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A Precise Calculation of Delayed Coincidence Selection Efficiency and Accidental Coincidence Rate

A model is proposed to address issues on the precise background evaluation due to the complex data structure defined by the delayed coincidence method, which is widely used in reactor electron-antineutrino oscillation experiments. In this model, the effects from the muon veto, uncorrelated random background, coincident signal and background are all studied with the analytical solutions, simplifying the estimation of the systematic uncertainties of signal efficiency and accidental background rate determined by the unstable single rate. The result of calculation is validated numerically with a number of simulation studies and is also applied and validated in the recent Daya Bay hydrogen-capture based oscillation measurement.


I. INTRODUCTION
The delayed coincidence method is used very broadly in nuclear and high energy physics experiments. The existence of a delayed signal greatly mitigates the critical requirement on low random background levels. The experimental observation of neutrino oscillation is one of the important breakthroughs in studying the Standard Model physics in the past half century. Recently three neutrino experiments, Daya Bay, RENO and Double CHOOZ [1][2][3] adopted this technique again to measure the neutrino mixing angle θ 13 precisely. Those anti-electron neutrinos from reactors are distinguished by detecting the positron and neutron from an inverse beta decay (IBD): p +ν e → e + + n. The whole signal is composed of a prompt positron signal and a delayed neutron capture signal, where the prompt positron has a detectable energy between 1 to 10 MeV and the delayed neutron spends about 29 µs in thermalization to be captured in the gadolinium (Gd)loaded scintillator region where several photons of total energy 8 MeV are released.
In the past year, the Daya Bay Experiment [1] and the RENO Experiment [2] have reported their first θ 13 measurements with delayed neutron capture signals on Gd. So far the accidental background is one of the dominant backgrounds. They both quoted the uncertainty for this background is at the 1% level. Their calculation methods are the same and can be found in their public presentations and papers, e.g. [1] and [2]. The formula for accidental background number, N accidental , is repeated here for discussion: In a low random background level environment, i.e. the number of neutron-like signal (N n−like ) and the rate of positron-like signal (R e + −like ) are both low, or with a very small coincidence window (T coincidence ), all the single signals will be very well separated in time on average, for which this simple formula is valid. But when the random background rate is high, it will become more likely to have other cases, like three-fold coincidence events. In this special physics context, the visible energy ranges of the prompt positron and the delayed neutron overlap, giving this formula an intrinsic bias, i.e. R e + −like will be higher than the real R e + −like by including n − like signals.
The Double CHOOZ [3] experiment estimated the accidental background by shifting the coincidence window by 1000 µs, which gave a larger statistical uncertainty of about 10%. There is also an additional systematic uncertainty in this off-window method: because of the spallation products produced by muons, the accidental background rate is lower when further in time after the previous muon, i.e. in the off-window.
For these three experiments, the Gd-loaded liquid scintillator region is enclosed by a pure liquid scintillator region which has an almost equivalent mass as the Gd region. There are proposals to use the hydrogen neutron capture signal to further enhance the neutrino signal statistics, and to even aim at the comparable precision level as the Gd capture result. The effect mentioned above may only have marginal effect on the published θ 13 measurement since the accidental background itself is small. The neutron capture on hydrogen will release a 2.2 MeV photon and the average capture time is around 200 µs. Apparently the delayed neutron signal energy is significantly lower than that in Gd-loaded scintillator which would allow more neutron-like singles. In addition, the coincident window chosen will be much longer. Correspondingly the accidental background will be much higher. This necessitates a precise analytic delayed coincident signal efficiency and accidental coincident background rate calculation.
Another interesting issue is that the full data-taking time is chopped into slices by cosmic muons. Some complex muon veto cuts are used in these analyses, i.e. IBD events are rejected if they tightly follow a muon. The length of veto time relies on the amount of the muon energy deposited or the location of the muon hits. It is imaginable that while the muon rate is high, there is no acceptance for a delayed coincident signal or an accidental coincident background. But how exactly the muon rate is going to affect the signal selection etc. is not transparent.
In this paper, we propose a data model to depict the full time sequence of all kinds of possible events in Sec. II, and with a possible signal selection scheme, the acceptance of delayed coincidence signal, the rate of accidental background and other coincidence possibilities are calculated analytically in Sec. III. In Sec. IV a high statistics simulation is done to verify the model and predictions.

A. Data model
A model is built up to demonstrate different types of events. Although the context used is closer to the physics of the reactor neutrino experiments, there is no loss of generality. In the following we use e and n to represent the positron and neutron from an IBD reaction respectively, s for random background (or singles) and µ for muon. The features of the event types are described below: 1) The most interesting event is the delayed coincident event including a prompt signal, e, and a delayed signal, n. The event rate of IBDs is assumed to be R IBD . The time between the prompt and the delay signal can be complex. For the three neutrino experiments mentioned above, the neutron capture cross section varies with its momentum, so that an exponential distribution is only a pool approximation for the neutron capture time. Here an abstract form P delay (t) is used to represent the delayed signals' time distribution.
2) Random signals, or singles, include those decays from residual radioactive nuclei in the detector or other non-correlated detector noise. Random signals occur with a uniform distribution, or equivalently the time interval between two random signals follows an exponential distribution with an average value of 1/R s , where R s is the singles rate.
3) Cosmic Ray muons are always present. They themselves are usually easy to identify because of their high energy deposit. The spallation products of muons, however, can cause a brief increase of random signal rate, which causes a trouble for the delayed coincidence event searching. Therefore a passive veto window is applied for each muon. The chance to produce spallation products depends on how far a muon track is from the sensitive region and the energy deposit. Based on this knowledge, different muon veto window lengths can be chosen to maximize the live time and at the same time have a good control of spallation background.
These different types of signals in the full data-taking time axis are shown in Fig. 1. First we discuss two muon events, A and B, on the full time axis. Muon A is supposed to be closer to the sensitive region of the detector than muon B, and so a longer veto window is applied to muon A.
A pair of delayed coincidence signals (a positron and a neutron) occurs between muon A and B, where T c is the coincidence window used to search for delayed coincidence events and is usually at the scale of the average arrival time of the delayed signal. To ensure the spallation background of muons will not contaminate the search for the prompt signal as well as the delayed signal, the time between a prompt signal and the next muon event cannot be smaller than T c . The total dead time introduced by a muon is the veto time plus T c . After muon B, a single signal occurs, since no delayed signal is found within T c after the single signal. Finally we show a case where two muons, C and D, are very close to each other in time such that their dead times overlap. In this example all coincident signals and singles are all well separated. We will discuss situations of overlap in the following sections.

B. Simplification
The first attempt at simplification is to project all signals from the full time axis to the live time axis as illustrated in Fig. 1 The important assumption making this analytic calculation possible is to postulate that the net muons are uniformly distributed on the live time axis and the time interval follows an exponential distribution with an average value of 1/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 . For the three reactor neutrino experiments we took as examples, R s is about 4 orders of magnitude higher than R IBD .

C. Searching for delayed coincidence events and other backgrounds
In this discussion we will use the simplest method to search for delayed coincidence pairs: as depicted in Fig. 1 for the positron and neutron, whenever a signal appears and is not within a muon dead time, a window will be opened for a duration of T c during which a second signal will be searched for. Two random signals can form an accidental coincidence background if their separation is less than T c .
Except to predict the event rate of delayed coincidence events and accidental background, other types of combinations are also interesting to check. The following cases will be calculated: 1-fold coincident events: s, e, n.

A. Two Steps of Calculation
The calculation of these event rates is divided into two steps: a) determine the probability of a type of signal to start a coincidence searching window; b) determine the probability that there is a second or third signal in the searching window for 2-fold or 3-fold coincidences or there is no other signal for 1-fold events. In the following sections the starting probabilities of a random background signal (P s−start ), positron (P e−start ) and neutron (P n−start ) will be calculated first and then followed by all kinds of event rates.

B. Starting probability
On the live time axis, each signal except a muon can start a coincident searching window, as long as it is not in the previous coincidence searching window. Note that on the live time axis, R IBD and R s are exactly the same as on the full time axis. First let us consider how a single event may start a searching window in different situations. In the formulas below, t µ is the time from a signal event to its previous net muon event.
Case a) As shown in the top panel of Fig. 2, when t µ < T c , if there is no other signal between the single event and the muon, a searching window will be started by the single event. The probability of this situation is: where P (0|R s t µ ) is the probability that there is no other single event in between, which is calculated as a Poisson distribution with a mean of R s t µ and count k = 0, and the second term P (t µ ) gives the probability of finding a muon at t µ before the random background being considered, which is calculated according to an exponential distribution with a rate of R µ .
Case b) While t µ T c , if there is no other signal in a T c window before the single event being considered, a searching window will be started, just as the panel b of Fig. 2 shows. The starting probability of this situation is: where the first term corresponds to no signals in T c and the second term is to ensure t µ T c .
Case c) In case b) when t µ > T c , if there is another scenario that the target single can start a searching window as depicted in the panel c of Fig. 2. There is a random signal, s1, in front of the target single event; however it is occupied by a previous searching window started by s2. The probability of this situation is: where the first integral gives the probability to find the first random background signal s1 at time t in front of the target single event, the second term calculates the probability of having at least one random background s2 in the early t window, that is T c away from the target single event, and the last integral just gives the probability of finding a muon at some time larger than T c + t.
Finally the starting probability is, There should be higher order corrections after P c , but, with the example parameters used in the simulation presented later, they are estimated to be five orders of magnitude smaller than these leading terms.
The starting rate is R s−start = R s · P s−start .
As for IBD signals, it is where R IBD ≪ R s and there is a detection efficiency ε e for positrons. Then the rate of prompt signals, i.e. positrons starting a time window, is similar to the rate of random backgrounds starting a time window: The situation of a neutron starting a searching window is a bit complicated, because by design it is always possible that there is a prompt positron signal ahead of it. It is only possible when the positron is failed to be detected. However the method to calculate P n−start is the same as above and the final result is listed in the appendix. Then the event rate with neutrons as a start is:

C. Construct an Event
After the starting probability or starting rate of a type of signal is known, the rate of accidental background, detectable IBD pairs and other cases can be calculated.
For example, the accidental background rate, i.e. the single-single combination in one T c , is just the rate of one single event to start a searching window multiplied by the probability of a second single event appearing in the same window: where R(ss) is used to describe the event rate of ss-type backgrounds and P (1|R s T c ) is the Poisson probability of one count with a mean of R s T c . IBD events are not explicitly required to be vetoed from this with R IBD ≪ R s .
The IBD event rate can be obtained in a similar fashion: where ε n|e is the efficiency of neutron detection after a positron is detected, and the random background is explicitly required to be outside this searching window. The complete IBD event detection efficiency can be expressed as: The results of other combinations are listed in the appendix.

IV. VERIFICATION WITH MONTE CARLO
Since the muon veto cut is quite complicated, a high statistics Monte Carlo simulation (1E10 events) is done to verify the assumption and the predictions. Three types of events are produced: IBD, random background, muon. They are generated on the full time axis according to three uniform distributions. Their event rates on the full time axis are chosen to be very similar to those of recent reactor neutrino experiments. Usually a veto time of 400µs is applied for each muon, but 0.05% of them can be shower muons or closer to the sensitive region, for which a one-second long veto is applied. The neutron capture time distribution P delay (t) is replaced with a simple exponential distribution with rate = λ. The detection efficiencies of the prompt and delayed signals are also included. All the parameters used are summarized in Tab   are not distinguished, as well as sns and ssn.

V. CONCLUSION
We brought up a complete mathematical model for the signals distribution on the full time axis for delayed coincidence experiments, and further projected this complicated model onto the live time axis. An analytic calculation was done by assuming all the net muons are uniformly distributed on the live time axis. The intrinsic relative uncertainties for all combinations' rates APPENDIX All 1-fold, 2-fold and 3-fold coincident event rate calculation formulas are listed in Tab. III, where the first column indicates the combination concerned and the second column shows the expression for calculation. P (k|m) is used to describe a Poisson distribution with a mean of m and a count of k. The delayed signal follows a simple exponential distribution with time constant λ.