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I∂/ ψ = ψ¯W Λ+ i∂/ Λ+ ψW 32 (150) Since Λ is space-dependent through the fields Φ(x), the derivative also acts on Λ, giving rise to additional terms. After some straightforward algebra, one finds ψ¯W Λ+ i∂/ Λ+ ψW = ψ¯W (i∂/ + γ µ Vµ + γ µ γ5 Aµ ) ψW (151) with 1 + ξ ∂µ ξ + ξ∂µ ξ + 2 i + ξ ∂µ ξ − ξ∂µ ξ + = 2 Vµ = (152) Aµ (153) ξ = ei τ Φ(x) 2fπ ⇒ U = ξξ (154) We do not need to transform the potential of the linear sigma-model, V (π, σ), since it vanishes on the chiral circle due to the constraint condition (139).
This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics Division of the Department of Energy, under Contract No. DE-AC03-76SF00098. D. D. Drell, Relativistic Quantum Fields, Mc Graw Hill, New York, 1965.  P. Ramond, Field Theory, Addison-Wesley, New York, 1989.  E. Henley and W. Thirring, Elementary Quantum Field Theory, Mc Graw Hill, New York, 1962. D. H. , 1983.  S. Weinberg, Phys. Rev. Lett. 18 (1967) 507. J. Sakurai, Currents and mesons, University of Chicago Press, Chicago, 1969.
Assuming that the pion mass vanishes. The explicit breaking of chiral symmetry is introduced by terms of the form ∼ tr(U + + U ) and and the simplest symmetry breaking is δLXχSB = fπ2 m2π tr(U + + U ) 4 1 4 − m2π Φ2 + O(Φ4 ) 2 (163) which to leading order in the pion-fields corresponds to a pion mass-term (the constant term does not contribute to the dynamics). Again, one can have many symmetry breaking term involving the above structure, such as tr(U + + U ), tr(∂µ U + ∂ µ U )tr(U + + U ) . .