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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 = Q γ μ Q ; (2) j μ a = Q γ μ τ a Q ; (3) j μ 5 = Q γ μ γ 5 Q ; (4) j μ 5 a = Q γ 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 = u d ; Q L = u L d L = 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 = u L d L γ μ ( τ 1 + i τ 2 ) u L d L (8) = Q L γ μ ( τ 1 + i τ 2 ) Q L (9) = Q 1 + γ 5 2 γ μ ( τ 1 + i τ 2 ) 1 γ 5 2 Q (10) = 1 2 Q γ μ (1 γ 5 ) ( τ 1 + i τ 2 ) Q (11) = 1 2 Q γ μ τ 1 Q + i Q γ μ τ 2 Q Q γ μ γ 5 τ 1 Q i Q γ μ γ 5 τ 2 Q (12) = 1 2 j μ 1 + i j μ 2 j μ 5 1 i j μ 5 2 (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.
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