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Unformatted text preview: IPHO 2003 Solution Theoretical Question 3 (prelimary) p.1/5 Solution- Theoretical Question 3 Part A Neutrino Mass and Neutron Decay (a) Let ) , ( 2 e e q c E c , ) , ( 2 p p q c E c , and ) , ( 2 v v q c E c be the energy-momentum 4- vectors of the electron, the proton, and the anti-neutrino, respectively, in the rest frame of the neutron. Notice that q q q E E E p e p e , , , , , are all in units of mass. The proton and the anti-neutrino may be considered as forming a system of total rest mass c M , total energy c E c 2 , and total momentum c q c . Thus, we have v p c E E E + = , v p c q q q + = , 2 2 2 c c c q E M- = (A1) Note that the magnitude of the vector c q is denoted as q c . The same convention also applies to all other vectors. Since energy and momentum are conserved in the neutron decay, we have n e c m E E = + (A2) e c q q - = (A3) When squared, the last equation leads to the following equality 2 2 2 2 e e e c m E q q- = = (A4) From Eq. (A4) and the third equality of Eq. (A1), we obtain 2 2 2 2 e e c c m E M E- =- (A5) With its second and third terms moved to the other side of the equality, Eq. (A5) may be divided by Eq. (A2) to give ) ( 1 2 2 e c n e c m M m E E- =- (A6) As a system of coupled linear equations, Eqs. (A2) and (A6) may be solved to give ) ( 2 1 2 2 2 c e n n c M m m m E +- = (A7) ) ( 2 1 2 2 2 c e n n e M m m m E- + = (A8) Using Eq. (A8), the last equality in Eq. (A4) may be rewritten as ) )( )( )( ( 2 1 ) 2 ( ) ( 2 1 2 2 2 2 2 c e n c e n c e n c e n n e n c e n n e M m m M m m M m m M m m m m m M m m m q-- +-- + + + =-- + = (A9) Eq. (A8) shows that a maximum of e E corresponds to a minimum of 2 c M . Now the rest mass c M is the total energy of the proton and anti-neutrino pair in their center of mass (or momentum) frame so that it achieves the minimum IPHO 2003 Solution Theoretical Question 3 (prelimary) p.2/5 v p m m M + = (A10) when the proton and the anti-neutrino are both at rest in the center of mass frame. when the proton and the anti-neutrino are both at rest in the center of mass frame....
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