenthe positron is failed to be detected. The method to calculate P n − start is the same as above and is not shown. The eventrate with neutrons as a start is expressed as R n − start = R IBD · P n − start . (8) D. Construct an event
After the starting probability and the starting rate of a typeof signal are known, the rate of accidental background, de-tectable IBD pairs and other cases can be calculated. The accidental background rate R ( ss ) is just the rate of onesingle to start a searching window multiplied by the probabil-ity of a second single event appearing in the same window R ( ss ) = R s − start · P (1 | R s T c ) , (9)where P (1 | R s T c ) is the Poisson probability of one count witha mean of R s T c . IBD events are not explicitly required to beexcluded from this because R IBD ≪ R s .The detectable IBD event rate R ( en ) can be obtained in asimilar way: R ( en ) = R e − start · ε n | e · Z T c P delay ( t ) dt · P (0 | R s T c )= R IBD · P s − start · ε e · ε n | e · Z T c P delay ( t ) dt · P (0 | R s T c ) , (10)where ε n | e is the efficiency of neutron detection after apositron is detected, the integral is the time cut efficiency, andthe random background is explicitly required to be outside thissearching window in the last term. The complete IBD eventdetection efficiency can be expressed as: ε IBD = R ( en ) R IBD . (11)It can be found that ε e , ε n | e , and time cut related efficiency P delay ( t ) integral can all be factored out and studied sep-arately. The details of P delay ( t ) will not affect the selec-tion. The result also applies to other non-IBD correlated back-grounds. The two-fold selection efficiencies of them are thesame to IBD events. The results of other combinations aresimilar and not shown. IV. VERIFICATION WITH MONTE CARLO
Since the muon veto cut, etc. were quite complicated, highstatistics ( × events) Monte Carlo simulation studieswere done to verify the predictions. Three types of eventswere produced: IBD, single, and muon. They were generatedon the full-time axis according to three uniform distributions.Several sets of parameters were tried according to the real caseas in [8]. The muon rate on the full-time axis was set to thehighest muon rate of 200 Hz as in the Daya Bay near site,since any muon rate lower than this would have a less signif-icant effect if the model or simplification failed. Usually aveto time of 400 µs was applied for each muon, but 0.05%of them can be shower muons, for which a one-second longveto was applied. The shower muon fraction 0.05% is close tothe real case, and the shower muon dead time takes up ∼ P delay ( t ) was rep-resented with a simple exponential distribution with rate λ .The detection efficiencies of the prompt and delayed signalswere also included. One set of parameters used is summarizedin Table I. R s Muon rate Veto time Shower fraction Shower veto R IBD ǫ e ǫ n | e T c /λ
50 Hz 200 Hz 400 µ s 0.05% 1 s 0.1 Hz 1 0.8 400 µ s 200 µ sTABLE I: Monte Carlo simulation parameters. From left to right, they are single rate, muon rate, muon veto window, shower muon fraction,shower veto window, IBD rate, prompt signal detection efficiency, conditional detection efficiency of neutron, coincidence window and averageneutron capture time. Rate [Hz] R ( s ) R ( ss ) R ( se ) R ( sn ) R ( sss ) R ( ens ) R ( sse ) R ( sns ) + R ( esn ) + R ( ses ) + R ( ssn ) Mea. 48.0833 0.96165 0.0010460 0.0001108 0.009626 0.001342 2.10E-5 2.27E-6Sta. Err. 0.0015 0.00022 7.2E-6 2.3E-6 2.2E-5 1.1E-5 1.4E-6 4.7E-7Pred. 48.0856 0.96171 0.0010499 0.0001093 0.009617 0.001330 2.10E-5 2.19E-6Diff. -1.5 -0.27 -0.54 0.65 0.41 1.1 0 0.17Rate [Hz] R ( e ) R ( en ) R ( es ) R ( sen ) R ( ess ) R ( n ) R ( ns ) R ( nss ) Mea. 0.029657 0.066447 0.0005991 0.0008715 6.91E-6 0.012719 0.0002616 3.11E-6Sta. Err. 0.000038 0.000057 5.4E-6 6.6E-6 5.8E-7 2.5E-5 3.6E-6 3.9E-7Pred. 0.029647 0.066525 0.0005930 0.0008735 5.93E-6 0.012770 0.0002554 2.55E-6Diff. 0.26 -1.4 1.1 -0.30 1.7 -2.0 1.7 1.4TABLE II: Measured event rates of simulation sample and their predictions for all one-, two-, and three-fold cases. For each type of combina-tion, Mea. gives the measured result, Sta. Err. is its statistical error, Pred. is the predicted value, and Diff. is (Mea.-Pred.)/Err. The situation ens is not distinguished with esn in the analysis, as well as sse with ses , and sns with ssn . The measurements with the simulated sample and the pre-dictions are listed in Table II, where the discrepancies are allwithin a 3- σ range. It was found that the net muon we definedstill follows Poisson statistics on the live-time axis and the netmuon rate on the live-time axis and the real muon rate on thefull-time axis are the same within the statistical uncertainty. V. APPLICATION
In the recent Daya Bay nH analysis [8], the method wasimplemented with real data. The muon rate or net muon ratecan be measured with data, and both were observed to be sta-ble throughout the period. A precise single rate can also beextracted. In the Daya Bay case, there are several kinds ofreal coincidence events, IBD, and Li/ He, fast neutron and
Am- C backgrounds, as pointed out earlier, and their to-tal rates were known to be four orders of magnitude lower thanthe singles rate. So an upper limit was calculated as a triggerrate of everything excluding muons. And a safe lower limitwas calculated by further rejecting all correlated-like eventsif two triggers were too close in time and in distance. (Al-though the correlated background rates of
Bi-
Po-
Pband
Bi-
Po-
Pb were high, they were rejected beforethe multiplicity selection by a 1.5 MeV energy cut for everytrigger.) Taking the average of the upper and lower limits, aprecise estimation of R s was obtained with a systematic of0.18%, 0.16% and 0.05% for the three sites of the Daya Bayexperiment, respectively, which reflected the real coincidenceevent fractions at each site. After an estimation of all corre-lated events is obtained, an iteration can further improve theaccuracy. For Daya Bay, R s was observed unstable in two as-pects. R s decreased slowly at a rate ( < . %/day), becausethe singles were originally from residual radioactivity in thedetector and after it was sealed, the total number could de-crease. Another effect was the instant increase after muons, since muons may introduce some spallation products. Witha veto window of a few hundreds of micro seconds, some ofthem can still survive. With these values as input, the IBDefficiency and accidental backgrounds rate, etc. can all be cal-culated. The uncertainties in IBD detection efficiency and ac-cidental background can be directly derived according the sys-tematic uncertainty in the R s and its variance as a function ofreal time and as a function of the time to previous muon.Validation with data is also possible. As in [8], the distancedistributions of all two-fold events were studied, as well asthe time distributions. It was known that only accidental co-incidences can have a large separation distance ( > m) ora long coincidence time ( > . ms). On the other hand, asingle sample can be selected and randomly combined to pre-dict the spectrum of the accidental background. With the nor-malization constant provided by the formulas above, the pre-dicted accidental spectrum can be compared with the data asin Fig. 2 of [8]. Given the high statistics of the Daya Bay data,the systematics of accidental backgrounds were validated to < . %, which is sufficient for the expected sin θ mea-surement precision. VI. CONCLUSION
A complete mathematical model was developed for the sig-nal and background distributions on the full-time axis fordelayed-coincidence experiments, for example, recent reactorneutrino experiments. It was then projected onto the live-timeaxis. An analytic calculation was done by assuming the netmuons are uniformly distributed on the live-time axis, and thereal correlated signals have a much smaller rate than the sin-gle rate. The intrinsic relative uncertainties for all combina-tions’ rates are at the − level. The predictions were ver-ified with high statistics simulation studies with realistic pa-rameters from the Daya Bay experiment. The model was alsoapplied to the Daya Bay nH analysis and validated to high pre-cision with data. With analytical expressions, it is convenientto consider the systematic uncertainties of the IBD detectionefficiency and the accidental background rate when the singlerate is unstable. In the future large reactor antineutrino exper-iments aiming to resolve the neutrino mass hierarchy [11–14],to achieve a longer attenuation length in liquid scintillator, thenH method is preferred over the nGd method, i.e. to not useGd-load liquid scintillator. The method presented here is di-rectly applicable for these studies. VII. ACKNOWLEDGEMENT
The work is supported in part by the Ministry of Scienceand Technology of China (Grant No. 2013CB834302), theNational Natural Science Foundation of China (Grant No.11235006 and 11475093), the Tsinghua University InitiativeScientific Research Program (Grant No. 2012Z02161), andthe Key Laboratory of Particle & Radiation Imaging (Ts-inghua University), Ministry of Education. [1] Y. Abe et al. (Double Chooz Collaboration), Phys. Rev. Lett. , 131801 (2012).[2] Y. Abe et al. (Double Chooz collaboration), Phys. Rev. D ,052008 (2012).[3] Y. Abe et al. (Double Chooz collaboration), Phys. Lett. B ,66 (2013).[4] J. K. Ahn et al. (RENO Collaboration), Phys. Rev. Lett. ,191802 (2012).[5] F. P. An et al. (Daya Bay Collaboration), Phys. Rev. Lett. ,171803 (2012).[6] F. P. An et al. (Daya Bay collaboration), Chinese Phys. C ,011001 (2013).[7] F. P. An et al. (Daya Bay collaboration), Phys. Rev. Lett. ,061801 (2014).[8] F. P. An et al. (Daya Bay collaboration), arXiv:1406.6468 (2014).[9] B. Pontecorvo, Sov. Phys. JETP , 429 (1957) and , 984(1968).[10] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. ,870 (1962).[11] S.T. Petcov and M. Piai, Phys. Lett. B , 94 (2002).[12] S. Choubey, S.T. Petcov and M. Piai, Phys. Rev. D , 113006(2003).[13] J. G. Learned, S. T. Dye, S. Pakvasa and R. C. Svoboda, Phys.Rev. D , 071302(R) (2008).[14] L. Zhan, Y. Wang, J. Cao, and L. Wen, Phys. Rev. D ,111103(R) (2008).[15] S. Abe et al. (KamLAND Collaboration), Phys. Rev. C81