# Why the contact terms in the Ward identity vanish due to the invariant Noether currents?

The picture below is a screenshot of Srednicki's QFT textbook.

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$$j^{\mu}$$ is the current associated with the $$U(1)$$ gauge symmetry; $$j_{A}^{\mu}$$ is the current associated with the axial $$U(1)$$ global symmetry (which is going to be proved anomalous).

I understand that both $$j^{\mu}$$ and $$j_{A}^{\mu}$$ are invariant under both $$U(1)$$ gauge symmetry and axial $$U(1)$$ symmetry, but I don't understand why this will lead to the vanishing of contact terms in the Wad identities (i.e. the right-hand side of eqs.(76.17-76.19) are zero)? Hope someone can give me an answer. Thank you!

• The right hand side of the ward identity is the contact terms. But each contact term is the variation of the operator under the U(1) gauge(axial) symmetry times a delta function. Since the currents themselves are invariants under the symmetry, the contact terms vanish. – Anonjohn Jan 11 '20 at 5:29
• @Anonjohn. Thanks. Now I understand. I go back to eq(22.26) of the book and check the form of contact term. Yes, the contact term contains the ''variation of the operator'' which in this case is zero since it is a Noether current. – youyou Jan 11 '20 at 5:39
• @Anonjohn (The Noether current j1 associated with symmetry G1 happens to be also invariant under transformation G2) – youyou Jan 11 '20 at 5:46