= Solution
Let $\mathcal G_R=\langle\mathcal O_{1,R}(x_1)\cdots\mathcal O_{n,R}(x_n)\rangle$ at separated points, with <composite operators> chosen in a basis without operator mixing. Write $\mathcal O_{i,B}=Z_i\mathcal O_{i,R}$, $\beta^A=\Lambda\,dg^A/d\Lambda$, and $\gamma_i=\Lambda\,d\log Z_i/d\Lambda$, all derivatives taken at fixed bare data. Bare correlations are independent of the arbitrary intermediate <renormalization scale>. Hence \b[the Callan–Symanzik equation] is
$$
\boxed{\left(\Lambda\frac{\partial}{\partial\Lambda}+\sum_A\beta^A\frac{\partial}{\partial g^A}+\sum_{i=1}^n\gamma_i\right)\mathcal G_R=0.}
$$
The explicit scale derivative holds positions and renormalized parameters fixed; the <renormalization-group beta functions> describe their running; and the <anomalous dimensions> account for renormalization of each insertion. Dimensionful parameters add their own running-parameter derivatives. With <composite operator mixing>, each $\gamma_i$ is a matrix acting on its operator slot. The displayed sign follows the stated convention for $Z_i$. Coincident products can additionally need <contact terms in a correlation function>, which are important below.
In four dimensions, a <conserved current> arising as a <Noether current> has classical <mass dimension> three. Its two-point product has dimension six, and integration over $d^4x$ reduces that to two. The transverse tensor $k_\mu k_\nu-\delta_{\mu\nu}k^2$ already has dimension two. Thus \b[the scalar coefficient has dimension zero]:
$$
\boxed{[G_0]=0.}
$$
The assumed vanishing <anomalous dimension> protects the individual current, but does not remove the logarithmic short-distance divergence of its product with another current.
Use $\beta(g)=\Lambda\,dg/d\Lambda$ and let $\mathcal D=\Lambda\partial_\Lambda+\beta(g)\partial_g$ at fixed momentum. Bare-scale independence gives $\mathcal D G_0=0$ at fixed $\Lambda_0$. Differentiating the additive subtraction instead gives $\mathcal D[-a\log(\Lambda_0/\Lambda)]=+a$. Hence \b[the renormalized <conserved-current two-point function> obeys an inhomogeneous equation]:
$$
\boxed{\left(\Lambda\partial_\Lambda+\beta(g)\partial_g\right)G_{\mathrm{ren}}(k^2,g,\Lambda)=a.}
$$
This is the <Callan-Symanzik equation with additive renormalization>. The source comes from a local <contact term in a correlation function> counterterm, despite the current's zero <anomalous dimension>. A homogeneous equation would miss this contribution.
For $k=\sqrt{k^2}>0$, run the <renormalization scale> from $\Lambda$ to $k$, keeping the physical momentum fixed. Along the <renormalization-group flow>, $dG_{\mathrm{ren}}/d\log\mu=a$, so
$$
G_{\mathrm{ren}}(k^2,g(\Lambda),\Lambda)=C(g(k))-a\log\frac{k}{\Lambda},\qquad C(g)=G_{\mathrm{ren}}(k^2,g,k).
$$
Dimensional analysis makes $C$ independent of $k$ in this massless problem. <Asymptotic freedom> means $g(k)\to0$ as $k/\Lambda\to\infty$. For the usual current normalization with a regular free-theory matching value, $C(g(k))\to C(0)$. Consequently \b[the leading ultraviolet behaviour at this one-loop subtraction accuracy] is
$$
\boxed{G_{\mathrm{ren}}(k^2,g(\Lambda),\Lambda)=-\frac a2\log\frac{k^2}{\Lambda^2}+C(0)+o(1).}
$$
In particular the leading coefficient is $-a/2$, with its sign fixed by the subtraction convention in the question. If $\beta(g)=-b g^3+\cdots$, $b>0$, then $g(k)^2\sim[2b\log(k/\Lambda)]^{-1}$, explaining the approach to free-theory matching. Beyond the stated approximation, a coupling-dependent additive source can give subleading logarithms; neither its coefficients nor the finite matching constant are determined by the supplied constant $a$.
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