IMPACTOS AMBIENTALES

5.3.1

Overview

A set of software tools to model the decays of neutrino parents in the T2K sec- ondary beamline, written in C++, has been developed to study the effectiveness of the ratio and beam matrix methods in correcting for systematic errors in the beam properties. The input to the package is a sample of neutrino parents from the T2K beam MC (jnubeam)1, giving the positions and momenta of the parents at the point of decay. The package uses the physics of each decay type, and the phase space coordinates of the decay, to calculate (En, Ef, Pn, Pf) as defined in Section 5.2.2, for each parent. En andPnare calculated using a random target position in the near detector in order to obtain an average over the flux; the far detector is treated as point-like. The software then uses the energies and proba- bilities for each parent to generate a beam matrix or ratio vector, which can be used to extrapolate a near detector flux to the far detector.

The probabilities to decay towards the detectors are calculated analytically, rather than using a Monte Carlo method and choosing only neutrinos passing through the detector. It is therefore possible to use each parent only once, al- lowing the errors on the calculated beam matrix elements or ratios to be properly evaluated.

5.3.2

Two-body decays

We will first consider the case of pion decays, π+ _{→ µ}+_ν

µ. Since the pion is spinless and the decay is two-body, the neutrino is produced isotropically and mono-energetically in the pion rest frame. From elementary Special Relativity, the probability for a neutrino produced by a pion decaying at position xπ with momentump_π, to pass through area ds centred on the pointxν is

dP ds = dP dcosθR × dcosθR dcosθL × dcosθL ds = 1 2× 1 γ2 π(1−βπcosθL)2 × 1 2π(xν −xπ)2 , (5.5)

where the lab decay angle cosθL is given by

cosθL=

(xν −xπ)·pπ

|xν −xπ| |pπ|

, (5.6)

andθRis the rest decay angle. (βπ, γπ)are the usual relativistic boost parameters. Note that since this decay is isotropic, there is a trivial angular dependence in the centre-of-mass frame, _dcosdP_θR = 1₂.

Similarly, the neutrino energy in the lab may be calculated by boosting from the pion frame:

Eν = m2 π−m2µ 2mπ 1 γπ(1−βπcosθL) (5.7) The two-body kaon decayK+ _→µ+νµ is exactly analogous to pion decay, and can be handled using the above method with mπ replaced by mK.

5.3.3

Ke3

decays

is not mono-energetic in the kaon rest frame, since there are three decay products. The neutrino energy distribution cannot be calculated exactly, since the interaction has a hadronic component, so the neutrino energy is selected using a form-factor parameterisation of the Dalitz plot density for the decay. The parameterisation used is an approximation of that preferred by [7], giving the joint probability density for the muon and pion energies in the kaon rest frame as

ρ(Eπ, Eµ)∝ f+(0) 1 +λ+ t m2 π × 2EeEν −mK _m2 K+m2π −m2µ 2mK − Eπ , (5.8)

where t is the squared momentum transfer to the leptonic system, and we use

f+(0) = 0.98GeV2, λ+= 0.0286. These parameter values are chosen to match

the model used by thejnubeam MC.

5.3.4

Muon decays

The case of muon decays is somewhat more complicated, because the muon has spin-1₂ and so the decay is not isotropic. The distribution of electron neutrinos from the decay µ+ _→e+_ν

eν¯µ, in the µcentre-of-mass frame, is given by d2P dxdcosθR = 6x2(1₋x) (1₋pcosθ) where x= 2E rest ν mµ , (5.9) where p is the muon spin polarisation, and θR is the angle at which the νe is emitted, relative to the muon polarisation axis [68]. We can easily derive the single derivative _dcosdP_θR, and the decay probability can then be calculated by substituting this into (5.5). The energy distribution in (5.9) is used to select a rest-frame

energy, using an acceptance-based MC — a value of x is chosen randomly, and accepted with a probability equal to the ratio of the probability density at this energy to that at the most probable value of x.

The same procedure can be used for the ν¯µ from the decay; in this case the decay distribution is given by

d2P

dxdcosθ =x

2

[(3₋2x)₋p(1₋2x) cosθ] (5.10) and the energy distribution depends on cosθ.

Note that since the three-body decays are not mono-energetic for a given lab decay angle, and also because the solid angle subtended by the near detector is significant, identical parents may contribute to different elements of the beam matrix. Over a large sample of parents, the method used will average out these possibilities to give the correct distribution.