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# hw21 - a(5 pts Show that 1 β α =-1 β 2 α 2 β 3 β 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 difference Δ m 2 between the two neutrino mass states given that the flux of electron neutrinos as they hit Earth is only about 1/3 of the electron neutrino flux 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 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
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Unformatted text preview: a. (5 pts) Show that 1 β ( α ) =-1 β 2 α 2 + β 3 β 2 2 α + analytic terms in α. b. (10 pts) De²ne the Λ-parameter by Λ 2 = μ 2 c exp -α μ Z α dα β ( α ) with α and c some constants. Show that one can choose α and c such that ln μ 2 Λ 2 = 1 β 2 α μ + β 3 β 2 2 ln( β 2 α μ ) + o ( α μ ) . (1) c. (10 pts) Solve Eq. (1) above for the running coupling α μ ≡ α ( μ 2 ) in terms of ln( μ 2 / Λ 2 ) assuming that ln( μ 2 / Λ 2 ) ± 1 and keeping all terms which decrease with increasing μ 2 less rapidly than ln-3 ( μ 2 / Λ 2 ). d. (10 pts) Show that Λ 2 = μ 2 e-1 β 2 α μ ( β 2 α μ )-β 3 /β 2 2 [1 + o ( α μ )] (2) and prove that this Λ 2 is μ-independent. 1...
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