Calculating the Probability for Neutrino Oscillations 1 Neutrino Oscillations Student ecture Series for MiniBooNE Darrel Smith Embry-Riddle University July 18, 001 The weak eigenstates that we normally observe ν µ, ν e can oscillate between each other if they are comosed of an add-mixture of mass eigenstates ν 1, ν. If the weak eigenstates are rotated by an angle θ with resect to the mass eigenstates Fig. 1, then a matrix equation can be written that relates the weak eigenstates to the mass eigenstates see below. For examle, using the matrix equation below, the ν e state can be written as ν e = cos θ ν 1 + sin θ ν, where θ is called the mixing angle. νe ν µ cos θ sin θ = sin θ cos θ ν1 ν ν µ ν ν e θ ν 1 Figure 1: The weak eigenstates are rotated by an angle θ with resect to the mass eigenstates ν 1 and ν to allow mixing i.e., oscillations between the ν µ and ν e. The mass eigenstates ν 1, ν have masses m 1 and m and both have momentum. 1
ook at the time evolution of the ν µ state ν µ t = 0 = ν µ = sin θ ν 1 + cos θ ν 1 ν µ t = sin θ ν 1 e i E 1 t h + cos θ ν e i E t h where E 1 = c + m 1c 4 and E = c + m c 4 and 1 =. 3 Some Aroximations et h = c = 1. Then E 1 = + m 1 and E = + m Also, the neutrinos are assumed to be relativistic: γ = E m o c = c + m oc 4 m o c 1 3 then m o 4 E = + m o = 1 + m o/ + 1 m o 5 where we use the binomial exansion: 1 + x n 1 + nx + nn 1! x + and kee just the first two terms. The energy of the two mass eigenstates can be written aroximately as: E 1 + 1 m 1 and likewise E + 1 m. 6
4 How does the ν µ roagate in time? ν µ t = sin θ ν 1 e i + 1 m 1 t + + cos θ ν e i 1 m t 7 ν µ t = e i + 1 m 1 t sin θ ν 1 + cos θ ν e +i 1 m 1 m t 8 Now, for some definitions and substitutions: m = m 1 m and t = x c = x and e iz = e i + 1 m 1 t 9 Now, we have: ν µ t = e iz sin θ ν 1 + cos θ ν e +i 1 m x 10 5 What is the robability for ν µ ν e? To calculate the robability for a ure ν µ state to oscillate into a ν e state, we must square the quantum mechanical amlitude that describes this transition. Recall from section 1 that So, now we can write the amlitude as: P ν µ ν e = ν e ν µ t 11 ν e = cos θ ν 1 + sin θ ν 1 ν e ν µ t = e iz sin θ cos θ + sin θ cos θ e i m x 13 where we use the relationshi ν i ν j = δ ij. Taking the absolute value squared, we find that: 3
P ν µ ν e = ν e ν µ t = = e +iz e iz sin θ cos θ 1 + e i m 1 x + e i m x Since the neutrino is relativistic, we can also make the substitution: =, and likewise, we will make the substitution x =. P ν µ ν e = 1 m sin θ 1 cos. 14 Using the trigonometric relation 1 cos θ/ = sin θ, we can write the above equation as P ν µ ν e = sin θ sin m 4. 15 Now, we can write the argument of the second sin term above so it s dimensionless by introducing the aroriate number of h s and c s. m 4 m c 4 4 hc et s write the above quantity in units that are convenient for an exerimental hysicist. We would like the variables in the above equation to have the following units: m c 4 ev, meters, and MeV. If we substitute hc with 197 ev nm, we can write the quantity in arenthesis as m c 4 m c 4 10 6 MeV/eV = 1.7 m 4 hc 4 197 ev nm 10 9 m/nm Eν Finally, we can write Eq. 15 in its more familiar form: P νµ ν e, E = sin θ sin 1.7 m. 16 If neutrino oscillations occur, the mixing robability sin θ and the mass difference m are determined by nature. Physicists can robe different regions of m by adjusting the distance between the neutrino source and the detector as well as the neutrino energy 4
0.03 Probability ν µ ν e sin θ 0.0 0.01 sin θ = 0.006 50 100 150 00 50 300 meters Figure : Neutrino oscillations ν e aearance as a function of length for a monoenergetic beam of ν µ s. There is a maximum robability to observe ν e interactions at 50, 150, and 50 meters.. If a mono-energetic beam of neutrinos is roduced e.g., 40 MeV and the mixing arameters suggested by SND are correct e.g., sin θ = 0.006 and m 1 ev, then it s ossible to lot the oscillation of ν µ ν e using Eq. 16. A small fraction of the initial ν µ beam sin θ aears as ν e s as shown in Fig.. Maximal mixing occurs if sin θ is 1 i.e., θ = 45 o. In the case of atmosheric neutrinos, it is susected that maximal mixing occurs. If this were the case, the eaks in Fig. would oscillate between 0 and 1.0 on the vertical axis assuming all the atmosheric neutrinos were monoenergetic which is not the case in real life. The /E term is the quantity of interest when exloring different mass regions. In the SND exeriment, was about 30 meters and E was about 30 MeV giving an /E of 1. If MiniBooE is going to exlore the same m and sin θ region, then its /E must be similar to the SND value. In the case of MiniBooNE, the neutrinos travel about 500 m and have energies on the order of 500 MeV. So, roughly seaking, the MiniBooNE exeriment is designed to exlore the same m region as SND, but with higher sensitivity i.e., down to lower values of sin θ. 6 Real Neutrino Beams In the revious section, we discussed where the ν e aearance occurs as a function of distance when we have a monoenergetic neutrino beam. In reality, most neutrino sources have 5
a range of energies which will tend to wash out the distribution shown in Fig.. To investigate the changes to the amlitude and oscillation length, we exlore two ossibilities for neutrino energy distributions a narrow band distribution, and a wide band distribution. In the first case, we assume that the neutrino momentum is well-defined, that is, / < 5%. The neutrino energy distribution is shown on the left-hand side of Fig. 3 while the resulting oscillation robability is shown as a function of length on the right hand side. The oscillation as a function of length P νµ ν e is calculated by convoluting the energy distribution fe on the left side of Fig. 3 with the oscillation robability P νµ ν e, E. P νµ ν e = P νµ ν e, E fe de = sin θ sin 1.7 m fe de 0 0 E "Good" laces to locate the detector 0. 0.15 0.1 0.05 Narrow Band Neutrino Beam f nb E 0 40 60 80 MeV Pν µ ν e 0.005 0.00 0.0015 0.001 0.0005 50 150 50 350 meters Figure 3: Neutrino oscillations ν e aearance as a function of length for a monoenergetic beam of ν µ s. There is a maximum robability to observe ν e interactions at 50, 150, and 50 meters. The convolution integral was done using Mathematica 4.0 Next we investigate the oscillation robability when using a wide band neutrino source, f wb E. A wide band neutrino beam ν µ is shown in Fig. 4 resulting from muon decays. The energy distribution is described by the following function: dn de = E 1 E 3E max where E max is 5.8 MeV for muon decay. This function is shown on the left-hand side of Fig. 4. Once again, convoluting this this energy distribution with P νµ ν e, E, we obtain the oscillating function seen on the right-hand side of Fig. 4. Notice that both the amlitude and the eaks of the oscillation P νµ ν e are shifted. What about Mini-BooNE? As a homework assignment, one can use the energy distribution of ν µ s roduced by π + decays-in-flight and convolute this distribution to obtain the 6
0.035 0.03 0.05 0.0 0.015 0.01 0.005 Broadband f bb E 0.00 0.0015 0.001 0.0005 10 0 30 40 50 EMeV 50 100 150 00 50 300 350 meters Figure 4: Neutrino oscillations ν e aearance as a function of length for a wide band beam of ν µ s. Note the ronounced oscillations from Fig. 3 are diminished due to the broadband energy distribution of neutrinos roduced by decay-at-rest muons. Furthermore, the maxima are ushed to 40, 160, and 90 meters. oscillation function P νµ ν e. This would give a first order calculation of the oscillation robability one might observe from the Mini-BooNE exeriment, assuming the m and sin θ solutions from the SND exeriment are correct. A more recise calculation of the oscillation robability would require a full Monte-Carlo simulation of the roduction of neutrinos along with the geometrical and detector efficiencies of the Mini-BooNE exeriment. This full-scale Monte Carlo simulation is underway. 7