Problem Set 5 Solutions

# Hence the matrix elements for the corresponding

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Unformatted text preview: 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 ﬁrst note that the vector current in QCD is conserved, while there is an axial-vector anomaly. Hence the matrix elements for the corresponding currents between vacuum and a pion state are given as: < 0| j µ a (x) |π b (p) > < 0| j µ 5 a (x) |π b (p) > =0 = − i pµ fπ δ ab e−i p·x (14) (15) We may simplify the leptonic current and write it as: lL γ µ νL = 1µ l γ (1 − γ 5 ) ν 2 1 (16) Using the above simpliﬁcations, the matrix element (in position space) takes the form: ￿ + i T = < l ν| i d4 x ∆L |π + > ￿ 4 GF = −i √ d4 x < l+ ν | (ν L γ µ lL ) (uL γµ dL )† |π + > 2 ￿ √ = − i 2 GF (u(q ) γ µ (1 − γ 5 ) v (k )) d4 x ei (k+q)·x < 0| (uL γµ dL )† |π + (p) > (17) (18) (19) We can evaluate the integrand in (19) using the relations (13), (14) and (15), so as to obtain: < 0| (uL γµ dL )† |π + (p) > = 1 √ i pµ fπ e−i p·x 2 (20) Thus we obtain: iT ∴ iM = fπ GF u(q ) ￿p (1 − γ 5 ) v (k ) (2π )4 δ (4) (k + q − p) = GF fπ (u(...
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