I am reading a paper, written by G. Chalmers and W. Siegel - https://arxiv.org/abs/hep-th/9606061, where they discuss the action of self-dual Yang-Mills theory, which in light-cone formalism is obtained by preserving only one of the triple vertices with the gluons of following helicity (++-), or in Lorentz invariant form the action has the following form: $$ S = \int_{M} \mathcal{P} \wedge F. $$ I adopt the notation of https://arxiv.org/abs/1711.10026. There $\mathcal{P}$ is an anti self-dual field ($\star \mathcal{P} = -\mathcal{P}$), which leads to the following equation of motion for the gauge field $A_\mu$ : $$ \star F = -F $$ Which is simply the self-duality equation for the field $F$. This theory has only two kind of amplitudes:
Tree amplitudes with one field $\mathcal{P}$ and any number of external gluons.
1-loop amplitude with no field $\mathcal{P}$ and any number of external gluons.
The expression for the 1-loop amplitude is given by rather a simple expression: $$ \mathcal{A}(1^+ \ldots n^+) = -\frac{i}{192 \pi^2}\sum \frac{[i j] \langle j k \rangle [k l] \langle l i \rangle}{\langle 1 2 \rangle \ldots \langle n 1 \rangle} $$ Which significantly resembles the famous $\textbf{MHV}$ amplitude.
The theory concerned is conformal on the classical level. However, at the quantum level there can be a conformal anomaly, generating the scale.
It is mentioned in the discussion section, that there is a possibility that the aforementioned one-loop contribution may by generated by a local term in the effective action by an introduction of an external field.
Do I correctly understand, that they speak about some contribution of form $\phi T_{\mu}^{\mu}$, where $\phi$ is kind of dilaton field?
The supersymmetric generalization of action of self-dual Yang-Mills action emerges as a low-energy of $\mathcal{N}=2$ open string theory. Then the authors say:
Explicit calculations in string theory [23], however, have indicated thevanishingof all one-loop graphs with more than three external lines in all N=2 string theories.These string results are in direct contradiction with field theory. This suggests somesubtlety was missed, possibly signalling the presence of an anomaly in the worldsheettheory describing the string.
Is there any progress in solving this issue since that time? Or any results for the anomalies in $\mathcal{N}=2$ string theory? I would be grateful for links or discussions in this area.