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the sudbury neutrino observatory ( sno ) was a large water cherenkov detector optimized for detecting solar neutrinos created from the @xmath1b reaction in the main @xmath2 fusion chain .
in addition to solar neutrinos , the sudbury neutrino observatory was also sensitive to high - energy muons that traverse the volume of the detector .
a small fraction of these events are neutrino - induced muons from atmospheric neutrinos , while the large remaining fraction come from cosmic rays created in the upper atmosphere .
it is possible to discriminate between these muon sources by looking at the angular distribution of incoming muons .
the combination of large depth and the relatively flat topography in the vicinity of the detector attenuates almost all cosmic ray muons entering the detector at zenith angle @xmath3 .
the study of muon events in the sno detector provides measurements of the absolute flux of atmospheric neutrinos and constraints on the atmospheric neutrino mixing parameters @xmath4 and @xmath5  @xcite . while the latter measurement is more strongly constrained by other experiments  @xcite ,
the former is unique to the sno experiment .    to facilitate a clean measurement of the zenith distribution of muons entering the sno fiducial volume ,
an accurate understanding of the muon reconstruction algorithm is necessary .
this includes both the angular and spatial resolution of high - energy muons which enter the detector . determining the accuracy of the muon tracking reconstruction algorithm
, however , relies almost entirely on monte carlo simulations .
although the detector response to muons was benchmarked against selected cosmic - ray data , there is not an external calibration source that can provide a consistency - check to the accuracy of the reconstruction algorithm .
this is in sharp contrast to the case for sno s response to neutrons and low energy electrons , which was calibrated with multiple sources to a precision of @xmath6  @xcite .
we present in this paper a means by which the sno experiment was able to calibrate its muon tracking algorithm via the use of an external muon tracking system .
the external muon system ( emus ) allowed sno to simultaneously reconstruct selected cosmic - ray events in two independent systems , thereby providing a cross - check on the tracking algorithm .
the emus experiment ran for a total of 94.6 live days during the last phase of the sno experiment .
this paper is divided as follows : section [ sec : sno ] describes the main sno experiment , section  [ sec : fti ] describes the sno muon reconstruction algorithm , section  [ sec : emus ] describes the characteristics of the emus apparatus , section  [ sec : data_selection ] describes the criterion for accepting events , and finally section  [ sec : recon ] discusses the analysis used to calibrate the sno tracking algorithm against data taken with the emus system .
the sno detector consisted of a 12-meter - diameter acrylic sphere filled with 1 kiloton of d@xmath7o .
the 5.5-cm - thick acrylic vessel was surrounded by 7.4 kilotons of ultra - pure h@xmath7o encased within a barrel - shaped cavity , 34 m in height and 22 m in diameter .
a 17.8-meter - diameter geodesic structure surrounded the acrylic vessel and supported 9456 20-cm - diameter photomultiplier tubes ( pmts ) pointed toward the center of the detector .
a non - imaging light concentrator was mounted on each pmt to increase the effective photocathode coverage to 54%  @xcite .
the detector is described in detail elsewhere  @xcite .
sno was located in the vale creighton mine in ontario , canada at a depth of 2.092 km ( @xmath8 meters water equivalent ) with a flat overburden . at this depth ,
the muon rate incident over the geodesic sphere and integrated over the seasonal variation is @xmath9/day across an impact area of 216 m@xmath10@xcite .
muons entering the detector produce cherenkov light at an angle of 42@xmath0 with respect to the propagation direction of the muon . cherenkov light and light from delta rays produced by the muon illuminate an average of 5500 pmts , whose charge and timing information are recorded .
the amplitude and timing response of the pmts were calibrated _ in situ _ using a light diffusing sphere illuminated by a laser at six distinct wavelengths @xcite .
this laser ball calibration was of particular relevance to the muon fitter because it provides a timing and charge calibration for multiple photon hits on a single pmt .
other calibration sources used in sno are described elsewhere @xcite .
data taking in the sno experiment was subdivided into three distinct phases for measurement of the solar neutrino flux . in the first phase , the experiment ran with pure d@xmath7o only .
the solar neutral current reaction was observed by detecting the 6.25 mev @xmath11-ray following the capture of the neutron by a deuteron . for the second phase of data taking , approximately 0.2% by weight of purified nacl
was added to the d@xmath7o volume to enhance the sensitivity to neutrons via their capture on @xmath12cl . in the third and final phase of the experiment ,
40 discrete @xmath13he and @xmath14he proportional tubes were inserted within the fiducial volume of the detector .
this enhanced the neutron capture cross - section to make an independent measurement of the neutron flux , by observing neutron capture on @xmath13he in the proportional counters .
results from the measurements of the solar neutrino flux for these phases have been reported elsewhere  @xcite .
the sno muon reconstruction algorithm fits for a through - going muon track based on the charge , timing , and spatial distribution of triggered pmts . using a maximum likelihood method , the fitter is able to determine a variety of muon tracking parameters , including the muon s propagation direction , impact parameter with respect to the center of sno , the total deposited energy , and a timing offset .
the likelihood is defined as : @xmath15 \label{eq : fti_lh}\ ] ] where @xmath16 is the number of detected photons , @xmath17 is the probability of @xmath16 photoelectrons being detected for @xmath18 expected number of detected photoelectrons , @xmath19 is the probability of seeing charge @xmath20 given @xmath16 photon hits , and @xmath21 is the probability of observing a pmt trigger at time @xmath22 given @xmath16 photon hits .
the heart of the fitter lies in the first probability term , which is calculated based on monte carlo simulations .
muons were simulated at discrete impact parameter values with random directions through the detector .
these simulations were used to create lookup tables for how many photoelectrons are expected to be detected by a pmt at a given position with respect to a muon track with a given impact parameter .
the second term further refines the fit by including the charge information from the pmts , and allows an estimate of the total energy deposited by the muon , correcting for offline pmts and the neck of the detector .
this probability was calculated by simulating multiple photon hits on all of the pmts in sno . for a given number of photon hits ,
the resulting charge distribution is modeled as an asymmetric gaussian with the widths extracted from simulations .
this fit model agrees well with the simulations for many photon hits , and acceptably for few photon hits ( see figure  [ fig : q_prob ] ) .
the third term in the likelihood refines the fit by including the pmt timing . for each pmt
, the time residual can be calculated as : @xmath23 where @xmath24 is the recorded time on a given pmt , @xmath25 is the time offset term in the likelihood fit , @xmath26 is the distance the muon travels within the detector before emitting the cherenkov photon , @xmath27 is the speed of light in vacuum , @xmath28 is the distance the cherenkov photon traveled , and @xmath29 is the average speed of light in d@xmath7o / h@xmath7o medium ( 21.8 cm / ns )
. the cherenkov photon is assumed to have an angle of @xmath30 with respect to the muon track , making @xmath26 and @xmath28 well - defined .
the probability of the time residual is modeled as a gaussian centered at zero with corrections to include estimates of prepulsing and late light as a function of the number of photon hits .
the sno muon fitter maximizes the likelihood function for the impact parameter , direction , deposited energy , and timing offset using the method of simulated annealing with downhill simplex  @xcite . after determining the parameters that maximize the likelihood , a set of data
quality measurements are used for background rejection .     the angular difference ( as defined in eq .
[ eq : thetamr ] ) of monte - carlo muon tracks through the sno detector ( solid histogram ) .
the angular distribution is fit to the function outlined in eq .
[ eq : xtwogaus1 ] ( solid line ) .
the results from the fit are given in table  [ tab : muon_recon ] . ]    the muon fitter is found to have good reconstruction accuracy on simulated muons .
figure  [ fig : mcang ] shows the angle @xmath31 between the monte carlo generated muon direction ( @xmath32 ) and the reconstructed muon direction ( @xmath33 ) : @xmath34 this is fit to a weighted double gaussian function : @xmath35 \label{eq : xtwogaus1}\ ] ]    the additional @xmath36-dependence is introduced in order to account for the phase space available .
the fit parameters are summarized in table [ tab : muon_recon ] .
although the tails are non - gaussian , this fit gives a reasonable estimate for the uncertainty for the angular resolution .
figure  [ fig : mcimp ] shows the impact parameter reconstruction accuracy .
the distribution is fit to the sum of two gaussians :    @xmath37 \label{eq : twogaus1}\ ] ]    with the fit parameters also summarized in table [ tab : muon_recon ] .
monte carlo studies show that the reconstruction accuracy of the muon direction and impact parameter are uncorrelated .     the impact parameter difference of monte - carlo muon tracks through the sno detector ( solid histogram ) .
the distribution is fit to the function outlined in eq .
[ eq : twogaus1 ] ( solid line ) .
the results from the fit are given in table  [ tab : muon_recon ] . ]
.[tab : muon_recon ] accuracy of the muon fitter based on monte carlo simulations .
fit parameters for mean ( @xmath38 ) , widths ( @xmath39 and @xmath40 ) , and relative weight ( @xmath41 ) are given in equations [ eq : xtwogaus1 ] and [ eq : twogaus1 ] . [ cols="^,^,^,^,^",options="header " , ]     figure  [ fig : ang_noshift ] shows the results of applying the two assumptions to the 30 reconstructed emus - sno events .
the data are fitted to the functional forms of equations  [ eq : xtwogaus1 ]  and  [ eq : twogaus1 ] . due to the small number of events , the weights and relative widths of the secondary gaussians
are fixed to their values from the earlier simulations .
we find that the angular width is @xmath42 .
the impact parameter bias is @xmath43 cm , while fit impact parameter width is @xmath44 cm .
the combined data from the sno detector and the external muon system have demonstrated that the sno muon reconstruction algorithm is accurate to the level needed by the neutrino - induced atmospheric flux analysis .
the emus analysis places a constraint on the angular reconstruction to better than @xmath42 and on the impact parameter bias to better than @xmath43 cm .
the latter constraint is in good agreement with other methods using cosmic - ray data in sno  @xcite .
we believe the method employed here is a unique , low - cost way to explicitly verify the validity of muon track reconstruction for deep underground experiments .
this research was supported by : canada : natural sciences and engineering research council , industry canada , national research council , northern ontario heritage fund , atomic energy of canada , ltd .
, ontario power generation , high performance computing virtual laboratory , canada foundation for innovation ; us : dept . of energy , national energy research scientific computing center
; uk : science and technology facilities council ; portugal : fundao para a cincia e a tecnologia .
we would like to thank the university of indiana , los alamos national laboratory , and k. eitel for loan of equipment to make the measurement possible .
we would also like to thank the sno technical staff for their strong contributions and vale ( formerly inco ) for hosting this project .
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to help constrain the algorithms used in reconstructing high - energy muon events incident on the sudbury neutrino observatory ( sno ) , a muon tracking system was installed . 
 the system consisted of four planes of wire chambers , which were triggered by scintillator panels . 
 the system was integrated with sno s main data acquisition system and took data for a total of 95 live days . using cosmic - ray events reconstructed in both the wire chambers and in sno s water cherenkov detector , 
 the external muon tracking system was able to constrain the uncertainty on the muon direction to better than 0.6@xmath0 .    
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