Comment on SU(16) grand unification.

In a recent paper on SU(16) grand unification, because of the presence of intermediateenergy gauge groups containing products of U(1) factors which are not orthogonal among themselves, the renormalization-group treatment has a few small errors. I correct it. I emphasize that one should not switch gauge couplings at the various thresholds. It is easier, and it avoids errors, to use throughout the gauge couplings of the standard model, and compute at each threshold, in the usual way, the extra contributions to the beta functions from the extra non-decoupled fields. I also point out that the SU(16) grand unification theory, due to the large number of scalars present in it, is not asymptotically free. It becomes a strong-coupling theory at energies only slightly larger than the unification scale. This is a Comment on the paper of Ref. [1], which is on the Grand Unified Theory (GUT) SU(16). The points made in this Comment are however also relevant to the GUT SU(15), notably its analysis in the paper of Ref. [2], and also other GUTs. Moreover, the first two paragraphs of this Comment are of a general nature, and one does not need to have read any of the above papers in order to understand them. In the context of GUTs, when intermediate-energy gauge groups containing products of U(1) factors arise, people usually use the set of U(1) factors which is more intuitive (which has some simple physical interpretation). Those U(1) factors are usually not, however, orthogonal relative to the active (non-decoupled) set of fields at each energy scale. Orthogonality is defined as follows: two U(1) charges are orthogonal if the sum over all the active fields of the product of their values, weighted by the factors 11/3 for gauge-boson fields, −2/3 for fermion fields, and −1/3 for complex scalar fields, vanishes. This non-orthogonality leads [3] to errors in the renormalization-group (RG) treatment of the theory. More or less erroneous conclusions may then be extracted from the RG analysis; I have given [3] as examples two GUTs, the first one based on the group SO(10), where the error has led only to small quantitative imprecisions in the results of the RG analysis, the second one based on the group SU(8), where the error has led to qualitatively false conclusions being extracted from the RG analysis. I find the same error in Ref. [1], this time in the context of the GUT SU(16). It is a much more complex case. The breaking chain of SU(16) in Ref. [1] has several intermediate-energy gauge groups,