Belle II 實驗第一級觸發器中二維軌跡探測器之實現

Implementing the 2D track reconstruction for the Level 1 trigger of the Belle II experiment

盛子安(Tzu-An Sheng)

指導教授 ： 張寶棣

國立臺灣大學/理學院/物理學研究所/碩士(2018年)

https://doi.org/10.6342/NTU201802022

摘要

位處日本筑波的B介子工廠：KEKB正負電子加速器與Belle實驗，透過研究B介子衰變中，弱作用之電荷對稱宇稱破壞的現象，奠定了小林──益川理論的實驗基礎，並且促成2008年的諾貝爾物理獎。為了從稀有衰變中探究粒子物理標準模型以外的新物理，此工廠正升級為SuperKEKB加速器與Belle II實驗，將加速器瞬時亮度提升至8×10^35 cm^−2 s^−1（原先的40倍）。然而在Belle II偵測器中，資料擷取的速率上限僅為每秒3萬次，並不能紀錄新亮度之下所有的對撞事例。實際上，具研究價值的Υ介子、B介子及τ子等事例僅佔所有對撞事件的數個百分比。另外還有許多偵測器反應並非對撞事件，而是源自加速器中帶電粒子簇的散射、同步輻射、或是粒子與真空管線中殘餘空氣分子碰撞等背景雜訊。為了在資料擷取的速限之下盡可能紀錄所有珍貴的事例，Belle II實驗勢必得仰賴一套基於硬體的即時觸發系統，提供高效率、低延遲、無死區時間的事例判別，使資料擷取系統得以忽略背景事例，不至受到掣肘。

由於多數背景事例不會在碰撞點附近產生具高橫向動量的帶電粒子，這樣的粒子便成為判別背景事例的關鍵。因此，Belle II實驗將帶電粒子軌跡觸發器重新改造，以因應加速器亮度提升。在高能加速器實驗中，透過辨認帶電粒子通過偵測器時，在數十處感應線留下的電流訊號，我們得以重建帶電粒子的空間軌跡。由於偵測器內通有縱向磁場，我們亦可藉螺旋軌跡推知粒子的動量。掌握帶電粒子的數量、動量等資訊，並輔以量能器能量團與帶電粒子軌跡的空間對應關係，便能輕易地區分目標事例與背景事例的差別。

帶電粒子留下的電流訊號經過數位化後成為擊打訊號，輸入至軌跡觸發器。軌跡觸發器首先將偵測器中相鄰的擊打訊號組成區段擊打訊號。每個帶電粒子軌跡由最多9層的區段擊打訊號所組成，其中5層包含了三維的粒子螺旋軌跡在偵測器橫段面上所投影出的二維圓弧軌跡訊號。這5層訊號的幾合位置透過共形變換以及霍夫變換後，在軌跡參數空間中形成許多三角函數曲線。藉由尋找參數空間中4條以上來自不同層的曲線交點，可知幾何空間中四層以上共圓弧的區段擊打訊號，與該圓弧所對應的粒子橫向動量之大小及方向。將橫向動量與剩餘四層包含粒子螺旋軌跡縱向資訊的區段擊打訊號結合後，即可推知完整的三維軌跡。

前述尋找區段擊打訊號、尋找二維軌跡及尋找三維軌跡的步驟皆由各別的硬體模組所實現。另外,軌跡觸發器還包含了整合最前端感應線擊打訊號的模組。各模組之間由光纖傳輸連接。本論文著重於將上述由二維區段擊打訊號尋找二維軌跡的演算法,以現場可程式化邏輯閘陣列實現。實現後的邏輯延遲為11個時脈周期(相當於350奈秒,不包含傳輸所需的延遲)。透過測量宇宙射線事例,並與更精密的軟體軌跡重建方法比較後,我們推估對於所有橫向動量在0.5GeV以上、與碰撞點徑向距離小於1公分、含有4個以上區段擊打訊號、並且不受前端模組錯誤影響的所有軌跡,二維軌跡尋找效率在一個標準差之下的信心區間完全落在98%以上。

本論文同時紀錄了二維軌跡擬合的實現方法。這個方法利用軌跡偵測器中由高能帶電粒子碰撞氣體分子游離出的電子，以及由該電子游離出的次級電子在電場中的的飄移速度，通過測量飄移時間，推算出更精確的軌跡區段擊打位置，並且以最小平方法擬合得出更精密的二維軌跡。由於這個步驟將會併至更後端的三維軌跡擬合模組中實現，並且包含大量需藉由查表實現的運算步驟，因此在不喪失計算精確度的前提下降低記憶體用量便成為最大的挑戰。我們發展了複和式的查表方法，並利用三角函數的對稱性減低記憶體用量。另外，本論文也包含數項對建立光纖傳輸資料流穩定性的改善。尤其透過以特定時間間隔重置位於晶片同一側的光纖收發器，我們得以在更高的傳輸速率下提升建立傳輸資料流的穩定性。

關鍵字

Belle II 實驗 ； 模式辨認 ； 粒子軌跡 ； CP 破壞 ； 觸發器 ； 現場可程式化邏輯閘陣列

並列摘要

The Belle experiment at the KEKB collider in Tsukuba, Japan is a B meson factory designed to operate at a center-of-mass energy of 10.58 GeV, the mass value of Υ(4S). It is undergoing an upgrade that will boost its instantaneous luminosity to 8×10^35 cm^−2 s^−1 (40 times higher than before), whereas the maximum acceptable event rate for the data acquisition system is only 30kHz. Most of the detector responses arise from the scattered particles with other particles in the accelerated bunch, or with the residual gas molecules in the vacuum beam pipe. Furthermore, only a few percent of the total number of e+ e− collisions correspond to Υ, B or τ events. The rest are considered backgrounds and must be either suppressed or prescaled in real time without losing too many signal events. To achieve this goal, a hardware-based online trigger system with good background suppression, high efficiency, low latency and no dead time is indispensable.

In experimental particle physics, tracking refers to the pattern recognition process that searches for the trajectories of charged particles by analyzing the traces they leave on the detector. Once the trajectory, or the track, is reconstructed, the momentum and the charge is also determined. High-precision tracking provides crucial information for telling signals from backgrounds, since most background events don't produce charged particles with enough transverse momenta near the collision point. As a result, the track trigger in Belle II is redesigned to accommodate the dramatic increase of luminosity and background rate.

The track trigger starts from relating adjacent wire hits in space and in time from a drift chamber, grouping them into maximally 9 segments of a track. Out of the 9 segment hits, 5 are groups of sense wires parallel to the beam axis, and thus their positions contain information of the track projected onto the 2-dimensional plane perpendicular to the beam axis. The track trigger then detects the coincidence of several axial track segments by transforming their radial and angular positions to a parameter space with a conformal map followed by a Hough map, and looking for their intersections there. Each segment in one layer of the detector cylinder contributing to the track is extracted. Afterwards, it fits these positions with the drift length, and reconstructs the track's projection in the plane perpendicular to the beam axis. Finally, by combining the 2D track information with the remaining track segments which contains the information of longitudinal position, the vertex position along the beam axis is reconstructed. Each of these steps is a separate module in the track trigger system. This thesis focus on implementing the steps of finding and reconstructing the 2D track using an algorithm developed by our collaborator.

The 2D tracker module is implemented on 4 printed circuit boards with field programmable gate array (FPGA) and 10 Gbps optical I/O connection to both upstream and downstream modules. It has a latency of 11 data clocks (352 ns) excluding the transmission time. The lower bound of the 1-σ confidence interval of its tracking efficiency is measured to be more than 98% for cosmic ray tracks with radial impact parameters smaller than 1 cm, pt > 0.5 GeV, with at least 4 track segment hits, and coming from regions with expected track segment finding efficiency.

This thesis also outlines the implementation of the 2D fitter, which involves fitting an arc to the positions of the axial track segment hits corrected by their drift lengths. As the fitting contains many fixed-point arithmetic operations implemented as look-up tables, it is crucial to reduce the usage of the block RAM while maintaining similar arithmetic precision. Composite look-up tables, which increase the precision in the worst-performing part of the arithmetic function's range by sacrificing the unnecessary precision in other parts, are developed to meet the requirement. Lastly, several improvements are made to stabilize the buildup process of the optical transmission data flow. In particular, an automatic way to reset different optical transceivers on the same side of the die, separated with an adjustable time interval, is tested to make the buildup more stable at the full 10 Gbps lane rate.

並列關鍵字

Belle II ； tracking ； CP violation ； trigger ； FPGA

參考文獻

[1] M. Huschle et al. “Measurement of the branching ratio of B ̄ → D (∗) τ − ν ̄ relative to B ̄ → D (∗) l − ν l ̄ decays with hadronic tagging at Belle”. In: Phys. Rev. D92.7 (2015), p. 072014. DOI: 10.1103/PhysRevD.92.072014. arXiv: 1507.03233 [hep-ex].

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[5] Paul AM Dirac. “The quantum theory of the electron”. In: Proc. R. Soc. Lond. A 117.778 (1928), pp. 610–624.