Constructing Cell^3D

To integrate the information in each readout layer, the information from the corresponding Cell^2Ds in both layers is utilized to establish a novel data structure known as Cell^3D. Specifically, Cell^3Ds were systematically constructed across the transverse section of the ECAL, with a radius equivalent to the module’s Molière radius. Along the longitudinal direction of the ECAL, all Cell^2Ds falling within the transverse span of Cell^3D are amalgamated into Cell^3D, with each Cell^2D being included only once. The energy of Cell^3D is determined by the cumulative energy of the Cell^2Ds it encompasses, and the position of Cell^3D is defined as its geometric center.

Searching Seed^3D and constructing temporary Cluster^3D

When an electromagnetic particle strikes the ECAL, it radiates energy outward from the impact point, typically resulting in the formation of Cell^2D and Cell^3D with the highest local energy deposition near the impact point. Therefore, as illustrated in Fig. 2a, the initial step in clustering is to iterate through all Cell^3Ds and Cell^2Ds in specific layers to identify the Cell^3D and Cell^2D layers that exhibit the highest local energy deposition. For the local maximum Cell^2D, the corresponding Cell^3D is considered a Seed^3D. Simultaneously, the local maximum Cell^3D is considered as Seed^3D. seed denotes the initial step in the clustering process.

To prevent faker Seed^3Ds during the seeding process, all identified Seed^3D s must satisfy a transverse momentum cut, typically defined as greater than 50 MeV in this framework. The threshold value of this cut can be lowered if required to investigate phenomena related to soft photons or electrons. All Seed^3Ds passing through the cut are labeled as final Seed^3Ds and stored.

As a result of transverse particle showering, the energy is not fully contained within Seed^3D. Hence, it is imperative to encompass a specific range of Cell^3Ds around Seed^3D to guarantee optimal coverage of all energy deposits from the particles. In this study, we followed the methods described in Ref. [34], which uses a fixed-size window to incorporate Cell^3Ds. The specific procedure is as follows: Centered around the Seed^3Ds, the Cluster^3Ds are formed by incorporating all Cell^3Ds within a 3×3 window around the Seed^3Ds, as illustrated in Fig. 2a. Additionally, the Cell^2Ds from all layers encompassed by the Cell^3Ds are also included as members of the Cluster^3Ds. However, when the Seed^3D s are located at the boundary of the region, special treatment is required because of the different types of Cell^3Ds in different regions. Around Seed^3D, within a radius of 1.5 times the size of Seed^3D, Cell^3Ds belonging to other regions are also included. In this study, a detailed description of the process is not provided. This process may lead to Cluster^3D at the boundary containing several Cell^3Ds, which may be greater or less than nine, and its shape may be irregular. This requires separate handling of Cluster^3D at the boundary in the Cluster^3D correction process described later.

In the subsequent steps, further layered modifications were made to the Cell^2Ds included in Cluster^3D. Therefore, Cluster^3D obtained in this section is referred to as a temporary Cluster^3D.

Searching Seed^2D and reconstructing Cluster^2D

Because of the incident angle of the particles and the rotation of certain modules, as the shower evolves longitudinally, the energy centroid of the shower varies across different layers. This results in Cell^2D with the highest local energy in each layer, not always being encompassed within Seed^3D, as shown in Fig. 2c.

To identify the Seed^2D in each layer, we iterate through the Cell^2Ds of each layer in the temporary Cluster^3D and select the Cell^2D with the highest energy as the Seed^2D of that layer. Moreover, the overlap of Cluster^2Ds results in an increased accumulation of energy in shared Cell^2Ds, which may cause the energy of a shared Cell^2D to exceed that of a Seed^2D, leading to the misidentification of a shared Cell^2D as a Seed^2D. Therefore, before searching for Seed^2D, energy splitting must be performed on all overlapping Cluster^2Ds, as shown in Fig. 2d, and is discussed in detail in Sect. 3.2.3.

Subsequently, Cluster^2Ds are formed with Seed^2Ds as the center, and all Cell^2Ds within the Molière radius of the module centered on Seed^2Ds in the same layer are included in the Cluster^2Ds. The raw energy, position, and timestamp of the Cluster^2Ds were computed using the following equations: $\begin{array}{c}{E}_{\text{raw}}={\displaystyle \sum _{i=1}^n}{E}_i\mathrm{,}\\ r_{\text{raw}}=\frac{{\displaystyle {\sum }_{i=1}^n{E}_i\times r_i}}{E_{\text{raw}}};r=x\text{or}y\mathrm{,}\\ t_{\text{raw}}=t_{{\text{Seed}}^{\text{2D}}}\mathrm{,}\end{array}$ (1) where n represents the number of Cell^2Ds in Cluster^2D, and ri and Ei represent the position and energy of Cell^2Ds, respectively.

Finally, a new Cluster^3D containing Cluster^2Ds in all layers is constructed. The raw energy of Cluster^3D is computed using the following formula: $E_{\text{raw}}^{3D}={\displaystyle \sum _{i=1}^n}{E}_i\mathrm{,}$ (2) where Ei represents the energy of the Cell^2Ds, and n represents the number of Cell^2Ds in all the layers incorporated in Cluster^3D. As shown in Fig. 2b, the energy deposition distribution of the incident particles depends on their type and energy of the incident particles. Therefore, unreasonable Cluster^3Ds can be filtered based on the energy ratio of Cluster^2Ds in each layer. The position information of Cluster^2Ds is considered as a point of the incident particle momentum direction on the corresponding layer, with the time information serving as the timestamp for this coordinate point. Additionally, the energy information is regarded as the total deposited energy of the particles in that layer. Reconstructing and correcting the information of the particles along the momentum direction layer-by-layer helps improve the position resolution, which will be discussed in detail in Sect. 4.

Energy correction

The objective of energy correction is to correct the energy of Cluster^3D to match that of the incident particles. Errors in reconstructing the energy of an incident particle typically stem from the following sources:

Intrinsic error of ECAL readout cell: It comes from the response linearity of sensitive materials, thermal noise in electronic systems, sampling errors in analog to digital converters (ADC), etc.

Calibration error of the readout cell: The shower development in the longitudinal direction is energy related, and the proportion of physical processes dominated in energy deposition changes at different stages of shower development. This leads to changes in the sampling fraction of the ECAL, which ultimately affects the calibration of the readout cell.

Leakage of energy: It is due to the incomplete deposition of particle energy in ECAL and the use of finite-sized windows during clustering.

Fitting error of calibration and correction parameters: It is typically expressed as a constant term in the energy resolution.

In the layered framework of this study, the energy correction of Cluster^3Ds was divided into two steps, as shown in Fig. 3a and Fig. 3b. The initial correction of the energy of Cluster^3D was performed based on the energy ratios of Cluster^2Ds in different layers. As shown in Fig. 3a, in the example based on PicoCal, we first correct the bias between the reconstructed Cluster^3D energy and true energy based on the energy ratio of the front layer to the back layer. Subsequently, as illustrated in Fig. 3b, further correction of the energy of Cluster^3Ds is conducted based on the total energy of Cluster^3Ds, which reduces the bias in the low-energy region.

Position and time correction

When a particle passes through the ECAL, it showers and deposits energy along its momentum direction. The center of gravity of the deposited energy in each layer can be considered a point in the direction of the particle momentum. In the layered reconstruction framework, the position of Cluster^2D is regarded as the reconstructed position of the center of gravity. Furthermore, Cluster^2Ds also provide a timestamp, which is related to the hit time of the particle, for the layers to which they belong. For the position and time correction in this study, layered correction is applied to Cluster^2Ds firstly, followed by the utilization of the corrected information from Cluster^2Ds to calculate the information of Cluster^3Ds.

The purpose of position correction is to correct the reconstructed position of Cluster^2D as accurately as possible to the center of gravity of the energy deposited by the particles in each layer.

For Cluster^2D, the x/y position information was derived from the energy-weighted positions of the Cell^2Ds. Based on the model described in Ref. [35], we define Δr_rec as the position of Cluster^2D minus the position of Seed^2D, and Δr_true as the position of the true transversal energy barycenter minus the position of Seed^2D, in each layer. This study explored the relationship between Δr_rec and Δr_true. As shown in Fig. 3g and Fig. 3h, the shape of this relationship is like an S, so we also call the process of correcting the x/y coordinates “S correction”. The shape of S is affected by the location of Seed^2D. When Seed^2D is positioned at the boundaries of the region, the introduction of a varying cell size Cell^2D and the presence of installation gaps affect the impact on the S shape. This results in a different S shape for Cluster^2Ds located at the boundary of the region compared with those inside the region, as illustrated in Fig. 3g and Fig. 3h.

The z coordinate of the center of gravity of the deposition energy is typically used to evaluate the depth of the shower. Owing to the typically small granularity and thick layers of ECAL in the longitudinal direction, the direct use of the z-coordinate information of Cell^2D to reconstruct the z-coordinate of the gravity center of the deposition energy in each layer introduces substantial uncertainty. Therefore, the reconstruction of the z-coordinate is typically performed based on the energies of the incident particles. As shown in Fig. 3c, and Fig. 3d, the shower depth (the difference between the z coordinate of the shower and the z coordinate of the front face of the module) is logarithmically related to the incident particle energy, because the pair production of electrons dominates the energy deposition [36]. This is the rationale behind the term "L correction" for this correction step. By leveraging this correlation, we can deduce the z-coordinates of Cluster^2D based on the energy of the incoming particles.

For the Cluster^2Ds, the positions obtained by the above position correction were also projected onto the front surface of ECAL and then used in subsequent steps to calculate the position information of the Cluster^3Ds.

In this study, the timestamp of the Seed^2Ds were used as the timestamp for Cluster^2Ds in the preliminary study. The purpose of time correction is to determine the time difference between the timestamps of Cluster^2D in each layer and the moment when the particle reaches a specific reference plane. This time difference is employed to correct the timestamp of Cluster^2D to an accurate time on a designated reference plane. The front surface of ECAL was used as the reference plane for time correction in this study. This time difference is related to the energy of the incident particle, and the correction results are shown in Fig. 3e and Fig. 3f. Furthermore, if there is a rotation of the module, it results in a longitudinal positional difference of Cell^2Ds at different transverse positions in the same layer. This results in a time difference for the particles to reach different Seed^2Ds on the same layer. Hence, it is necessary to compensate for this time difference based on the longitudinal position difference of Seed^2Ds as follows: $\begin{array}{l}{t^{\prime }}_{{\text{Seed}}^{\text{2D}}}=\left(t_{{\text{Seed}}^{\text{2D}}}+\frac{z_{{\text{Seed}}^{\text{2D}}}-z_{\text{layer}}}{v_z}\right)\mathrm{,}\hfill \end{array}$ (3) where ${t^{\prime }}_{{\text{Seed}}^{\text{2D}}}$ represents the compensated time of Seed^2D and vz represents the velocity of the particle along the direction of the beam pipe.

After obtaining the corrected position and time information of Cluster^2Ds, the time and position information of Cluster^3Ds are obtained by weighting the time or position information of Cluster^2Ds as follows: $\begin{array}{c}V=r\text{or}t\mathrm{,}\\ W_i\left(V\right)=\frac{1}{Re{s}_i{\left(V\right)}^2}\mathrm{,}\\ V_{{\text{Cluster}}^{\text{3D}}}=\frac{{\displaystyle {\sum }_{i=1}^n\left(V_{{\text{Cluster}}_i^{\text{2D}}}\times W_i\left(V\right)\right)}}{{\displaystyle {\sum }_{i=1}^n{W}_i\left(V\right)}}\mathrm{,}\end{array}$ (4) where r represents the position, t represents the time, i represents the number of layers, and Resi represents the resolution of r or t in layer i, as shown in Fig. 6b and c.

Splitting of overlapping clusters

When two Cluster^3Ds overlap in an event, as shown in Fig. 4a, it is necessary to conduct energy splitting on the shared Cell^2Ds to ensure accurate reconstruction of the energy and position of the Cluster^3Ds. In the layered reconstruction framework, energy splitting of the overlapping Cluster^3Ds is performed at the Cluster^2D level. The general logic of the algorithm is as follows. First, determine if the two Cluster^3Ds share any Cell^2Ds. If they do, distribute the energy of the shared Cell^2Ds between the respective Cluster^2Ds. Upon completing the energy splitting of Cell^2Ds, reevaluate the information of Cluster^2Ds based on the updated energy of Cell^2Ds, and remake the necessary corrections.

Fig. 4Full-screen View

(a) Two overlapping clusters shared the same cell. (b) The transverse shower profile, where the y axis is the fraction of the energy of the Cell^2Ds to the energy of the Cluster^2Ds, and the x axis is the distance between the center of the Cell^2Ds and the center of the Cluster^2Ds

Currently, the energy splitting is determined by the transverse shower profile obtained from MC truth information. This study provides a layered description of the transverse shower profile at the Cell^2D level. As shown in Fig. 4b, the transverse shower profile is represented by the distance of Cell^2Ds from Cluster^2Ds on the x-axis and the energy fraction of Cell^2Ds relative to Cluster^2Ds on the y-axis.

When Cell^2D is shared by two Cluster^2Ds, the distribution of Cell^2D energy to each Cluster^2D is evaluated in two steps. First, the energy fraction of the shared Cell^2D corresponding to each Cluster^2D is calculated based on the distance between Cell^2D and Cluster^2D, as depicted in Fig. 4b. Second, the estimated energy from each of the two Cluster^2Ds to Cell^2D is computed using the fraction established in the first step and the energy of the Cluster^2Ds. At this juncture, the total estimated energy from the two Cluster^2Ds to the shared Cell^2D exceeds that of the shared Cell^2D. The energy of the shared Cell^2D is then distributed between the two Cluster^2Ds, using the calculated estimated energy as the weighting factor. Subsequently, the energy and position of Cluster^2D are recalculated, and the aforementioned procedures are repeated. Upon stabilizing the splitting weights of the shared Cell^2D for the two Cluster^2Ds, this iterative process completes and finalizes the energy splitting of the shared Cell^2D.

The precision of the energy fraction contributed by Cell^2D to each Cluster^2D relies heavily on the accuracy of the transverse shower profile. As depicted in Fig. 4b, even when the distances between Cell^2Ds and their respective Cluster^2Ds are identical, the energy contribution from Cell^2Ds to Cluster^2Ds can vary significantly and even display multiple peaks, particularly at distances of approximately 0.5 [Cell]. Therefore, it is essential to categorize the data points in the left-hand plot in Fig. 4b to achieve narrower fraction ranges corresponding to the same distance within a single category, as well as a singular peak in the energy fraction.

The classification methods used in this study are listed in Table 1 and the classification results are shown in the right plot of Fig. 4b. First, whether this Cell^2D is a Seed^2D should be determined. This is because the random scattering direction of the initial electron pairs affects which Cell^2D near the hit point has the opportunity to receive more energy. A Cell^2D deposited with the maximum energy was constructed as a Seed^2D in the reconstruction process. Additionally, because the readout unit (Cell^2D) is not circular, the position of the hit point relative to the edges or corners of Seed^2D affects the energy fraction of Seed^2D when the distance to the hit point remains unchanged. Finally, the position of the hit point and the Cell^2Ds relative to Seed^2D in the positive or negative direction of the particle’s transverse momentum also affect the energy fraction, which can be described as being relatively close to or far from the beam pipe in LHCb. In practical operations, the hit position of the particles in each layer is substituted by the position of Cluster^2Ds.

Searching the second Seed^2D in Cluster^2Ds

During the reconstruction process, a merged π⁰ indicates that the generated photon pair was reconstructed as one Cluster^3D in ECAL. This indicates that energy deposition from π⁰ produces only one local maximum energy Cell^3D. For the unlayered-reconstruction framework in some experiments [37, 38] where layered readout information is lacking, it is common practice to select one of the non-seeded cells in the cluster with the highest energy cell as the second seed. A new cluster is then constructed around the second seed as the center. After splitting the energy of the shared cells between the new and original clusters, the overlapping clusters are referred to as subcluster pairs of the original cluster.

As shown in Fig. 5a, a merged π⁰ does not necessarily result in "merged" Cluster^2D in each layer. According to the cell-energy-splitting algorithm described above, the success rate of splitting increased as the number of cells shared by the two clusters decreased.

Furthermore, it is necessary to base the selection of the second Seed^2D on the energy of Cell^2Ds. This is because the energy of Cell^2Ds are also related to the distance from the hit point of the photon. When one of the photons resulting from the decay of a π⁰ has a much higher energy than the other photon in non-Seed^2D Cell^2Ds, the Cell^2D closer to the higher-energy photon may have a higher energy than the Cell^2D closest to the lower-energy photon. Solely focusing on the energy of Cell^2D may result in misidentifying the second Seed^2D.

Hence, motivated by the aforementioned reasons, the layered reconstruction framework described in this study involves a two-step process for identifying the second Seed^2D to enhance the accuracy of splitting Cluster^2D and ultimately improve the efficiency of π⁰ reconstruction.

The first step is to search for Cell^2D, except for Seed^2D, which has the highest energy E’ in Cluster^2D as Cell^2D’. The definition of E’ is given by the following equation: $E^{\prime }=\frac{E\left({\text{Cell}}^{\text{2D}}\right)}{Frc}\mathrm{,}$ (5) where Frc is the estimated ratio of Cell^2D obtained from the relationship shown in Fig. 4b. The second step is to determine whether there is any local energy maximum Cell^2D other than Seed^2D in the neighboring Cell^2Ds of Cell^2D’ determined in the first step. If, then Cell^2D with this local energy maximum is taken as the second Seed^2D; otherwise, Cell^2D’ is taken as the second Seed^2D. Subsequently, the second Seed^2D is used to construct a new Cluster^2D and energy splitting for the shared Cell^2D and correction for Cluster^2D are performed as detailed in the previous sections.

Cluster^2D matching

After completing the preliminary algorithm, a sub-Cluster^2D pair was obtained for each layer. The algorithm described in this section aims to match the sub-Cluster^2Ds in different layers and finally obtain two sub-Cluster^3Ds. The subCluster^3D pair is regarded as a photon generated by the merged π⁰. The matching of Cluster^2D on different layers directly affects the accuracy of the reconstruction of the final sub-Cluster^3D. The utilization of multidimensional information from different readout layers facilitates a more accurate Cluster^2D matching.

First, the energy of the sub-Cluster^2Ds in each layer was used for prematching. Because the ratio of the deposition energy in each layer is related to the energy of the incident particles[25], unreasonable matches can be filtered based on the energy ratios between Cluster^2Ds and the energy of Cluster^3D as shown in Fig. 2b. For example, in the case of a dual-layer ECAL, the energy of the Cluster^2Ds in the front layer is used to calculate the energy of the Cluster^2Ds in the other layers, and if the energy of the Cluster^2Ds in the other layers exceeds the calculated value by $3\sigma$, the pre-matching is considered a failure, and the matching result is filtered.

If there is more than one Cluster^2D in a certain layer that is prematched with the front layer, the final match is made based on the positional relationship. First, Cluster^2D in the front layer is connected to the initial vertex (typically, the zero point). The connection line is extended and projected onto these layers, and the closest pre-matched Cluster^2Ds to the projection point are matched with the front layer Cluster^2D to complete the matching and obtain sub-Cluster^3D.

Finally, to avoid the resolution of π⁰ being reconstructed in a merged model, Seed^2Ds in all layers of sub-Cluster^3D must be checked to determine whether they are included in the direct reconstruction Cluster^3D. If so, the current process of the merged π⁰ reconstruction is terminated, and a new process starts by skipping the next directly reconstructed Cluster^3D.

Energy resolution

The Fig. 6a illustrates the energy resolution versus the incident energy. This relationship can be described by the following equation: $\sigma (E)/E=\frac{a}{\sqrt{E}}\oplus b\mathrm{,}$ (6) where a represents the statistical fluctuations of the readout signal (e.g., photoelectrons), b represents the constant term errors owing to uncertainties in the calibration and correction, as well as inhomogeneities in the active material, and ⊕ represents the sum of squares. The energy resolution and bias as functions of energy illustrate the stability of the layered reconstruction framework. In addition, as shown in Figs. 7, the layered reconstruction algorithm demonstrates a better energy resolution for particles with large incident angles. However, because of the forward detector design and layout of PicoCal [33], there are fewer single-photon events with large incident angles. Therefore, this improvement was not significantly reflected in the energy resolution as a function of the incident particle energy.

Fig. 6Full-screen View

(Color online) The resolution and bias relative to the true value in regions 4–7. σ and μ are derived from the Gaussian fitting, where σ represents the standard deviation (used to denote the resolution) and μ is the mean value (used to denote the bias). (a) The energy resolution and bias, where ΔE is the difference between the true energy and the reconstructed energy, and E is the true energy. (b) The position resolution and bias of front and back layers, where Δr is the difference between the true position and the reconstructed position. (c) The time resolution and bias, where Δt is the difference between the true time and the reconstructed time. (d) The position resolution and bias, where Δr is the difference between the true position and the reconstructed position

Fig. 7Full-screen View

(Color online) The energy resolution versus angle θ in regions 4-7. Here, θ is the angle between the particle motion direction and beam direction. ΔE represents the difference between the true and reconstructed energies, and E represents the true energy. σ is the standard deviation derived from the Gaussian fitting

Position resolution

The position resolution is a critical parameter of the ECAL and is, to some extent, equivalent to the angular resolution. The positional resolutions are shown in Fig. 6b and d, respectively, where Front and Back represent the front and back layers, respectively, r represents the position. The development stages of the shower and readout layers change as the particles undergo showering and deposit energy along their momentum direction. However, the tendency of the reconstructed Cluster^2D’s raw position, relative to the true position, varied between the different layers. Essentially, if we only have the energy-weighted position information of Cell^2Ds located in different layers, the raw position will spread out in the transverse plane owing to different tendencies. Consequently, utilizing an unlayered-reconstruction algorithm and an overall correction parameter for reconstructing the transverse position will lead to degradation of the position resolution owing to this spreading effect. In contrast, the reconstruction in this study effectively resolved the previously mentioned issue and improved the positional resolution, as illustrated in Fig. 6d. This enhancement results in an improvement of approximately 0.5 mm in the high-energy region, representing a 33% increase compared with the unlayered-reconstruction algorithm.

Time resolution

Time is essential for event reconstruction and data analysis in high-luminosity environments. Within the layered reconstruction framework, time information can be provided in the form of Cluster^3D or along the longitudinal direction using Cluster^2D, as shown in Fig. 6c. The integration of layered time information provides new analytical perspectives for future physical investigations. Ongoing research also focuses on exploring the application of time information for reconstruction, correction, and analysis.