Conformal invariance and vector operators in the \(O(N)\) model (Q2283164)

Not mentioned in the title but aparent from the introduction, the publication entitled ``Conformal Invariance and Vector Operators in the \(O(N)\) Model'' deals with the renormalisation flow at the fix points. In general, conformal invariance means the invariance of angular relations, and this invariance is of central importance for the renormalisation flow because only under this invariance the different, in the ideal case orthogonal, scalings can be disentangled. This at least remains unmentioned in the publication but from my point of view is important to keep in mind. Conformal invariance is found to be directy related to scale invariance in the case that the integrated scale dimension is strictly larger than \(-1\). This theorem is criticised by mathematicians but seems to be rigidly valid in physics. The authors then spend the first half of the publication in arguing for the validity of this theorem and showing (mostly numerically) that this condition is indeed satisfied for the \(O(N)\) model for all kinds of values of \(N\), from \(N=0\) (which is difficult to understand) to the large \(N\) limit, and for different space-time dimensions, starting from \(d=2.5\) for technical reasons. As the dependencies on both quantities turn out to be smooth, a sudden fall-off to integrated scale dimension of \(-1\) and, therefore, the breakdown of conformal symmetry is not expected. In the second part the authors apply their considerations to the Ising universality class, again defending the statements against criticism of a previous publication on this subject. The argumentation is straightforward but quite difficult to understand for a non-expert in this field. In general, including also five appendices (the last one not mentioned in the introduction), the publication presents a lot of details on relevant and less relevant operators and their scaling behaviour. I am convinced that the authors were able to dispel doubts on the simple criterion for the essential feature of conformal invariance of renormalisation flows close to fix points.