Muon Tracking with the fastest light in the JUNO Central Detector
NNoname manuscript No. (will be inserted by the editor)
Muon Tracking with the fastest light in the JUNO Central Detector
Kun Zhang · Miao He · Weidong Li · Jilei Xu
Received: date / Accepted: date
Abstract
The Jiangmen Underground Neutrino Observatory(JUNO) is a multi-purpose neutrino experiment designed tomeasure the neutrino mass hierarchy using a central detec-tor (CD), which contains 20 kton liquid scintillator (LS) sur-rounded by about 18,000 photomultiplier tubes (PMTs), lo-cated 700 m underground. The rate of cosmic muons reach-ing the JUNO detector is about 3 Hz and the muon inducedneutrons and isotopes are major backgrounds for the neu-trino detection. Reconstruction of the muon trajectory in thedetector is crucial for the study and rejection of those back-grounds. This paper will introduce the muon tracking al-gorithm in the JUNO CD, with a least squares method ofPMTs’ first hit time (FHT). Correction of the FHT for eachPMT was found to be important to reduce the reconstructionbias. The spatial resolution and angular resolution are betterthan 3 cm and 0.4 degree, respectively, and the tracking effi-ciency is greater than 90% up to 16 m far from the detectorcenter.
Keywords
JUNO, Central Detector, Muon Tracking, Firsthit time, Least squares method
The Jiangmen Underground Neutrino Observatory [1,2] is amultiple purpose neutrino experiment to determine neutrinomass hierarchy and precisely measure oscillation parame-ters, using reactor antineutrinos from Yangjiang and Tais-han nuclear power plants. Fig. 1 shows the schematic view
Kun Zhang · Miao He · Weidong Li · Jilei XuInstitute of High Energy Physics, Chinese Academy of Sciences, Bei-jing 100049, ChinaE-mail: [email protected] Miao HeKun Zhang · Weidong LiUniversity of Chinese Academy of sciences, Beijing 100049, China
Fig. 1
Schematic view of the JUNO detector of the JUNO detector. Twenty kiloton LS is contained ina spherical vessel with the radius of 17.7 m as the centraldetector (CD). The light emitted by the LS is watched byabout 18,000 20-inch PMTs installed in the water pool, withthe photocathode at a radius of 19.5 m, with more than 75%optical coverage. There is a top tracker made of plastic scin-tillator bars, on top of the water pool.In the JUNO detector, cosmogenic radioactive isotopes –especially Li and He – and fast neutrons are serious corre-lated background sources to reactor antineutrinos, which canbe efficiently rejected by sufficient time veto after the taggedmuons. For example, the time window to veto Li and He isrequired to be no less than 1.2 s, according to their lifetimes.However, according to the Monte Calor simulation, the rateof muon reaching CD is about 3 Hz with the mean energyof about 215 GeV. If the full LS volume is vetoed, there isalmost no live time in the detector. Since the vertex of the a r X i v : . [ phy s i c s . i n s - d e t ] M a r Kun Zhang et al. cosmogenic isotopic is correlated with the primary muon inboth time and space, the most effective way is to only veto acylindrical volume along the muon trajectory instead of thefull detector. This approach requests precise tracking of themuon. The top tracker above CD can also track muons butit has small coverage and can only measure 25% of the totalcosmic muons, therefore, muon reconstruction in the CD isnecessary for the entire 4 π solid angle.This paper introduces the reconstruction of muons in theJUNO CD with the time signal of PMTs. The algorithm isdescribed in detail in Sec. 2, including the parameterizationof a track, the prediction and correction of the first hit time(FHT), and the minimization with the least squares method.The reconstruction performance is shown in Sec. 3 and theconclusion is in Sec. 4. x inj , y inj , z inj , t inj , θ p , φ p , and l trk . x inj , y inj , z inj and t inj are the position andtime of the muon injecting into the LS with the center of theLS ball as the origin of coordinates. θ p and φ p are the direc-tion of the muon track in spherical coordinates, and l trk isthe length of muon trajectory in LS. Muons that are gener-ated outside the detector and stopped inside are very rare inJUNO and are not considered in this paper. For muons goingthrough the detector, the injection and outgoing points canbe fixed at the surface of the LS sphere thus x inj , y inj , z inj and l trk can be replaced with another two parameters θ inj and φ inj .Then the FHT of PMT at position (cid:174) R i can be predicted bythe tracking parameters with a proper optical model: T prei = f (cid:16) (cid:174) R i ; θ inj , φ inj , t inj , θ p , φ p (cid:17) (1)Details about the model will be discussed in Sec. 2.2.Then with the predicted FHT and the observed value, the χ can be built as: χ = (cid:213) i (cid:32) T prei − T obsi σ i (cid:33) , (2)where T obsi is the observed (from data or MC) value ofFHT and σ i represents the error of FHT for the i th PMT.The reconstructed track parameters are obtained by min-imizing the χ function. Minuit2 in root package is usedto minimize χ , and initial values of the parameters are re-quired. To get them, the position of the PMT which has theearliest FHT is regarded as the injecting point and its FHT Fig. 2
Schematic view of fastest light. (cid:174) R is the injecting position ofthe track into the LS, (cid:174) R i is the position of the i th PMT, and (cid:174) R c is theposition where the fastest light emitted for the PMT (cid:174) R i as the injecting time, and the charge center of all PMTs (cal-culated by Eq. 3) is calculated as the muon trajectory centerin LS. With this information, the initial values of all the pa-rameters can be inferred. (cid:174) R = (cid:205) i q i (cid:174) R i (cid:205) i q i (3)where q i and (cid:174) R i are the charge and position of the i th PMT seperately.2.2 Fastest light modelAs shown in Fig. 2, when a muon travels through the cen-tral detector, energy is deposited in the LS and scintillationlights are emitted isotropically along the muon track. Thereare also Cherenkov lights, however, most of them are ab-sorbed by the LS and re-emitted as scintillation lights. Asa result, only less than 1% of lights detected by PMTs car-ries the directional information, which is extremely difficultto separate from scintillation lights. Therefore, all lights aretreated as isotropic in our reconstruction.After emission, the optical photon travel in the LS andwater, and there is a certain probability for it to reach a PMT.For a specified PMT, among all the photons hit on it, theearliest one is defined as the fastest light of this PMT, andthe time of the fastest light hit on the PMT is defined as thefirst hit time (FHT).Given the time of muon injection t inj , the time of a pho-ton arriving at a certain PMT can be calculated as [3]: t = t inj + t muon + t photon , (4)where t muon is the muon propagation time before thelight emission, and t photon is the time of flight of this photon. uon Tracking with the fastest light in the JUNO Central Detector 3 Defining the refractive index of LS as n LS , the muon veloc-ity as c µ , and the light velocity as c , it is straight forward toget t muon = lc µ , (5)and t photon = (cid:12)(cid:12)(cid:12) (cid:174) R i − (cid:16) (cid:174) R + l ˆ V (cid:17)(cid:12)(cid:12)(cid:12) c / n LS . (6)Here l means the distance of the muon travelling beforeit emits that photon, and ˆ V is the unit vector of the muontrack direction. Eq. 4 can be rewritten as t = t inj + lc µ + (cid:12)(cid:12)(cid:12) (cid:174) R i − (cid:16) (cid:174) R + l ˆ V (cid:17)(cid:12)(cid:12)(cid:12) c / n LS , (7)and the minimum of t can be obtained from the partialderivative with respect to l : ∂ t ∂ l = c µ + n LS c · ∂ (cid:12)(cid:12)(cid:12) (cid:174) R i − (cid:16) (cid:174) R + l ˆ V (cid:17)(cid:12)(cid:12)(cid:12) ∂ l = c µ − n LS c · cos θ, (8)where (cid:174) R ci = (cid:174) R i − ( (cid:174) R + l ˆ V ) means the vector from theemitting point to the PMT and ˆ R ci is its unit vector, and θ is the angle between ˆ R ci and ˆ V . The calculation relatedwith the vector’s derivative in Eq. 8 can refer to Appendix A.Considering for ultra-relativistic muons, c µ ≈ c , to minimize t , there is: cos θ = n LS (9)This means that for each PMT, the position where thefastest light emitted can be determined by the angle θ , whichis the same as the emission angle of the Cherenkov light.There is an exception that the calculated point R c is out ofLS. In this case, the fastest light should come from the in-jecting point of muon into the LS R .2.3 Residual of the first hit timeMonte Calor simulation was done based on Geant4 with theJUNO detector geometry. A muon with 200 GeV kinetic en-ergy was simulated going through the detector. The simula-tion software is a part of the JUNO offline software, and allthe geometry parameters and optical properties used in thedetector simulation are from JUNO Yellow Book[2].The first hit time of each PMT obtained from MC truthwas subtracted by the predicted time of the fastest light un-der the optical model in Sec. 2.2 (using the track parametersin MC truth), and the distribution ∆ FHT is shown in Fig. 3.
FHT [ns] D -15 -10 -5 0 5 10 15 E n t r i e s Fig. 3
Residuals of FHTs of a MC muon event, with the TTS of PMTset as 3 ns. ∆ FHT = T obs − T pre There is not only fluctuation of ∆ FHT, but also an overallshift of the central value. There are a few reasons as below: – The scintillating light is not emitted instantaneously. In-stead, the decay time follows a double-exponential dis-tribution, with the time constants of 4.93 ns and 20.6 nsfor the fast and slow components respectively. – Because of different refractive index between LS andwater, there should be reflection and refraction at theboundary of the LS ball, which are ignored in the op-tical model. In particular, when the light emission pointis at the edge of the LS sphere, there is a region in whichall PMTs can not see the light because of the total re-flection. This effect is not included in the optical modeleither. – There is an intrinsic transient time spread (TTS) of pho-toelectrons in the PMT. In this study, σ TTS was assumedto be 3 ns, i.e. a 3 ns Gaussian smearing was added tothe hit time of each photoelectron in MC. – Scintillating lights are not infinite, and their emitting pointsare discrete along the track after all, which means thatthe fastest light for a given PMT does not have to comefrom the predicted position. Assuming the dE/dx of muonis 0.2 MeV/mm, and the average number of photoeletronscollected by all PMTs correlated with every MeV energydeposit in the CD center is about 1,200, considering thetotal number of PMT (20 inch) is about 18,000, along thetrack about × . / ≈ . p.e./mm ( p . e . ≡ photoelectron ) can hit on each PMT on average. Thismeans that on average the muon emits a photoeletron hitfor one PMT when flying every / . ≈ . mm long,which can be considered as the mean uncertainty of theposition at which the fastest light is emitted, and thecorresponding uncertainty of FHT is about mm / c µ ≈ . ns, which is acceptable. From another view, if wewant the uncertainty of FHT smaller than 1 ns, we needthe muon emitting 1 p.e. when flying every c µ × ns foreach PMT. And for a trajectory through the CD center,with the track length of about 35,000 mm, this means Kun Zhang et al.
Fig. 4
Definition of correcting factors d and a that mm /( c µ × ns ) ≈
120 p . e . are needed for eachPMT. This is a strict PMT selecting condition when re-constructing and 100 p.e. cut condition is used in theperformance study in Sec. 3. – For a cosmic muon, there are multiple photoelectrons ineach PMT. Assuming all hits are from the same pointsource and the probability density function (PDF) of thesingle hit time, after the time-of-flight subtract, is f ( x ) ,then the PDF of FHT in case of n photoelectrons is: F ( t , n ) = n f ( t ) (cid:16)∫ + ∞ t dx f ( x ) (cid:17) n − . However, for a muon,photons may be from any point along the track, thus thePDF of FHT can not be analytically expressed.As a result, an additional correction to the predicted FHTis needed, as well as its fluctuation.2.4 Correction to the first hit timeIn this study, MC muon samples were used to correct the firsthit time of each PMT, taking into account the full detectorgeometry, with all optical parameters taken from Ref. [2]. Inreality, there is a top tracker which can detect a small frac-tion of cosmic muons going through the LS ball with verygood tracking resolution, which can be used as a calibrationsource to tune MC.According to the analysis above, the FHT biases stronglydepend on the number of Photoelectrons collected on eachPMT. In addition, the relative position between the PMT andthe track also have significant impacts on the FHT distribu-tion. We defined some parameters as below to identify eachPMT.- ddd : as shown in the left panel of Fig. 4, the distance be-tween the injecting point to the projection of the PMTfalling at the track, which ranges from -19.5 m (whenthe track is at the edge of the LS ball, half of PMTs havenegative d ) to 37.2 m (when the track goes through thecenter of CD).- aaa : the azimuth angle of each PMT, in the plane perpen-dicular to the track, which ranges from to π (the area − π ∼ can be combined into that of ∼ π for the sym-metry). - DDD ((( iiisssttt ))) : the distance between the track and the CD cen-ter.- qqq : the number of Photoelectrons.Apparently, with a given d , q and a are correlated, thuswe tried two different combinations a - d and a - d to studythe correction of FHT. The correcting factors ∆ T pre , definedas the mean of ∆ FHT = T obs − T pre , as a function of q - d and a - d , are shown in Fig. 5 (a) and (b), respectively.From Fig. 5(a), we can see that the correcting factor be-comes small when q increases as expected, because of thefirst hit selection. On the other hand, given a fixed q , the cor-recting factor changes with d , because the detected photonshave different emission points along the track. In Fig. 5(b),the left top region corresponds to the total reflection areawhere the first hit can not come from the predicted emis-sion point thus have a latency, while the top middle regionis furthest from the track so the PMTs have least q .With these samples we can also obtain the standard de-viation of FHT in every bin, which can be treated as theeffective error of FHT σ eff i in the χ function Eq. 10. χ = (cid:213) i (cid:32) T prei + ∆ T prei − T obsi σ eff i (cid:33) (10)Since both d and a rely on the tracking parameters, inprinciple, they have to be re-calculated in every iterationduring the minimization process. However, the variation ofthe correcting factors and the effective error of the FHTmakes the minimization unstable and difficult to converge.Therefore, to solve this technical problem in practice, weonly run the correction and minimization twice and we foundthe results are acceptable. α whichis the angle between the reconstructed track and the MCtruth one, and ∆ D which means the error of reconstructed D ( ist ) .Fig. 6 compares the different performances with differ-ent FHT correcting methods. We can see the mean α and D ( ist ) without any correction can reach several degrees anddozens of centimeters respectively. And the only q basedcorrecting method can not improve the performance too much.Correcting methods “corr1d XXX” means correcting theFHT by the parameters XXX one by one with relevant one-dimensional correcting curves, while the “corr2d XXX” meth-ods are correction by all the two correcting-parameters at the uon Tracking with the fastest light in the JUNO Central Detector 5 F H T [ n s ] D -10-5051015d [m]-10 -5 0 5 10 15 20 25 30 35 p . e . ] · q [ (a) ∆ FHT vs q and d F H T [ n s ] D -10-8-6-4-202d [m]-10 -5 0 5 10 15 20 25 30 35 a [r a d ] (b) ∆ FHT vs a and d Fig. 5 ∆ FHT vs the correcting factors. Samples are MC muon tracks5.5 m away from the CD center with initial momentum 200 GeV andtime resolution 3 ns same time with a corresponding two-dimensional correctingmap.It is easy to see that the four kinds of correction methodsby 2 parameters perform better. And the best one is ”corr2da-d” method, with average α angle smaller than . ◦ andaverage ∆ D ( ist ) smaller than 1 cm.3.2 Reconstruction resolution and efficiencyIn this section, the spatial and angular resolution and thetracking efficiency are shown using the correction method”corr2d a-d”.Fig. 7 shows the reconstruction resolution and the effi-ciency. The “injecting x,y,z” means the position of the trackinjecitng into the LS. And the resolutions are figured out by agaussian fitting on data in every bins. From Fig. 7 (a) we cansee the spatial resolution is better than 3 cm and the angularresolution is better than 0.4 degree, up to 16 m far from thedetector center. Muons going through the edge are very dif-ficult to be reconstructed because the injection and outgoingpoints are too close to be separated. In Fig. 7 (b), success-ful reconstruction is defined with all the 5 track parametersare in the range of 5 times of their standard deviation. Withthe distance less than about 14 meters of the track to the CDcenter, the tracking efficiency can be greater than 90%, and [m] mc D0 2 4 6 8 10 12 14 16 [ d e g ] a M ea n no correctioncorr1d qcorr1d d,a,d,acorr1d q,d,q,dcorr2d q-dcorr2d a-d (a) Mean value of α vs true D ( ist ) [m] mc D0 2 4 6 8 10 12 14 16 D [ c m ] D M ea n -50050100150 no correctioncorr1d qcorr1d d,a,d,acorr1d q,d,q,dcorr2d q-dcorr2d a-d (b) Mean value of biases of reconstructed D ( ist ) vs true D ( ist ) Fig. 6
Performances of reconstruction with various FHT correctingmethods. Samples are MC data with PMTs’ TTS of 3ns with tracks close to the edge of the LS, the efficiency willdecrease about 10%.
We have developed a reconstruction algorithm of muon inJUNO CD based on the least square method and the fastestlight model, but which has biases on account of the inaccu-rate prediction of the FHT. Considering that the top-trackerin JUNO above the CD can track some tracks downwardsfrom top, we can use them to obtain the FHT-correctiondata vesus related observed correcting-factors, and then withwhich correct the predicted FHT in the reconstruction. Withthis algorithm, we can effectively reconstruct the muon trackin the CD almost without biases and the spatial and angularresolution can be better than 3 cm and 0.5 degree respec-tively. And in most cases the tracking efficiency can be bet-ter than 90%. More studies on the fastest light model will bedone to get better prediction of the FHT and thus to improvethe reconstruction performances.
Acknowledgements
This work is supported by National Natural Sci-ence Foundation of China (Grant No. 11575226, 11605222), Joint LargeScale Scientific Facility Funds of NSFC and CAS (Grant No. U1532258)and the Strategic Priority Research Program of the Chinese Academyof Sciences (Grant No. XDA10010900) Kun Zhang et al. S p a ti a l [ c m ] injecting x D injecting y D injecting z D [m] mc D0 2 4 6 8 10 12 14 16 A ngu l a r [ d e g ] qD fD (a) Reconstruction spatial and angular resolution [m] mc D0 2 4 6 8 10 12 14 16 T r ac k i ng E ff ec i e n c y (b) Reconstruction efficiency Fig. 7
Reconstruction resolution and efficiency with correctionmethod “corr2d a-d”. In order to do comparison easily, the x axes areshifted a little for ∆ injecting-y and ∆ injecting-z in the top figure. Appendix A