Theorem: Complete ambiguity and its dimension

Every extension in Assumption ass:class is uniquely of the form

with real $c,\tau ,w$.

Conversely, these formulas give all the required extensions. For coupling-dependent families, $c,\tau ,w$ can be taken independent of $\xi$, and then the improvement identity holds.

The coefficients $c,\tau$ are even in $\beta$, and $w$ is odd. Consequently:

1. for a generic charged sector $\beta \notin \{0,1/2\}$, the contact space has dimension $3$;

2. at a self-conjugate charged sector $\beta =0$ or $1/2$, $w=0$, and the dimension is $2$;

3. for the real neutral field, there is no charged current and the dimension is $2$.

The dimension is for fixed background and field parameters, not for arbitrary functions of those parameters.

Proof. The difference of two extensions is supported on $\mathcal{W}$. The structure theorem for distributions supported on a submanifold expresses it as a finite sum of transverse delta derivatives with coefficient distributions on $\mathcal{W}$ Dang (Section 8) Hormander (Theorem 2.3.5). Translation invariance forces those coefficient distributions to be constants times the longitudinal Lebesgue distribution.

A transverse derivative of order $m$ obeys

The scaling-degree bounds and (eq:AHclass) therefore allow only order $0$ for the field-square difference, order $2$ for stress, and order $1$ for current. For example, a stress term proportional to ${\delta }_{\mathcal{W}}$ has $(E+4{)}^2{\delta }_{\mathcal{W}}=4{\delta }_{\mathcal{W}}$, and is excluded.

Rotation covariance and the reflection/charge-conjugation conditions give

with no mixed blocks. The parity-odd symmetric transverse tensor is excluded by the even charge-conjugation behaviour of stress combined with meridian reflection.

For the current, longitudinal Lorentz covariance excludes a constant longitudinal vector. Rotation covariance allows ${\partial }_i{\delta }_{\mathcal{W}}$ and ${ϵ}_i{}^k{\partial }_k{\delta }_{\mathcal{W}}$. The radial term is even under meridian reflection, whereas the current must reverse under charge conjugation. The simultaneous requirements exclude the radial term. Thus the current difference is precisely the term in (eq:allJ).

Stress conservation gives $v+z=0$, so write the transverse block as $v^{\prime }{K}_{ij}{\delta }_{\mathcal{W}}$. Its trace is

The trace equation in (eq:classWard) requires

Put $\tau =u-nqc$. Then $v^{\prime }=2\tau -nqc$, which is exactly (eq:allT) by (eq:Iblocks) and (eq:Hcontact). The parameters are unique because the listed contact generators are linearly independent.

Conversely, the contact tensors in (eq:allF)–(eq:allJ) satisfy all the symmetries, scaling conditions, and Ward equations. The improvement identity follows by subtracting the same formula at $q=0$. At a self-conjugate sector, oddness forces $w=0$.