hw21 - a. (5 pts) Show that 1 ( ) =-1 2 2 + 3 2 2 +...

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Homework Set No. 1, Physics 880.02 Deadline – Tuesday, April 28, 2009 1. (10 pts) Assuming for simplicity that there are only two neutrino families (electron and muon neutrinos), estimate the mass squared diFerence Δ m 2 between the two neutrino mass states given that the ±ux of electron neutrinos as they hit Earth is only about 1/3 of the electron neutrino ±ux as they leave the Sun (that is 2/3 of electron neutrinos oscillate into muon ones). Assume that the distance between the Earth and the Sun is 1 . 5 × 10 11 m, neutrinos have the energy of about 1 MeV and the mixing angle θ is 45 . 2. In class we have discussed the renormalization group (Callan-Symanzik) equation ± μ 2 ∂μ 2 + β ( α μ ) ∂α μ ² M ( Q 2 2 , α μ ) = 0 for a dimensionless observable M in zero-mass QCD. Suppose the beta-function is exactly β ( α μ ) = - β 2 α 2 μ - β 3 α 3 μ in some renormalization scheme.
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Unformatted text preview: a. (5 pts) Show that 1 ( ) =-1 2 2 + 3 2 2 + analytic terms in . b. (10 pts) Dene the -parameter by 2 = 2 c exp - Z d ( ) with and c some constants. Show that one can choose and c such that ln 2 2 = 1 2 + 3 2 2 ln( 2 ) + o ( ) . (1) c. (10 pts) Solve Eq. (1) above for the running coupling ( 2 ) in terms of ln( 2 / 2 ) assuming that ln( 2 / 2 ) 1 and keeping all terms which decrease with increasing 2 less rapidly than ln-3 ( 2 / 2 ). d. (10 pts) Show that 2 = 2 e-1 2 ( 2 )- 3 / 2 2 [1 + o ( )] (2) and prove that this 2 is -independent. 1...
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This note was uploaded on 07/28/2011 for the course PHYSICS 880.02 taught by Professor Kovchegov during the Winter '09 term at Ohio State.

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