One-loop calculations for a translation invariant non-commutative gauge model

Abstract

In this paper we discuss one-loop results for the translation invariant non-commutative gauge field model we recently introduced in ref. [1]. This model relies on the addition of some carefully chosen extra terms in the action which mix long and short scales in order to circumvent the infamous UV/IR mixing, and were motivated by the renormalizable non-commutative scalar model of Gurau et al. [2].

On the problem of renormalizability in non‐commutative gauge field models – a critical review

“…(6) cannot simply be generated by contracting F µν with θ µν [25] but requires 'fine tuning' of the action. Several generalizations of the mechanisms working in scalar theory to Yang-Mills type models have been presented [21,22,23,25,26,27], but proofs of renormalizability are still missing, and it is not obvious that all problems can be solved. At least, up to now there is no model which features a suitable term to absorb the one-loop divergence Eqn.…”

“…However, their approach was unsuccessful: On the one hand the functional integrals they obtained lacked sufficient positivity, and on the other hand the related Gribov problem [73] was not solved. Nonetheless we may suggest to use the MSA and try to handle the Gribov problem using a soft breaking mechanism [26,74] similar to the one present in the Gribov-Zwanziger action [75,76,77], and see if renormalizability can in principle be achieved.…”

“…• the construction of a renormalizable gauge model, or more precisely, a rigorous proof of renormalizability of one of the promising candidates [21,22,23,24,25,26,27].…”

One-loop structure of the U(1) gauge model on the truncated Heisenberg space

“…The latter was implemented for the scalar field theory in [13]. Several generalizations were studied for the gauge fields, for example models defined in [33,34] and [36,37]; however, the complexity of the actions prevented the complete analysis so it remains unclear which nonlocal operators could render the gauge theory renormalizable. Our present result shows that −1 terms appear in quantization even in a local version of gauge theory.…”

Gauge fields on non‐commutative spaces and renormalization

“…where the operator t 2 p denotes a second order Taylor expansion with respect to p around p = 0 (but keepingp = 0 and independent), and we introduced the abbreviation N := k 2 + a 2 k 2 . The integral (9) may eventually be carried out further by using the decomposition [26,27]…”