Problem Set 5 Solutions

Problem Set 5 Solutions - QFT 3 Problem Set 5 1 Peskin...

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QFT 3 : Problem Set 5 1.) Peskin & Schroeder 19.2 Weak decay of the pion. (a.) In this section we are working with the Lagrangian: L = 4 G F 2 ( l L γ μ ν L )( u L γ μ d L )+ h.c. (1) We need to express the hadronic part of the operator for semileptonic weak interactions in terms of the following quark currents: j μ = μ Q ; (2) j μa = μ τ a Q ; (3) j μ 5 = μ γ 5 Q ; (4) j μ 5 a = μ γ 5 τ a Q ; (5) where τ a = σ a 2 represents the generators for SU (2). We may represent the quarks using the following matrix structure: Q = ° u d ± ; Q L = ° u L d L ± = 1 γ 5 2 Q ; (6) Q = ² ud ³ ; Q L = ² u L d L ³ = Q 1+ γ 5 2 ; (7) We may now use the above notation to rewrite the hadronic operator as follows: u L γ μ d L = ² u L d L ³ γ μ ( τ 1 + 2 ) ° u L d L ± (8) = Q L γ μ ( τ 1 + 2 ) Q L (9) = Q γ 5 2 γ μ ( τ 1 + 2 ) 1 γ 5 2 Q (10) = 1 2 μ (1 γ 5 τ 1 + 2 ) Q (11) = 1 2 ´ μ τ 1 Q + i μ τ 2 Q μ γ 5 τ 1 Q i μ γ 5 τ 2 Q µ (12) = 1 2 ´ j μ 1 + ij μ 2 j μ 51 μ 52 µ (13) In going from eqn.(10) to (12) we have made use of some known properties of γ matrices: γ μ γ 5 = γ 5 γ μ and ( γ 5 ) 2 = 1. In order to evaluate the matrix element for pion decay, we Frst note that the vector current in QCD is conserved, while there is an axial-vector anomaly. Hence the matrix elements for the corresponding
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Problem Set 5 Solutions - QFT 3 Problem Set 5 1 Peskin...

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