Do the Ward identities contain contact terms in Euclidean QFT?

Question

In derivations of the Ward identities, I have never seen the signature of spacetime explicitly specified, so I'd always assumed they hold regardless of signature. However, the argument below seems to imply that they cannot hold in Euclidean QFT.

Consider a Euclidean QFT with Noether current $j_\mu$. The Ward identites with operator insertions $\mathcal{O}_i(x_i)$ are usually written as:

$$\tag{1}\left< \partial^\mu j_\mu(x) \mathcal{O}_1(x_1)\cdots \mathcal{O}_n(x_n) \right>=\sum_{i=1}^n \delta(x-x_i) \left<\mathcal{O}_1(x_1) \cdots \delta \mathcal{O}_i(x_i)\cdots \mathcal{O}_n(x_n)\right>,$$

where the angle brackets denote the vacuum expectation. In particular, the $n=0,1$ identities are:

$$\tag{2}\left<\partial^\mu j_\mu\right>=0$$

$$\tag{3}\left< \partial^\mu j_\mu(x) \mathcal{O}(y)\right>= \delta(x-y)\left<\delta\mathcal{O(y)}\right>.$$

However, in the path integral derivation of Eq. $(1)$, the past and future BCs are arbitrary (e.g. see the derivation in Polchinski's String Theory Vol 1 Sec 2.3). Vacuum BCs are not required. So in fact we can strip off the angle brackets to get operator equations. Note that this is unlike in Lorentzian signature, where time ordering would prevent us from simply stripping off the angle brackets. Now, $(2)$ and $(3)$ become:

$$\tag{4}\partial^\mu j_\mu=0$$

$$\tag{5} \partial^\mu j_\mu(x) \mathcal{O}(y)= \delta(x-y)\cdot\delta\mathcal{O(y)}.$$

Polchinski stresses that these do indeed hold as operator equations.

But $(4)$ and $(5)$ clearly cannot both be true as operator equations, at least not in the usual sense. If $(4)$ were true then the LHS of $(5)$ would equal zero. Something has gone wrong,

Have I missed something, or are the usual Ward identities false in Euclidean signature?

Answer 1

Euclidean correlators do also have an operator ordering prescription, typically radial ordering ${\cal R}$, which is analogous to the time ordering $T$ for Minkowskian correlators. The radial ordering ${\cal R}$ becomes covariant radial ordering ${\cal R}_{\rm cov}$ in the Schwinger-Dyson (SD) equations,

$$\left< \Omega \left| {\cal R}_{\rm cov}\left\{ F[\phi]\frac{\delta S[\phi;J]}{\delta \phi(x)}\right\}\right| \Omega \right>_J~=~i\hbar\left< \Omega \left| {\cal R}_{\rm cov}\left\{\frac{\delta F[\phi]}{\delta \phi(x)} \right\}\right| \Omega \right>_J ~, \tag{A} $$

i.e. radial-differentiations inside its argument should be taken after/outside the usual radial ordering ${\cal R}$, cf. e.g. this Phys.SE post. This in turn induces contact terms in the Ward identities.