Our Research

 

 

 

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     This research develops an empirical approximation of the appearance probability without assuming neutrino (or anti-neutrino) oscillations and takes into account most of the reactor distances.

 

Incorporating Multiple Reactor Distances

 

Number of Ep Counts equation: (For a derivation, look at the PowerPoint presentation attached in the Home page)

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20N%28E_%7B1%7D%3CE_%7Bp%7D%3CE_%7B2%7D%29%3D%20%5Csum_%7Bi%3D1%7D%5E%7BM%7D%5Cint_%7BE_%7B1%7D%7D%5E%7BE_%7B2%7D%7D%5Cint_%7B1.8%7D%5E%7B10%7D%5Cfrac%7BS_%7Bi%7D%28E_%7B%5Cnu%7D%29%7D%7B4%5Cpi%20L_%7Bi%7D%5E%7B2%7D%7D%5Csigma%28E_%7B%5Cnu%7D%29T%28E_%7Bp%7D%2CE_%7B%5Cnu%7D%29P_%7B%5Cnu%7D%28%5Cfrac%7BL_%7Bi%7D%7D%7BE_%7B%5Cnu%7D%7D%29dE_%7B%5Cnu%7DdE_%7Bp%7D

 

 

After a change of variables:

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20N%28E_%7B1%7D%3C%20E_%7Bp%7D%3CE_%7B2%7D%29%3D%5Csum_%7Bi%3D1%7D%5E%7BM%7D%5Cint_%7BE_%7B1%7D%7D%5E%7BE_%7B2%7D%7D%5Cint_%7Bl%3D0%7D%5E%7Bl%3D%5Cinfty%20%7DdldE_%7Bp%7D%5Cfrac%7B1%7D%7B4%5Cpi%20L_%7Bi%7Dl%5E%7B2%7D%7DS_%7Bi%7D%28%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29%5Csigma%20%28%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29T%28E_%7Bp%7D%2C%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29P_%7B%5Cnu%7D%28l%29

 

 

where,    N   =      http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20N%28E_%7B1%7D%3C%20E_%7Bp%7D%3CE_%7B2%7D%29%3D%5Csum_%7Bi%3D1%7D%5E%7BM%7D%5Cint_%7BE_%7B1%7D%7D%5E%7BE_%7B2%7D%7D%5Cint_%7Bl%3D0%7D%5E%7Bl%3D%5Cinfty%20%7DdldE_%7Bp%7D%5Cfrac%7B1%7D%7B4%5Cpi%20L_%7Bi%7Dl%5E%7B2%7D%7DS_%7Bi%7D%28%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29%5Csigma%20%28%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29T%28E_%7Bp%7D%2C%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29P_%7B%5Cnu%7D%28l%29 

 

                                                                          Q   =     http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20N%28E_%7B1%7D%3C%20E_%7Bp%7D%3CE_%7B2%7D%29%3D%5Csum_%7Bi%3D1%7D%5E%7BM%7D%5Cint_%7BE_%7B1%7D%7D%5E%7BE_%7B2%7D%7D%5Cint_%7Bl%3D0%7D%5E%7Bl%3D%5Cinfty%20%7DdldE_%7Bp%7D%5Cfrac%7B1%7D%7B4%5Cpi%20L_%7Bi%7Dl%5E%7B2%7D%7DS_%7Bi%7D%28%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29%5Csigma%20%28%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29T%28E_%7Bp%7D%2C%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29P_%7B%5Cnu%7D%28l%29

 

 

                                                                                                                                                                              P   =    http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20N%28E_%7B1%7D%3C%20E_%7Bp%7D%3CE_%7B2%7D%29%3D%5Csum_%7Bi%3D1%7D%5E%7BM%7D%5Cint_%7BE_%7B1%7D%7D%5E%7BE_%7B2%7D%7D%5Cint_%7Bl%3D0%7D%5E%7Bl%3D%5Cinfty%20%7DdldE_%7Bp%7D%5Cfrac%7B1%7D%7B4%5Cpi%20L_%7Bi%7Dl%5E%7B2%7D%7DS_%7Bi%7D%28%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29%5Csigma%20%28%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29T%28E_%7Bp%7D%2C%5Cfrac%7BL_%7Bi%7D%7D%7Bl%7D%29P_%7B%5Cnu%7D%28l%29

 

From this, we get:

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20N%28E_%7Bp%7D%29%3DQ%28E_%7Bp%7D%2Cl%29P%28l%29

 

After minimizing chi square, a way to invert the Q matrix is by:

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20%28Q%5E%7BT%7DV%5E%7B-1%7DQ%29%5E%7B-1%7DQ%5E%7BT%7DV%5E%7B-1%7DN%3DP

 

 

 

Forming the Q matrix:

 

      Test for l values ranging from 9-550 km per MeV

      If the Eν lies between 1.8 MeV-10 MeV, then the values are plugged into the Q equation

      If the Eν lies outside of that range, it does not contribute to the detector, so zero is inputted for that matrix element

      Obtain a different Q matrix for each reactor

      Superpose all the Q matrices

 

Binning:

 

     Why bin the Eps?

      The greater the counts per bin, the smaller the relative error

     Why bin the l's?

      More functions than unknowns

      A higher sum in each l column will provide for a smaller error

 

Definitions:

 

      Q0 = 76x1083 Q

      P0 = 1083x1 (no oscillations)

      N0 = 76x1

      Q0 = 17x11 binned Q0

      N0 = 17x1 binned N0

      P1 = 1083x1 (using oscillation formula)

      N1 = 76x1 (test N or real events data N)

      N1 =  17x1 binned N1

      C is boxed

      Y is what is outside the box in the same equation as C, excluding N and P

      R is a matrix containing the orthonormal eigenvectors for each eigenvalue

      D is a diagonal matrix containing all the eigenvalues of C

      D tilde inverse has the smallest eigenvalue element equal to zero

 

 

The Test

 

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5Clarge%20N_%7B0%7D%3DQ_%7B0%7DP_%7B0%7D

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%28%5Cthinspace%5Cboxed%7BQ%5E%7B%27T%7D_%7B0%7DV%5E%7B-1%7DQ%5E%7B%27%7D_%7B0%7D%7D%5Cthinspace%29%5E%7B-1%7DQ%5E%7B%27T%7D_%7B0%7DV%5E%7B-1%7DN%20%3D%20P

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20V%5E%7B-1%7D%3D%5Cbegin%7Bpmatrix%7D%20%5Cfrac%7B1%7D%7BN%5E%7B%27%7D_%7B0_%7B1%7D%7D%7D%20%26%200%20%26%200%20%26%20%5Ccdots%20%26%200%5C%5C%200%20%26%20%5Cfrac%7B1%7D%7BN%5E%7B%27%7D_%7B0_%7B2%7D%7D%7D%20%26%200%20%26%20%5Ccdots%20%26%200%5C%5C%200%20%26%200%20%26%20%5Cfrac%7B1%7D%7BN%5E%7B%27%7D_%7B0_%7B3%7D%7D%7D%20%26%20%5Ccdots%20%26%200%5C%5C%20%5Cvdots%20%26%20%5Cvdots%20%26%20%5Cvdots%20%26%20%5Cddots%20%26%20%5C%5C%200%20%26%200%20%26%200%20%26%20%26%20%5Cfrac%7B1%7D%7BN%5E%7B%27%7D_%7B0_%7B17%7D%7D%7D%20%5Cend%7Bpmatrix%7D

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20C%20%3D%20C%5E%7BT%7D

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20C%20%3D%20RDR%5E%7BT%7Ddbfc%3Bouvb%3Bv%20bjefjv%20bef%20vjbf%20kfnvfivnfevknf

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20C%5E%7B-1%7D%3DR%5E%7BT%5E%7B-1%7D%7DD%5E%7B-1%7DR%5E%7B-1%7D

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20%5Ctilde%7BC%7D%5E%7B-1%7D%3DR%5E%7BT%5E%7B-1%7D%7D%5Ctilde%7BD%7D%5E%7B-1%7DR%5E%7B-1%7D

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20N_%7B1true%7D%20%3D%20Q_%7B0%7DP_%7B1%7D

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20N%5E%7B%27%7D_%7B1observed%7D%20%3D%20N%5E%7B%27%7D_%7B1true%7D+%5Ceta_%7Bnoise%7D

 

 

 

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5Ctilde%7BC%7D%5E%7B-1%7DYN%5E%7B%27%7D_%7B1observed%7D%20%5Capprox%20P_%7B1%7D

 

 

 

Why omit the smallest eigenvalue?

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20P_%7B1%7D%5Capprox%5Ctilde%7BC%7D%5E%7B-1%7DY%28N_%7B1true%7D%5E%7B%27%7D+%5Ceta_%7Bnoise%7D%20%29

http://latex.codecogs.com/gif.latex?%5Cdpi%7B300%7D%20%5CLARGE%20P_%7B1%7D%5Capprox%5Ctilde%7BC%7D%5E%7B-1%7DYN_%7B1true%7D%5E%7B%27%7D+%5Ctilde%7BC%7D%5E%7B-1%7DY%5Ceta_%7Bnoise%7D

Hence, the smaller the eigenvalues get, the bigger the noise error gets

 

Average Estimated P(l) of the KamLAND Equation

 

This program is funded by the National Science Foundation through grant number PHY-1157044.  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.