Global Fit of Neutrino Oscillation Parameters

1. Global Fit of Neutrino Oscillation Parameters Student: Wei-Jiun Tsai Supervisor: Melin Huang, Pisin Chen

2. Part 1Neutrino Oscillation • In this part, I just give a global picture about how to calculate the transition probability from one neutrino flavor to another. • The state of a neutrino can be expressed either in the flavor eigenstate basis or in the mass eigenstate basis. The transformation between the two bases is • Because what I study is atmospheric neutrinos, this part is divided into two portions : (1) Neutrinos travel from the source (atmospheric layer) to the earth surface( propagation in vacuum). (2) Neutrinos travel from the earth surface to the detector (propagation in matter). (1.1) flavor basis mass basis , δ is the CP-violating phase, and U is a unitary matrix,

3. Part 1Neutrino Oscillation Now we solve the time-evolution equation of three-flavor neutrinos. The wave function of a neutrino propagating through a medium obeys Schrodinger equation (1) In vacuum : In the flavor eigenstate basis, (1.2) can be rewritten as (1.2) where Hamiltonian (1.3) where and Hamiltonian for a neutrino propagating in vacuum, expressed in mass eigenstate basis, and (1.4)

4. Part 1Neutrino Oscillation (2) In matter Because of weak interaction between neutrinos and matter, the Hamiltonian should be modified by including such a effect. By (1.5), (1.4) becomes (1.5) where The modified term from weak interaction for a neutrino propagating in matter, expressed in flavor eigenstate basis. (1.6) , Veis the potential for the charged-current interaction. Gf is Fermi’s constant, Ne is the electron number density and (1.7) Solving equation (1.4) and (1.7) computationally, one can get after neutrino travel through a certain distance. Function of (1.8)

5. Part 2Theoretical Yield/Rate Expectation • Yield is the number of events that are detected. The theoretical yield formula in general is expressed as and rate is defined by Variables in equation (2.1) and (2.2) are explained in detail on next page (2.1) Some certain incident zenith angle of ν (2.2) Function of neutrino oscillation parameters

6. Part 2Theoretical Yield/Rate Expectation For SNO and Super-K atmospheric neutrino, Φν : SNO uses Bartolνatmos flux distribution. Super-K uses Honda νatmos flux distribution. Pαβ : include ① ν propagation from atmosphere to the earth surface and ② ν propagation from the earth surface to detector For ① : Need to calculate For ② : Melin has the code I have to do. (C) ϵ : detection efficiency, can be found from published papers. (D) : total number of target nucleon. (E) tlive: total livetime. All are involved in SNO and Super-K atmospheric neutrino analysis Quasi-Elastic Scattering Deep-Inelastic Scattering Single meson production Coherent π production (F) : differential cross section Quantities needed to check: Quantities needed to calculate: Pαβ , theoretical yield or rate as functions of neutrino oscillation parameters

7. Part 3Analysis Method n : Eν energy bin. m : neutrino zenith angle bin. Depending on data distribution, we have two methods Dnmis measured yield. Ynmis theoretical yield. (1) If the data behave like a Gaussian distribution. σnmis standard deviation. (2) If the data behave like a Poisson distribution. By minimizing , one can find the best fit of neutrino oscillation parameters.

8. Part 4 Existing Experimental Data for global fit SNO  done Solar experiments Super-K  done Homestake done GNO  done Rate experiments Gallex done SAGE  done Combine together for global fit of neutrino oscillation parameters Brexino done CHOOZ  done Reactor experiments KamLAND done Accelerator νμ MINOS  done K2K  done Atmospheric νμ SNO on going Super-K on going I have to deal with them.