3.      Detector Simulations

  We wish to explore how well in principle a neutrino three-flavor unitarity 
test can be performed with a given muon-neutrino beam as a function 
of dataset size, and study which systematic uncertainties are likely to be important, 
and their impact. 

3.1  The "Perfect Experiment"

  It is useful to begin by considering a "perfect experiment" in which:

i)  The initial numu beam has no nue and no antineutrino component.

ii) The beam flux and spectrum, and the detector fiducial mass are 
    precisely known.

iii) The neutrino interaction cross-sections are precisely known.

iv) The detector has near-to-perfect performance, which we will take to 
    be:

    a) An energy threshold of 10 MeV for the detection and measurement
       of electrons, muons, charged and neutral pions and kaons.

    b) Perfect particle identification and energy resolutions for 
       electrons and pi0s, charged pions, and muons.

In our perfect experiment the sensitivity of the unitarity test is determined 
only by the statistical uncertainties, which we have calculated using a parameterization 
of the known beam flux and spectrum, together with a simulation of neutrino interactions 
in the detector. The simulation is used to determine the relevent detection efficiencies and 
contamination factors.We use the NEUGEN Monte Carlo code to simulate neutrino interactions, 
and clasify events as follows:

  i)   NC Events:  The total event energy is above threshold, but there are no electron 
       candidates and no muons above threshold.

  ii)  Muon CC Events:  One or more muons above threshold.

  iii) Electron CC Events: One or more electron candidatess above threshold, and no muons 
       above threshold.

The definition of an electron candidate will depend upon the detector technology. For a 
water cherenkov detector we will define an electron candidate as either an electron above 
threshold, or a pi0 with an energy exceeding 1 GeV. In practice the two daughter photons 
from these high energy pi0s produce cherenkov rings that overlap in the detector and 
cannot de distinguished from a single electromagnetically showering particle.

3.2  The K2K Experiment

  To identify the most important systematic uncertainties it is useful to compare the 
sensitivity of our "perfect experiment" with that of a realistic experiment. The K2K 
experiment is the first and only long baseline neutrino experiment at present, and is 
therefore a natural choice for comparison. The ultimate experimental sensitivity of 
the K2K experiment can only be determined by the K2K Collaboration. In the following 
we will use a simple model for the K2K detector performance. Although this is 
inadequate to precisely predict the eventual K2K sensitivity, it does enable us to 
identify the dominant sources of systematic uncertainty, and hence explore how the 
experimental results will depend upon the sizes of these systematics. We use the following 
parameterization of the K2K detector response:

    a) An energy threshold of 100 MeV for the detection and measurement
       of electrons, muons, charged and neutral pions and kaons.

    b) Energy resolutions given by:

       Electrons and pi0:  rms/ E  =  0.005  +  0.025 / sqrt(E)
       Charged pions and Muons:  rms / p =  0.03

In addition, our statistical precision will be based on a simulated data sample that 
corresponds to 4.8 * 10^19 protons on target. The NEUGEN prediction for the total number 
of muon CC events in a detector with a fiducial mass of 22 kt is xxxxx, which is consistent 
with the reported event sample (xxx events) from the K2K Collaboration. We have chosen to 
renormalize our predicted event rates by a factor of x, which yields a muon CC rate that 
corresponds to the K2K measurement. 



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 Fragment left over from previous version :



 The detector resolutions should be achievable with a liquid argon detector
 plus external rangestack for measuring high energy muons. We will therefore 
 take argon as the detector medium for our neutrino interaction simulations, 
 although in practice this choice has little impact on our results.  




iv) The detector has near-to-perfect performance, which we will take to 
    be:

    a) An energy threshold of 100 MeV for the detection and measurement
       of electrons, muons, charged and neutral pions and kaons.

    b) Energy resolutions given by:

       Electrons and pi0:  rms/ E  =  0.01  +  0.03 / sqrt(E)
       Charged pions:      rms/ E  =  0.05  +  0.20 / sqrt(E)
       Muons:              rms / p =  0.05

     c) Particle identification efficiencies above threshold of unity 
        (and hence no particle misidentification).








  NEUTRINO BEAMS AND BASELINES

 We will consider all the long-baseline beams and setup that exist or are 
 under construction, namely:


 1.  K2K:

       <Enu> (GeV)      1.3
       L (Km)           250
       nue at peak      1%  (not implemented)
       Mdet (kt)        22.5
       numu CC/yr       50
    
 2.  JPARC

 3.  NuMI on-axis

 4.  NuMI off-axis

 5.  CNGS