Neutrino amplitude decomposition, S matrix rephasing invariance, and reparametrization symmetry

Abstract

          The S matrix rephasing invariance is one of the fundamental principles of quantum mechanics that originates in its probabilistic interpretation. For a given S matrix which describes neutrino oscillation, one can define the two different rephased amplitudes 
            
              
            
            
              
                S
                αβ
                
                  Reph
                  −
                  1
                
              
              ≡
              
                e
                
                  i
                  
                    
                      
                        λ
                        1
                      
                      /
                      2
                      E
                    
                  
                  x
                
              
              
                S
                αβ
              
            
            $$ {S}_{\alpha \beta}^{\textrm{Reph}-1}\equiv {e}^{i\left({\lambda}_1/2E\right)x}{S}_{\alpha \beta} $$
           and 
            
              
            
            
              
                S
                αβ
                
                  Reph
                  −
                  2
                
              
              ≡
              
                e
                
                  i
                  
                    
                      
                        λ
                        2
                      
                      /
                      2
                      E
                    
                  
                  x
                
              
              
                S
                αβ
              
            
            $$ {S}_{\alpha \beta}^{\textrm{Reph}-2}\equiv {e}^{i\left({\lambda}_2/2E\right)x}{S}_{\alpha \beta} $$
          , which are physically equivalent to each other, where λk/2E denotes the energy eigenvalue of the k-th mass eigenstate. We point out that the transformation of the reparametrization (Rep) symmetry obtained with “Symmetry Finder” maps 
            
              
            
            
              
                S
                αβ
                
                  Reph
                  −
                  1
                
              
            
            $$ {S}_{\alpha \beta}^{\textrm{Reph}-1} $$
           to 
            
              
            
            
              
                S
                αβ
                
                  Reph
                  −
                  2
                
              
            
            $$ {S}_{\alpha \beta}^{\textrm{Reph}-2} $$
          , and vice versa, providing a local and manifest realization of the S matrix rephasing invariance by the Rep symmetry of the 1–2 state exchange type. It is strongly indicative of quantum mechanical nature of the Rep symmetry. The rephasing and Rep symmetry relation, though its all-order treatment remains incomplete, is shown to imply absence of the pure 1–3 exchange symmetry in Denton et al. perturbation theory. It then triggers a study of convergence of perturbation series.