Realistic nPE map with full electronic effects

One of the crucial components of the energy reconstruction in Ref. [9] is the nPE map denoted by ${\widehat{\mu }}^{\text{L}}(r\mathrm{,}\theta \mathrm{,}{\theta }_{\text{PMT}})$. This describes the expected number of LS photoelectrons per unit of visible energy. The visible energy is defined as PE_total/Y₀, where PE_total is the total number of PEs and Y₀ is the constant light yield defined in Ref. [17]. The definitions of r,θ, θ_PMT are shown in Fig. 1. A data-driven method for constructing the nPE map was introduced in [9], in which electronic effects were omitted. In a realistic case with full electronic effects, the observable of PMTs change from nPE to charge due to the charge smearing of every photoelectron. Dark noise also contribute to the total charge of each PMT. Consequently, the formula for ${\widehat{\mu }}^{\text{L}}$ must be modified accordingly after considering these two effects. The updated formulas are given by Eq. 1, ${\widehat{\mu }}^L\left(r\mathrm{,}\theta \mathrm{,}{\theta }_{\text{PMT}}\right)=\frac{1}{E_{vis}}\frac{1}{M}{\displaystyle \sum _{i=1}^M\frac{\frac{{\overline{q}}_i}{{\widehat{Q}}_i}-{\mu }_i^{\text{D}}}{D{E}_i}}\mathrm{,}{\mu }_i^{\text{D}}=DN{R}_i\times L\mathrm{,}$ (1) where i runs over PMTs with the same θ_PMT and DEi is the relative detection efficiency, with the mean relative detection efficiency normalized to 1. Photoelectrons from the LS have a strong temporal correlation, whereas dark noise photoelectrons occur randomly in time. The length of the full electronic readout window is L_FADC =1250 ns. To reduce the impact of dark noise, the signal window $\left[t_{\text{r}}^{\text{A}}\mathrm{,}{t}_{\text{r}}^{\text{B}}\right]$ is set according to the residual time distribution (see Eq. 4) of the positrons to exclude most dark noise. These dark-noise photoelectrons within the signal window contaminate the photoelectrons from the physical signal; thus, the expected number of dark-noise photoelectrons ${\mu }_i^{\text{D}}$ must be subtracted. Here, ${\mu }_i^{\text{D}}$ is proportional to the dark noise rate DNRi and the signal window length $L=t_{\text{r}}^{\text{B}}-t_{\text{r}}^{\text{A}}$, which was optimized to 280 ns by scanning its value from 160 to 540 ns and selecting the one with the best effective energy resolution [7]. In contrast, the average detected nPE ${\overline{n}}_i$ was estimated using $\frac{{\overline{q}}_i}{{\widehat{Q}}_i}$, where ${\overline{q}}_i$ is the average recorded charge inside the signal window and ${\widehat{Q}}_i$ is the expected average charge of one photoelectron. Except for the two changes in Eq. 1, the construction procedure for the nPE map is the same as that in Ref. [9].

Calibration of the effective refractive index of photons in LS

In addition to charge, another important observable of PMTs is the hit time of the photons. This can be approximately expressed as Eq. 2: $t_h^{\text{'}}=t_0+t_{\text{LS}}+t_{\text{tof}}+t_{\text{TT}}^{\text{'}}+t_d\mathrm{,}$ (2) The different components are shown in Fig. 3 with a simple illustration, where t₀ is the event starting time, t_LS is the scintillation time, which is governed by the LS optical properties, t_tof is the time of flight of photons propagating from the event vertex to PMTs, ${t^{\prime }}_{\text{TT}}$ is the transit time of PMTs which roughly obeys a Gaussian distribution with (μ_TT, σ_TT), and t_d is the delay time caused by the PMT readout electronics. Both μ_TT and t_d are PMT dependent, and the difference between PMTs can be calibrated and absorbed by ${t^{\prime }}_{\text{h}}$, resulting in the calibrated hit time t_h: $t_{\text{h}}=t_0+t_{\text{LS}}+t_{\text{tof}}+t_{\text{TT}}\mathrm{,}$ (3) where t_TT is described by the new Gaussian ($\overline{{\mu }_{\text{TT}}+t_{\text{d}}}$, σ_TT), and $\overline{{\mu }_{\text{TT}}+t_{\text{d}}}$ is the average value of all PMTs.

Fig. 3Full-screen View

Illustration of the components of photon hit time. t₀ is the event starting time, t_LS is the scintillation time, t_tof is the time of flight of photons propagating from the event vertex to PMTs, ${t^{\prime }}_{\text{TT}}$ is the transit time of PMTs, and t_d is the delay time caused by the PMT readout electronics.

Ref. [11] defines the residual time of PMT photon hits t_r as $t_{\text{r}}=t_{\text{h}}-t_{\text{tof}}-t_0=t_{\text{LS}}+t_{\text{TT}}\mathrm{.}$ (4) To calculate the time-of-flight of the photons in the JUNO CD, a constant effective refractive index was used in Ref. [11]. However, the propagation of photons in LS is complicated and includes various optical processes, as described in Sect. 2. The time-of-flight may not necessarily be proportional to propagation distance. Given that the calibration sources can be deployed at different positions in the CD, the calibration data can be used to calibrate the time-of-flight or, equivalently, the effective refractive index n_eff as a function of the propagation distance. For radioactive sources, the starting time t₀ of each event is unknown. However, the precision of t₀ can reach < 0.5 ns for a laser source [21]. Thus, a laser source was chosen to calibrate n_eff. Although the original optical photons of the laser source have a fixed wavelength, they are rapidly absorbed and re-emitted by the LS, resulting in a wavelength spectrum similar to that of other sources [21]. By deploying the laser source at different positions along the z axis of the CD with ACU, one can plot the distribution of t_r + t_tof = t_h - t₀ for each PMT with sufficient statistics of the laser events. Because t_LS depends only on the LS properties, it is the same for all PMTs. In addition, t_TT defined in Eq. 4 follows approximately the same distribution for the same type of PMTs. Thus, the shape of the t_h - t₀ distribution should be the same for the same type of PMTs and the t_tof term merely leads to a relative shift with respect to t_r, which can be measured by the peak of the t_h - t₀ distribution $t_{\text{peak}}=t_{\text{tof}}+t_{\text{peak}}^0\mathrm{,}$ (5) where $t_{\text{peak}}^0$ is the peak of the t_r distribution and is a constant with different values for each type of PMTs. We define the distance between PMT and the source position as d (Fig. 1). For PMTs of the same type and d, their t_peak should be aligned because t_tof is the same in the first-order approximation. Thus, their t_peak values were averaged to reduce small second-order fluctuations. Given that Dynode PMTs have much better time resolution than MCP PMTs, only Dynode PMTs are used to obtain a more precise t_h. By deploying the laser source at different positions, a large set of (t_peak, d) data points can be obtained, which can then be used to fit t_tof as a function of d. For convenience, the effective refractive index $n_{\text{eff}}(d)$ is used instead of $t_{\text{tof}}(d)$: $n_{\text{eff}}(d)=\frac{c\times (t_{\text{peak}}(d)-t_{\text{peak}}^0)}{d}\mathrm{.}$ (6) $t_{\text{peak}}^0$ can be approximately defined as the time interval of the sharp rising edge in the $t_{\text{h}}-t_0$ distribution. It was estimated to be approximately 3.3 ns by extrapolating this time interval for all $t_{\text{h}}-t_0$ distributions with different d values. The fitting function of the effective refractive index is given by $n_{\text{eff}}=n_{\text{eff}}^{\text{LS}}+\frac{A}{d}\mathrm{,}$ (7) where $n_{\text{eff}}^{\text{LS}}$ is the dominant term of the effective refractive index of liquid scintillator, and A is the coefficient of the additional term which is distance dependent. Figure 4 presents the fitting results. The best-fit value $n_{\text{eff}}^{\text{LS}}\approx 1.54$ was consistent with that in Ref. [11].

Fig. 4Full-screen View

(Color online) The fitting results of effective refractive index. Based on the timing error of the laser system (Fig.11 in [21]), the uncertainty of t_peak (Δ t_peak) is set as 0.5 ns. The error of n_eff is then calculated by cΔ t_peak/d, with negligible statistical uncertainty. For small values of d, the source is located near the north or south pole and the corresponding PMTs fall in the dark zone due to total reflection.

Construction of more accurate and realistic time PDF.

One caveat of the residual time PDF from Ref. [11] is that it was obtained from MC simulations. Any potential discrepancy between the MC and real data will degrade the vertex reconstruction. Furthermore, the PDF was constructed using only the events at the detector center. This simplification did not consider the PDF dependence on the vertex and was later found to cause a large vertex bias near the detector border in Ref. [11]. Based on the calibration-data-driven construction of the nPE map in Ref. [9], the same calibration data can be used to build a realistic time PDF. Moreover, various calibration positions allow for a more precise parametrization of the time PDF.

To construct the nPE maps in Ref. [9], ⁶⁸Ge and laser sources were used. As previously mentioned, the optical photons of the laser source are absorbed and re-emitted by the LS. However, the other particles must first transfer energy to the LS molecules. Consequently, the photon timing profile of the laser source is different from that of the positrons. In this study, we only use ⁶⁸Ge to construct the time PDF of the positrons. Ideally, an electron source can be used to describe the time PDF of photons originating from the kinetic energy of positrons. Although electron sources were not available, we compared the reconstruction performance using the time PDF of ⁶⁸Ge to that of ⁶⁸Ge and electrons to check its impact. The details are presented in Sect. 4.

As shown in Eq. 4, to calculate the residual time, the information of t₀ is required for every single event. In contrast to a laser source with high precision of t₀, this quantity of calibration events from radioactive sources is unknown. Reconstruction of t₀ for each event was attempted; however, the uncertainty was greater than 2 ns. Because the t_h - t_tof distribution for different events with a fixed vertex and energy should have approximately the same shape, a different t₀ would merely cause a relative shift in the distribution. The peak of the t_h - t_tof distribution t’₀ can be used as the new reference time instead of t₀ to align the different events. Eq. 4 can be modified as follows such that the distribution of the newly defined residual time t_r’ always peaks at 0. For convenience, the prime symbols t_r’ and t_0’ are omitted in the remainder of this paper. $t_{\text{r}}\text{'}=t_{\text{h}}-t_{\text{tof}}-{t^{\prime }}_0=t_{\text{LS}}+t_{\text{TT}}+\text{constant}\mathrm{.}$ (8) The PDF $P_{\text{T}}\left(t_{\text{r}}|r\mathrm{,}d\mathrm{,}{\mu }_{\text{l}}\mathrm{,}{\mu }_{\text{d}}\mathrm{,}k\right)$ describes the probability of the residual time of the first photon hit falling in $[t_{\text{r}}\mathrm{,}{t}_{\text{r}}+\delta t]$, where r is the radius of the event vertex, $d=|\overrightarrow{r}-{\overrightarrow{r}}_{\text{PMT}}|$ is the propagation distance from the previous section, μ_l and μ_d are the expected numbers of LS and dark noise photoelectrons inside the full electronic readout window, respectively, and k is the detected number of photoelectrons. The dependence of P_T on these parameters is discussed in the following subsections. For convenience, let us denote f(t) as the probability density of “photoelectron hit at time t" and F(t) as the probability of “photoelectron hit after time t" for a general case. Then, F(t) can be calculated as: $F(t)={\displaystyle {\int }_t^{L_{\text{FADC}}}}f(t^{\prime })\text{d}{t}^{\prime }\mathrm{.}$ (9)

3.3.1

Dependence on r and d

Optical processes such as absorption and re-emission or Rayleigh scattering are not negligible in an LS volume as large as JUNO and become more prominent as the photon propagation distance d increases. Meanwhile, the total reflection can significantly change the direction of the photons, which is more likely to occur for events with a larger radius r. Thus, f(t) of the LS photoelectron f_l(t) depends on d and r. This dependency can be addressed by deploying calibration sources at different positions with the ACU + CLS system. This data-driven construction of the time PDF does not require a comprehensive understanding of the properties of the liquid scintillator such as attenuation length and decay time.

One of the challenges in constructing a time PDF from the calibration data is that the number of calibration positions is limited. To address this challenge, 35 and 200 bins were set in the r- and d-directions, respectively. Examples of the construction results of the t_r PDF of one LS photoelectron are shown in Fig. 5. The left and right plots correspond to vertices in the central and edge regions, respectively. One can clearly observe the difference in the time PDF for different radii r. Moreover, the time PDF becomes wider as d increases, except in the total reflection region. Most of the detected photons in the total reflection region are scattered light, whose time PDF is relatively flat. Notably, the contribution from dark noise was subtracted and will be added independently in the next subsection.

Fig. 5Full-screen View

(Color online) Examples of the time PDFs at different positions. (a) Central region: the time PDF becomes wider as d increases. (b) Edge region: the time PDF also becomes wider as d increases, except in the total reflection region with 12 m < d < 16 m. Most detected photons in the total reflection region are scattered light, whose time PDF is relatively flat.

Once the time PDF of one LS photoelectron is obtained, it is straightforward to calculate the time PDF of n LS photoelectrons using Eq. 10, where $I_{\text{n}}^{\text{l}}$ denotes the normalization coefficient. $P_{\text{T}}^{\text{l}}(t|k=n\mathrm{,}r\mathrm{,}d)=I_{\text{n}}^{\text{l}}\times [f_l(t|r\mathrm{,}d)F_{\text{l}}^{n-1}(t|r\mathrm{,}d)]\mathrm{.}$ (10)

3.3.2

Adding dark noise contribution

Photoelectrons induced by PMT DN contaminate the photoelectrons from the signal particles in the LS. Their impact on the residual-time PDF must be carefully considered. As dark-noise photoelectrons occur randomly in time, their f(t) is simply $f_{\text{d}}(t)=1/L_{\text{FADC}}$. The probability of a DN photoelectron falling into $\left[t\mathrm{,}{L}_{\text{FADC}}\right]$ is given by Eq. 11: $F_{\text{d}}(t)=1-\frac{t}{L_{\text{FADC}}}\mathrm{.}$ (11) For any given PMT with the expected nPE from the LS μ_l, expected nPE from DN μ_d, and the detected nPE k, one can mathematically calculate the probability density of the first photon hit observed at time t. In the simplest case, where only one PE is detected by the PMT or k=1, $P_{\text{T}}(t|{\mu }_{\text{l}}\mathrm{,}{\mu }_{\text{d}}\mathrm{,}k=1)$ is given by $\begin{array}{c}{P}_{\text{T}}\left(t|k=\mathrm{1,}r\mathrm{,}d\mathrm{,}{\mu }_{\text{l}}\mathrm{,}{\mu }_{\text{d}}\right)=I_1\times [P\left(\mathrm{1,}{\mu }_{\text{l}}\right)P\left(\mathrm{0,}{\mu }_{\text{d}}\right)f_{\text{l}}\left(t|r\mathrm{,}d\right)\\ +P\left(\mathrm{0,}{\mu }_{\text{l}}\right)P\left(\mathrm{1,}{\mu }_{\text{d}}\right)f_{\text{d}}(t)],\end{array}$ (12) where I₁ is a normalization factor and $P(k_{\text{l}}\mathrm{,}{\mu }_{\text{l}})$ and $P\left(k_{\text{d}}\mathrm{,}{\mu }_{\text{d}}\right)$ are the Poisson probabilities of detecting k_l photoelectrons from the LS and k_d photoelectrons from DN, respectively, under the condition k=k_l+ k_d. Given k=1 in this case, both k_l and k_d can be either 1 or 0. Thus, one can easily see that the two terms correspond to the photoelectron emitted from LS or DN, respectively. The same method can be applied to the case where k≥2. The time PDF of the first hit of n photoelectrons is given by $\begin{array}{l}{P}_{\text{T}}\left(t|k=n\mathrm{,}r\mathrm{,}d\mathrm{,}{\mu }_{\text{l}}\mathrm{,}{\mu }_{\text{d}}\right)\approx I_{\text{n}}\times [P(n\mathrm{,}{\mu }_{\text{l}})P(\mathrm{0,}{\mu }_{\text{d}}))P_{\text{T}}^{\text{l}}\left(t|n\mathrm{,}r\mathrm{,}d\right)\\ +P\left(n-\mathrm{1,}{\mu }_{\text{l}}\right)P\left(\mathrm{1,}{\mu }_{\text{d}}\right)\left(P_{\text{T}}^{\text{l}}\left(t|n-\mathrm{1,}r\mathrm{,}d\right)F_{\text{d}}(t)+f_{\text{d}}(t)F_{\text{l}}^{n-1}\left(t|r\mathrm{,}d\right)\right)\\ +P\left(n-\mathrm{2,}{\mu }_{\text{l}}\right)P\left(\mathrm{2,}{\mu }_{\text{d}}\right)(P_{\text{T}}^{\text{l}}\left(t|n-\mathrm{2,}r\mathrm{,}d\right)F_{\text{d}}^2(t)\\ +2f_{\text{d}}(t)F_{\text{d}}(t)F_{\text{l}}^{n-2}\left(t|r\mathrm{,}d\right))]\mathrm{,}\end{array}$ (13) where I_n is the normalization coefficient. The three terms correspond to the cases where k_d=0, 1, and 2. Given that ${\mu }_{\text{d}}$ is rather small, terms with $k_{\text{d}}>2$ will be highly suppressed by the Poisson probability $P(k_{\text{d}}\mathrm{,}{\mu }_{\text{d}})$ and thus can be safely omitted to simplify the calculation. To verify this analytical approach, the calculated results of the time PDF containing dark noise contributions were compared with those obtained using the true information of k in the MC simulation, as shown in Fig. 6. The good agreement verified the validity of Eq. 12 and Eq. 13.

Fig. 6Full-screen View

(Color online) Time PDF of Dynode-PMT of central ⁶⁸Ge events considering dark noise. Red histograms are calculated by Eq. 13, while black histograms are obtained from MC simulation data.

Charge based maximum likelihood estimation

Ref. [9] presented the basic strategy for energy reconstruction; a likelihood function was constructed using the expected nPE $\mu ={\widehat{\mu }}_{\text{L}}\times E_{\text{vis}}$ and observed charge for each PMT. By maximizing the likelihood function, one could obtain the reconstructed energy. However, the expected nPE for each PMT depends strongly on the positron vertex, as indicated by Eq. 1. In Ref. [9], the vertex was assumed to be known; however, in real data, it must also be reconstructed. Thus, the vertex and energy can be simultaneously reconstructed using a likelihood function similar to that in Ref. [9].

This likelihood function utilizes only the charge information of PMTs and is referred to as the charge-based maximum likelihood estimation (QMLE). It is constructed using Eq. 15, which is the product of the probabilities of observing a charge qi when the expected nPE is μi for the i-th PMT. $\begin{array}{l}\mathcal{L}\left(q_1\mathrm{,}{q}_2\mathrm{,...,}{q}_N|r\mathrm{,}{E}_{\text{vis}}\right)\\ ={\displaystyle \prod _{\text{unfired}}}{e}^{-{\mu }_j}{\displaystyle \prod _{\text{fired}}}\left({\displaystyle \sum _{k=1}^{+\infty }{P}_Q}\left(q_i|k\right)\times P\left(k\mathrm{,}{\mu }_i\right)\right),\end{array}$ (15) where ${\mu }_i=E_{vis}\times {\widehat{\mu }}_i^L+{\mu }_i^D$ and P(k, μi) is the Poisson probability for detecting k photoelectrons. $P_Q(q_i|k)$ is the charge PDF of k photoelectrons that can be constructed by convolving the single photoelectron charge spectrum (SPES). Indices j and i run over all "unfired" and "fired" PMTs, respectively, with a PMT firing threshold of q>0.1 p.e. The index k ends when $P_Q(q_i|k)<1.e^{-8}$. The constraint power dramatically decreases for positrons in the central region of the CD. This was demonstrated in Ref. [13, 11] and can also be seen in the bottom-left plot of Fig. 7, where the vertex resolution in the central region is much worse than that in the border region. Moreover, the vertex bias of QMLE is large, as shown in the top-left plot of Fig. 7. An inaccurate vertex degrades the energy resolution for the simultaneous reconstruction.

Fig. 7Full-screen View

(Color online) Vertex reconstruction performances. The left, middle, and right columns correspond to the QMLE, TMLE, and QTMLE methods, respectively. The top row shows the vertex bias, and the bottom row shows the vertex resolution. The red band corresponds to the region outside the FV with 17.2 m < r < 17.7 m.

Time-based maximum likelihood estimation

The event vertex is strongly constrained by the time information of the PMTs. Similar to Ref. [11], a likelihood function can be constructed using the first hit time of PMTs and a more accurate and realistic time PDF P_T from Eq. 14. This likelihood function uses only the PMT time information and is referred to as time-based maximum likelihood estimation (TMLE). It can be constructed using Eq. 16, which is the product of the probabilities of observing the residual time of $t_{i\mathrm{,}\text{r}}$ when the expected t_tof is $d/\left(c/n_{\text{eff}}\right)$ for the i-th PMT. $\begin{array}{l}\mathcal{L}\left(t_{\mathrm{1,}\text{r}}\mathrm{,}{t}_{\mathrm{2,}\text{r}}\mathrm{,...,}{t}_{N\mathrm{,}\text{r}}|r\mathrm{,}{t}_0\right)=\\ {\displaystyle \prod _{T-\text{valid}\text{hit}}}\frac{{\displaystyle {\sum }_{k=1}^K{P}_{\text{T}}}\left(t_{i\mathrm{,}\text{r}}|r\mathrm{,}{d}_i\mathrm{,}{\mu }_i^{\text{l}}\mathrm{,}{\mu }_i^{\text{d}}\mathrm{,}k\right)\times P\left(k\mathrm{,}{\mu }_i^{\text{l}}+{\mu }_i^{\text{d}}\right)}{{\displaystyle {\sum }_{k=1}^{K}P}\left(k\mathrm{,}{\mu }_i^{\text{l}}+{\mu }_i^{\text{d}}\right)},\end{array}$ (16) where "T-valid" hit refers to those hits satisfying $-100<t_{i\mathrm{,}\text{r}}<500\text{ns}$ and $0.1\text{p}\mathrm{.}\text{e}\mathrm{.}<q_i^{\text{f}}<K=20\text{p}\text{.e}\mathrm{.}$ The definition range of the residual-time PDF was (-100, 500) ns. q^f denotes the total charge in the full electronic readout window. A cut-off value K was set for the detected nPE k to simplify the calculations. TMLE takes the reconstructed vertex and energy from the QMLE as the initial values and only updates the reconstructed vertex. Note that reference time t₀ is also a free parameter in the reconstruction. As shown in Fig. 7 and 8, the vertex bias and resolution of the TMLE are significantly improved with respect to the QMLE, especially in the central region of the CD. This is mainly because of the stronger constraint of the PMT time information.

Fig. 8Full-screen View

(Color online) Comparison of the vertex resolution as a function of the energy. The left plot shows the comparison among the three methods QMLE (Blue), TMLE (Red), and QTMLE (Black). The ratio is defined as Res_R(other)/Res_R(QMLE). The QTMLE method utilizing both charge and time information of PMTs effectively improves vertex reconstruction. The right plot shows the comparison between using ⁶⁸Ge+electron time PDF and ⁶⁸Ge time PDF in the QTMLE method, which will be discussed in Sect. 4.4. The ratio is defined as Res_R(⁶⁸Ge+e^- time PDF)/Res_R(⁶⁸Ge time PDF).

Charge and time combined maximum likelihood estimation

Given the likelihood functions from QMLE with charge information only and TMLE with time information only, it is straightforward to construct the charge and time combined maximum likelihood estimation (QTMLE), as Eq. 17, by multiplying Eq. 15 and Eq. 16. $\begin{array}{l}\mathcal{L}\left(q_1\mathrm{,}{q}_2\mathrm{,...,}{q}_N;t_{\mathrm{1,}\text{r}}\mathrm{,}{t}_{\mathrm{2,}\text{r}}\mathrm{,...,}{t}_{N\mathrm{,}\text{r}}|r\mathrm{,}{t}_0\mathrm{,}{E}_{\text{vis}}\right)=\\ {\displaystyle \prod _{\text{unfired}}}{e}^{-{\mu }_j}{\displaystyle \prod _{\text{fired}}}\left({\displaystyle \sum _{k=1}^{+\infty }{P}_Q}\left(q_i|k\right)\times P\left(k\mathrm{,}{\mu }_i\right)\right)\\ {\displaystyle \prod _{T-\text{valid}\text{hit}}}\left(\frac{{\displaystyle {\sum }_{k=1}^K{P}_{\text{T}}}\left(t_{i\mathrm{,}r}|r\mathrm{,}{d}_i\mathrm{,}{\mu }_i^l\mathrm{,}{\mu }_i^d\mathrm{,}k\right)\times P\left(k\mathrm{,}{\mu }_i^{\text{l}}+{\mu }_i^{\text{d}}\right)}{{\displaystyle {\sum }_{k=1}^{K}P}\left(k\mathrm{,}{\mu }_i^{\text{l}}+{\mu }_i^{\text{d}}\right)}\right).\end{array}$ (17) Because QTMLE uses both the charge and time information of PMTs to constrain the vertex, it yields the best vertex reconstruction performance among the three methods, which can be observed in the plot on the left side of Fig. 8. Across the entire energy range of interest, the vertex resolutions of TMLE and QTMLE are, on average, approximately 56% and 60% better than that of QMLE, respectively. Meanwhile, a more accurate vertex also leads to a more accurate energy, given the strong correlation between them. This is shown in Fig. 10, where the energy resolution of QTMLE is significantly better than that of the QMLE. The resolutions of the x-, y-, z-, and r-components of the vertex were similar, with a discrepancy less than 5%, as shown in Fig. 9. Meanwhile, the energy resolution was not sensitive to the resolution of θ and ϕ components, given the approximate spherical symmetry of the CD. Thus, only the radial resolution is presented in this study.

Fig. 10Full-screen View

(Color online) Comparison of the energy resolution with different vertex precisions. The bottom panel shows the ratio to the QTMLE method with ⁶⁸Ge time PDF. The blue dots correspond to the energy resolution of QMLE, which has the worst vertex resolution. The pink dots represent the energy resolution of QTMLE using true vertex, which is equivalent to an ideal vertex resolution of 0 mm. The energy resolution of QTMLE using ⁶⁸Ge + electron time PDF is also shown and will be discussed in Sect. 4.4.

Fig. 9Full-screen View

(Color online) Comparison of the x, y, z, and r resolution of the QTMLE method with ⁶⁸Ge time PDF. The bottom panel shows the ratio to the radial resolution.

The uniformity of the reconstructed energy using QTMLE for positrons with different energies is shown in Fig. 11. The QTMLE method yields excellent energy uniformity, and the residual energy nonuniformity is less than 0.23% inside the fiducial volume. Compared to the value of 0.17% in Ref. [9], which used a true vertex and did not include any electronic effects, one can see that the impact of vertex inaccuracy and electronic effects on energy non-uniformity is non-negligible but still under control.

Fig. 11Full-screen View

(Color online) Energy uniformity of QTMLE. The red band corresponds to the region outside the FV with 17.2 m < r < 17.7 m. The QTMLE method yields excellent energy uniformity. The residual energy non-uniformity is less than 0.23% within the FV for all the different energies.

Discussion

As mentioned previously, the energy deposition process of positrons in the LS usually consists of two parts, kinetic energy and annihilation with an electron emitting two gamma particles. During the construction of the expected nPE map of PMTs, Laser and ⁶⁸Ge were used to mimic the two parts, respectively. However, because the photons from the laser contain only the fast component, they cannot be used to describe the photon timing profile of the charged particles. Although electrons can mimic the kinetic energies of positrons, monoenergetic electron sources are unavailable. Consequently, only the ⁶⁸Ge source was used to construct the time PDF of PMTs in all previous studies. Pseudo electron calibration data were produced to check the impact of the accuracy of the time PDF on the vertex and energy reconstruction. The QTMLE method was used and a new set of time PDF was constructed using a weighted ⁶⁸Ge + electron-time PDF. The reconstruction results are compared with those obtained using the ⁶⁸ Ge-time PDF.

The right plot in Fig. 8 shows the comparison of the vertex resolution. The weighted ⁶⁸Ge + electron-time PDF was more accurate than the ⁶⁸Ge time PDF, and the corresponding vertex resolution was about 5% better. Fig. 10 compares the energy resolutions of the two cases. Despite the better vertex resolution obtained using ⁶⁸Ge + electron-time PDF, the energy resolution was almost the same as that obtained using the ⁶⁸Ge time PDF. To verify the impact of the accuracy of the vertex on the energy resolution, two additional cases, namely, QMLE and QTMLE using the true vertex, are also shown in Fig. 10. The black dots correspond to the energy resolution of QMLE, which has the worst vertex resolution. The pink dots represent the energy resolution of QTMLE using the true vertex, which has an ideal vertex resolution of 0 mm. By comparing these cases, it is clear that better vertex resolution leads to better energy resolution. Meanwhile, comparing the default case of QTMLE using ⁶⁸Ge time PDF to the ideal case of QTMLE using the true vertex, the impact of vertex inaccuracy on the energy resolution is approximately 0.6%.