Normalization
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Since I always have a real hard time with normalization factor here is a page where I summarized all what I know!
This page also include units and things like that.

Index:
Unsolved questions:
  • For T2KK 1yr = 130 days (1.12 E+7 sec) according to hep-ex 0106019 is that still true?
  • How is the steradian binning done in the flux tables? (cf atmospheric SK)?

Solved questions:
  • Factor of 1E+5 off in T2KK signal calculation (cf T2KK section)? See difference between 069023 and 061023
  • For T2KK signal why is the normalization factor different for nue and nuebar? : Not exactly sure why, but it taking care of when computing the absolute normalization. Something about the amount of neutrons/protons in water...
  • How to compare P.O.T vs seconds in order to divide T2KK flux by SK flux?: SK flux is given per second, where T2KK flux is given in 1E+21 POT which corresponds to 8.53E+6 seconds when we assume 40GeV protons and 0.75MW beam.
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Useful conversion factors:

1GeV =1.6 E-10 J
1MW = 1E+6 J/s
1Mton of water =2.6 E+35 neutrons
                       =6.02 E+35 nucleons
Avogadro's number = 6.02 E+23 mol-1
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T2KK


070326
Fixed one more bug 40/50 GeV protons issue:
Rubbia's flux were given for 1E+21POT per year, but with 40 GeV protons and a 0.75 MW beam, this corresponds only to 8.5E+6 sec per year.
In Ishitsuka's paper one year is assumed to be 130 days =1.12E+7sec
There I am missing a factor of 1.12/0.85 and the correct normalization is:

Event spectrum is given in : 1MW 1Mt 1yr so the conversion factor from the flux*xsec to the event spectrum should be:

 target*beam*xsec*flux = (6.02 E+35) * (1/0.75) * (1E-38) * (1E+6) * (1.12/0.85) = 10576


061023
New Calculation (following Okumura-san's prob2event_korea.kumac)
Flux is given in cm-2 per 1E+21 POT according to A.Rubbia's webpage
        (we assume that 1E+21POT are delivered in 1 yr)  TO BE CHECKED
Cross-section are given in 1E-38 cm2  according to the following plot
The cross-sections in that files already assume the fraction of neutron (protons) in H20
for neutrinos (antineutrinos)   
0.444 is the fraction of neutrons
0.566 is the fraction of protons (includint free protons)

Beam power assumed to be 0.75MW but I want 1MW

Mass of target 1 Mton of water = 1E+12 gr of water = 6.02 E+35 nucleons


Event spectrum is given in : 1MW 1Mt 1yr so the conversion factor from the flux*xsec to the event spectrum should be:
target*beam*xsec*flux = (6.02 E+35) * (1/0.75) * (1E-38) * (1E+6) = 8026.66

Question: why should the unit of flux be 1E+6 cm-2 per 1E+21 POT
Answer: This factor come from Okumura-san's kumac to make flux files from Rubbia's file (mkflux.kumac)

For background:
070418
Here is the correct way to deal with 40/50 GeV proton:
 As for the signal I want to consider one year =1.12E+7 sec but the flux files are given with 1 year=8.5E+6sec
 So the normalization factor that I get is: (for more details see 061023)

SK time = 100yr =3.15 E9 seconds
T2KK time = 1year of 1.12 E7 seconds

and that's it since I already put the flux in seconds before!!!

    T2KK/SK = ((8.5E+6)-1 s-1/s-1)  *  (1/10)*    (1/0.0225)* (1.12E7/3.15E9 )*  1E+6                    * 1/0.75
                            Flux time units         10MeVbins    Masses                    Time          1E-6 factor in T2KK beam

                    = 1/(8.5E+6)             *1/10     *1/0.0225    *3.71E-3      * 1E+6  * 1/0.75

                    =  2.47E-3

Note the T2K normalization number stated before was given at Kamioka, so I need to divide by 12.66 to compare with 2.47E-3

For 28 BG events, at 2.5 OA, correcting for time by 1.12/0.85, correcting for distance by 12.66
the factor I get is 2.33E-3, which is very close to the computer number.
If I consdider 23 BG, 2.0 OA then the factor is 1.5E-3, which is off by about a factor 1.6.
I think this is ok, and within the uncertainty of my input parameters.


070326
As for the signal I want to consider one year =1.12E+7 sec but the flux files are given with 1 year=8.5E+6sec
 So the normalization factor that I get is: (for more details see 061023)

    T2KK/SK = ((8.5E+6)-1 s-1/s-1)  *  (1/10)*    (1/0.0225)* {(1/100)    *(1.12/0.85) } *  1E+6                    * 1/0.75
                            Flux time units         10MeVbins    Masses                    Time                        1E-6 factor in T2KK beam

                    = 1/(8.5E+6)             *1/10     *1/0.0225    *1/100 *1.31       * 1E+6  * 1/0.75

                    =  9.18E-3

 In order to check that factor, I used our new flux files (properly normalized by Okumura-san) and used our old technique (described above) and I got a factor of 0.01976 =1.97E-2.
- But in Ishitsuk-as paper 50 GeV protons were assumed which might lead to a difference in the number of background events expected:
- In Maxim's talk we get 23 background events instead of 28 and therefore the normalization factor would become 1.61E-2.
and I think it is more correct to use this number.
- Using Reconstructed energy to assign likelihood weight in the mkbg.kumac gave 0.2156 before using 28 bg events so now
using reconstruced energy + Maxim's number of events I get 0.01821

 I am off "only" by a factor of ~2.0..... where is my mistake???



061023
 For background, we ran on atmospheric MC and just reweighted by the T2KK flux.  We also set the overall normalization knowing how many events where expect for T2K at Kamioka for 1MW 1Mt 1yr, and just rescale by the ratio of the distance squared to get the normalization in Korea.

Here is Okumura-san's email:
>          In Ishituka paper, background normalization is adjusted so that
>          28 events in 350-850MeV for (0.75x0.0225x5)MwMtonYr and 2.5
>          degree off-axis beam
>          so for 1MwMtonYr, 332 events are expected.
>          please determine normalization so that number of events in 350-850MeV
>          become 332 events for 2.5degree off-axis beam.

I would like to try to get it from first principles.

In that case the normalization factor is: ratio*posc*like_weight*0.01976
    - where ratio is the ratio of the fluxes, posc is the oscillation probability, like_weight is the likelihood weight
    - where 0.01976 is the normalization factor that I'd like to reproduce.

(NB I checked that factor by making event spectrum using the new flux files and it gives sensible results)


For that I need to know the units of the flux and cross section assumed in T2KK (done same as signal) and in atmospheric neutrino Monte Carlo (go see section about atmospheric neutrinos MC).

Assuming both cross-sections were the same (and I checked it is):

    T2KK
                Flux =   cm-2 per 1E+21 POT (And there is a factor of 1E-6 to keep track of)
                            in 10 MeV bins
                            1E+21 POT for a 0.75MW beam and  40GeV protons this is equal to 8.53E+6 seconds

                           POT=Power*time/Energy of protons
               time=POT*Energy/Power=1E+21*(40(GeV)*1.6E-10(J/GeV))/0.75E+6)=8.53E+6


      Mass =  1Mton = 6.02 E+35 nucleons
               Time =  I want to normalize it to 1 year
               Beam = I want a 1MW beam but I have 0.75 MW --> Factor of 1/0.75

     (Distance factor taken care of later, not need to worry about it now)
                
    SK

             Flux =         m-2 sec-1 sr-1 GeV-1
                    That is was is usually stated BUT in flux.F, where we get the Superk flux, we transform the units to:
                       cm-2 sec-1 sr-1 MeV-1
                The text says cm-2 sec-1 sr-1 MeV-1 but I think that the steradian integration is done in flux.F  through:
     flxb(i) = flxb(i) + fnbarfx03(energy,dir,solact,ip)*dd/1.E+7
 with dphi=2.*3.141592/float(nphi)
      dcos=2./float(ncos)
      dd=dphi*dcos

So I should NOT multiply the SK flux by 4pi. RIGHT?

              Mass = 0.0225 Mton = 0.0225*6.02E+35 nucleons
              Time = I have 100 yr of MC       


In order to be able to compare the fluxes I have to:
- divide the T2KK flux by 10 MeV to get rid of the binning (and get MeV-1 units) 


So the normalization factor that I get is:

    T2KK/SK = ((8.53E+6)-1 s-1/s-1)  *  (1/10)*    (1/0.0225)* (1/100)     *  1E+6                    * 1/0.75
                            Flux time units         10MeVbins    Masses       Time   1E-6 factor in T2KK beam

                    = 1/(8.53E+6)             *1/10     *1/0.0225    *1/100        * 1E+6  * 1/0.75

                    =  6.93E-3

In order to check that factor, I used our new flux files (properly normalized by Okumura-san) and used our old technique (described above) and I got a factor of 0.01976 =1.97E-2.

I am off "only" by a factor of ~2.8..... where is my mistake???


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FNAL

Signal:

I want the normalization for 2500 Kton MW 107sec: (which is 5 years of running for 300kton and 1year =1.7 E7 seconds)
Flux is given in neutrino/GeV/m2/POT at 1km from the target
Warning the units are per GeV but the binning is per 75MeV, so I need to multiply my flux by 0.075 to have something correct
 
Cross-section are given in 1E-38 cm2 in file  bnl14_numu_crsqe_flux.hbk (bnl14_nue_crsqe_flux.hbk)
    NB (In this case the fraction of neutrons in water is NOT included.. need to be taken care off later)

Beam power  1MW

Time 5*1.7 E+7 sec

Mass of target 300 kton of water = 0.3E+12 gr of water = 1.81 E+35 nucleons
 Note that sometimes we said 500 Kton and therefore    3.01 E+35 nucleons
 this number is ok if used with the 1.12 POT as exaplined below.

5yrs with 28GeV protons and 1MW beam = 1.90 E+22 POT
Note that in previous work it was sometimes considered 500Kton
and less running time and there for the following number is sometimes stated:
 2500 Kton MW 107sec corresponds to 1.12 E+22 POT (cf there)

Distance 1290km --> 6.01E-7

Normalization factor: 0.075 * 1E-42 * 6.01E-7 * 1.12E+22 * 3.01E+35 = 1.52 E+8



Background:


We are using the same technique than for T2KK where we use the SK MC in order to know how the likelihood behave, and then we just reweight the SK flux by the FNAL flux.

FNAL:
Flux is given in neutrino/GeV/m2/POT at 1km from the target
Warning the units are per GeV but the binning is per 75MeV, but I would have to divide by the binning to be able to compare with the SK flux. So in this case I just have to change GeV to MeV but I DO NOT need to multiply by the binning.
Cross-section 1E-38 cm2
Beam power  1MW
Time 5E+7 sec
Mass of target 500 kton
Distance 1290km --> 6.0E-7

And the number of POT in 1 second for a 1 MW beam with 28GeV protons is 2.23E+14.


SK:
Flux  cm-2 sec-1  MeV-1  (Look at the T2KK section about background for more details about that units of flux)
Cross-section 1E-38 cm2
Mass 22.5 kton 
Time 100 yr of MC  = 3.15E+9 seconds       


So

FNAL/SK = (1/1E+4)      * (1/1E+3)         *(5E+7/3.15E+9)   *(500/22.5)     *(2.23E+14)             *(6.0E-7)
                    m-2/cm-2   GeV-1/MeV-1      Time                    Mass                 POT to seconds       Distance

                = 4.71



 Here is a results using those factors  (eps)
Where the black line is signal + background for delta=135° and the red line is delta=0°
Green is the nue background, purple is the NC background and  blue is the CCnumu background
The integral of the 135° histogram is:1628
The integral of the 0° histogram is: 1185
The assumptions are:
 sin2(2theta13) = 0.04
2500KtonMW 10^7sec




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Atmospheric MC for Super-K

For the plots in the combined paper, the flux for atmospheric neutrino is given in: Flux*E2 (m-2, sec-1, sr-1 GeV)

But according to some piece of codes in neut (fxazitest.F and fxcosztest.F) that was changed to cm-2
I think this is true in order to match the cross-section which is given in cm2 but the stored flux table is still given in m2 .
Also in the ntuple, I think that the values for flxh etc are given in m2

In order to sort this out, I will try to reproduce a table from a Honda paper.
--->> Just plotting what's in the ntuple and applying the good cuts I am usually off by a factor of 10%.
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Old Calculations



060923
Old calculation (contains some errors, pointed in purple)

 Here are normalization that have been used in the T2KK analysis:

For signal:
    Flux is given in cm-2 per 1E+21 POT according to A.Rubbia's webpage
                                        NB: Assuming 1MW beam, 50GeV protons, then 1E+21POT corresponds to 8E+6 seconds
                                                From hep-ex 0106019 we say 1yr =130days=1.12 E+7 sec for T2K, I assume the same for T2KK.
In T2K the beam is assumed to be 0.75 MW not 1MW also, it should be 40GeV protons
Therefore 1E+21POT corresponds to 8.5E+6 sec 

    Cross-section are given in 1E-38 cm2  according to the following plot
                            The cross-section in that files already assume the fraction on neutron (protons) in H20
                            for neutrinos (antineutrinos)   
                            0.444 is the fraction of neutrons
                            0.566 is the fraction of protons (includint free protons)

    Event spectrum is given in : 1MW 1Mt 1yr so the conversion factor from the flux*xsec to the event spectrum should be:
        Assuming 1yr =130 days then 1yr =1.46E+21 POT
        We know 1Mton of water = 2.6 E+35 neutrons
            then event rate= flux * cross section = 1E-38 * 1.46 * 2.6E+35 =3.8 E-3
    I am off by a factor of 1E+5    See updated version to get factors right


        But what we did was normalize the number of events according to:
            
            *  normalize to 1Mtonx1yr event spectrum (from nunokawa-table)
            *    kam   nue   37148.4 ev/1Mton/1yr   (0-1.5GeV)
            *    kam   nueb  10874.8 ev/1Mton/1yr   (0-1.5GeV)
            *    korea nue    2932.3 ev/1Mton/1yr   (0-1.5GeV)
            *    korea nueb    858.4 ev/1Mton/1yr   (0-1.5GeV)
            *    sum in hid 991 in 0-1.5 GeV   7.723114    (OA2.5deg,fluxkorea25_hist.hbk)
            *    sum in hid 992 in 0-1.5 GeV   2.708750    (OA2.5deg,fluxkorea25_hist.hbk)           
            *
            add 991 991 201 $eval(2932.3/7.723114) 0       ---->>> =379.68
            add 992 992 202 $eval( 858.4/2.708750) 0

                    NB numbers for Korea are just scale by 12.66 which the distance ratio squared

This problem has no consequences because I trust Nunokawa's table but I'd like to understand!!!


For background: PLENTY OF ERRORS... DO NOT look

For background, we ran on atmospheric MC and just reweighted by the T2KK flux.  We also set the overall normalization knowing how many events where expect for T2K at Kamioka for 1MW 1Mt 1yr, and just rescale by the ratio of the distance squared to get the normalization in Korea.

Here is Okumura-san's email:
>          In Ishituka paper, background normalization is adjusted so that
>          28 events in 350-850MeV for (0.75x0.0225x5)MwMtonYr and 2.5
>          degree off-axis beam
>          so for 1MwMtonYr, 332 events are expected.
>          please determine normalization so that number of events in 350-850MeV
>          become 332 events for 2.5degree off-axis beam.

So, as for the signal, the normalization I get is correct, but I would like to try to get it from first principles.

In that case the normalization factor was: ratio*posc*like_weight*0.001180
    - where ratio is the ratio of the fluxes, posc is the oscillation probability, like_weight is the likelihood weight
    - where 0.00118 is the normalization factor that I'd like to reproduce.

For that I need to know the units of the flux and cross section assumed in T2KK (done same as signal) and in atmospheric neutrino Monte Carlo (go see section about atmospheric neutrinos MC).

Assuming both cross-sections were the same:
    T2KK flux=     cm-2 per 1E+21 POT  (assume=8E+6 sec for 060923 but becomes 8.5E+6 for 061023)
    SK flux=         m-2 sec-1 sr-1 (GeV-1 binning??)


The steradian is taking care of inside the ntuple, same about the binning. So to get the correct normalization factore I just have to take care of the area units and the time units:
 
So normalization factor = T2KK/SK = (m2 to cm2) / (1 yr to sec) = 1E+4/(8.5E+6) = 0.0011764
which is not exactly perfect but probably close enough.