2 (2) and sin
A.3 Helicity Suppression in Meson Decays
Take the pion decay shown in Equation A.2. In principle, ¯νe’s and e+’s could also have been produced but that channel is suppressed due to the weak interaction’s chiral selectivity – only left- handed particles and right-handed antiparticles interact.
Let us look at the decay in the rest frame of theπ−. The outgoing antineutrino must have right- handed chirality and, since the neutrino mass is negligible, right-handed helicity. Since the pion has spin-0, conservation of angular momentum requires that the outgoing lepton and antineutrino must have opposite spins:
←¯ν−l ←π− →−→
l−, (A.36)
meaning both the ¯νlandl− come out with right-handed helicity. If the leptonl−were massless, this
process would be completely forbidden since then its helicity and chirality would always be equal and the right-handedl− would couple to the weak force.
However, since the l− does have mass, its right-handed helicity state has a component with left-handed chirality.3 Let us take a Dirac spinor with right-handed helicity and look at its chiral projections:
u↑=PRu↑+PLu↑=1 2
1 + |p| E+m
uR+1
2
1− |p| E+m
uL. (A.37)
Since only the left-handed chiral component will contribute, the matrix element for the decay must be proportional to the coefficient of the left-handed projection:
M ∝1 2
1− |p| E+m
. (A.38)
Let us calculate this coefficient in the pion rest frame, M ∝ 1
2
1− |p∗l| El∗+ml
. (A.39)
From conservation of momentum and taking the neutrino mass as negligible,
|p∗l|=| −p∗ν|=|p∗ν|=Eν∗, (A.40) and from conservation of energy,
El∗=Eπ∗−E∗ν=mπ−Eν∗, (A.41)
and now substituting back into the matrix element:
M ∝ 1 2
1− Eν∗
mπ+ml−Eν∗
. (A.42)
Substituting in the expression forEν∗from Equation A.28 and taking parent p=π,
M ∝ 1 2
1−
m2π−m2l 2mπ
mπ+ml−m2m2π−mπ2l
(A.43)
∝ 1 2
1− m2π−m2l
m2π+ 2mlmπ+m2l
(A.44)
∝ 1 2
2mlmπ+ 2m2l
(mπ+ml)2
(A.45)
∝ ml
mπ+ml. (A.46)
3It must since the particle moves at less than the speed a of light, meaning a Lorentz-transformation can change the helicity.
134 Meson and muon decay kinematics So, the ratio of the branching ratios for producing a ¯νeand a ¯νµ is approximately:
P(π− →e−ν¯e)
P(π−→µ−ν¯µ) ≈ m2e/(mπ+me)2
m2µ/(mπ+mµ)2 ≈1×10−4. (A.47) The detailed calculation of the ratio of matrix elements and phase-space factors can be found in [155] and comes to
P(π−→e−ν¯e)
P(π−→µ−ν¯µ)= m2e(m2π−m2e)2
m2µ(m2π−m2µ)2 = 1.28×10−4, (A.48) which is quite close to the experimental value 1.23±0.02×10−4.
Appendix B
The Decay Pipe Systematic
Decay pipe production, or downstream production, refers to the neutrinos that come from the decay of hadrons produced in the decay pipe rather than in the target. The interest in studying it is that 14% of the Near Detector antineutrino events and 6% of the Far Detector antineutrino events come from parents produced in the decay pipe. Figure B.1 shows the vertex distribution (weighted to the number of events expected in the Near Detector) color coded for the different production regions. The decay pipe is in red. While the downstream production also exists for neutrinos, it is negligible in comparison to the much larger focused sample. The trouble with the sample is two-fold.
First, it does not produce the same spectrum in the Near and Far Detector (see Figure B.2) so the uncertainties in production cross-sections do not fully cancel between the two detectors. Second, the downstream production is modeled with GFluka [156], not Fluka05 [129, 130], our preferred
Figure B.1: The vertex of the production (immediate) parent of antineutrinos interacting in the Near Detector in the neutrino-mode beam. Note the log scales.
136 The Decay Pipe Systematic
Energy (GeV) iµ
True
Arbitrary Units
0 2 4
6 Low Energy Beam
Near Upstream Far Upstream
MINOS Preliminary Monte Carlo
0 5 10 15 20 30 40 50
Energy (GeV) iµ
True
Arbitrary Units
0 0.2 0.4 0.6 0.8 1 1.2
Low Energy Beam Near Decay Pipe Far Decay Pipe
MINOS Preliminary Monte Carlo
0 5 10 15 20 30 40 50
Figure B.2: The Near (color) and Far (black) Detector charged-current antineutrino spectra broken down by parent production region. The left shows the spectra from neutrino parents produced upstream and the right shows the spectra from neutrino parents produced in the decay pipe. The Near and Far spectra are scaled to the same number of events summed across the two histograms. Note that the decay pipe spectrum has much larger Near/Far differences.
and tuned hadroproduction model.1 As a consequence, this so-called downstream production is a significant systematic error for the analysis of neutrino-mode antineutrinos.
In order to constrain the uncertainty on decay pipe production, the Near Detector data is used.
The Near Detector Monte Carlo is first reweighted using a special set of beam tuning weights generated using the most recent version of the beam fits but without using the ¯νµ Near Detector spectrum (NA49 data is used as a constraint on theπ+/π−ratio). Then, the decay pipe component is scaled up or down in order to make the total number of Monte Carlo events match the data. This scale will then be taken as the systematic uncertainty on the decay pipe. This gives a ‘worst case’
systematic uncertainty – the true systematic error will almost certainly be less than this value. The range of the integral to get the total number of events starts at 0 and ends at 13 GeV. This value was chosen because it gives the largest systematic error band, consistent with the idea of bracketing the decay pipe systematic, rather than trying to estimate an exact 1σuncertainty.
However, there are systematic uncertainties on the Near Detector spectrum. In order to avoid having these uncertainties mask a discrepancy that might be due to the decay pipe, the above evaluation is done with all other systematics applied (see Section 5.4), where their effects are summed in quadrature. By looking at the two extremes of the systematic shifts, the possible range of decay pipe scalings allowed by the Near Detector data plus its systematics can be determined. This gives us a systematic uncertainty on the decay pipe events of−100% and +50%. The effect of this systematic on the Far-over-Near ratio can be seen in Figure B.6.
1For the antineutrino-mode analysis, the beam simulation does use a consistent hadroproduction model (see F), but since antineutrino-mode antineutrinos are focused, decay pipe production is not an important systematic in that sample.
Systematic Shift
SKZP +6.4% −6.4%
Combined MaRes +5.0% −4.3%
NuBar DIS 2 +4.4% −3.8%
NuBar Overall +3.8% −3.8%
Combined Overall +3.5% −3.5%
Backgrounds +2.7% −2.7%
Combined DIS 2 +2.3% −2.3%
Combined MaQE +2.0% −1.7%
NuBar Res +1.9% −1.9%
NuBar QEL +1.1% −1.1%
Combined DIS 3 +0.8% −0.0%
Table B.1: The systematic errors on the Near Detector, in order of their effect on the total number of Near Detector events. The systematics that shift events between bins but do not change the total number of events (e.g. energy shifts) all have very small effects (less than 1%) and are excluded from the table.
Reconstruced Energy (GeV)
Near Detector
0 2000 4000 6000
0 5 10 15 20 30 40 50
Shift=-0.27
Figure B.3: The Near Detector spectrum with no systematics applied. The data is the black points, the original spectrum, with only the SKZP weights applied is the dashed line, the non-decay pipe component is in blue, the scaled decay pipe is in red, and the scaled total spectrum is in black. The shift needed in order to match the Monte Carlo to the data in this case is−27%.
138 The Decay Pipe Systematic
Reconstructed Energy (GeV)
0 10 20 30 40 50
Systematic Error
0.6 0.8 1 1.2 1.4
Figure B.4: The Near Detector systematic error bands, obtained from adding the effects of all systematics in quadrature. The significant systematics are broken down in Table B.1.
Reconstruced Energy (GeV)
Near Detector
0 2000 4000 6000 8000
0 5 10 15 20 30 40 50
Shift= -1
Reconstruced Energy (GeV)
Near Detector
0 2000 4000 6000
0 5 10 15 20 30 40 50
Shift=0.539
Figure B.5: The Near Detector spectrum with all systematics applied. The left figure has all positive systematics and the right has all negative systematics. As in Figure B.3, the data is the black points, the original spectrum, with only the beam tuning weights applied is the dashed line, the non-decay pipe component is in blue, the scaled decay pipe is in red, and the scaled total spectrum is in black. The shifts needed in order to match the Monte Carlo to the data in these cases are−100% and +50%.
Reconstructed Energy (GeV)
Far/Near
0.2 0.25 0.3
10-3
×
0 5 10 15 20 30 40 50
Shift= -1
Reconstructed Energy (GeV) Shifted over Normal1.06
1.08 1.1 1.12 1.14 1.16
0 5 10 15 20 30 40 50
Reconstructed Energy (GeV)
Far/Near
0.2 0.25 0.3
10-3
×
0 5 10 15 20 30 40 50
Shift=0.539
Reconstructed Energy (GeV) Shifted over Normal0.94
0.95 0.96 0.97 0.98
0 5 10 15 20 30 40 50
Figure B.6: The Far-over-Near ratios for the two calculated systematic errors from Figure B.5. The effects on the Far-over-Near ratio give a sense of how significant this systematic will be for the final analysis.
140 The Decay Pipe Systematic