The Taylor–Lagrange scheme as a template for symmetry-preserving renormalization procedures

Abstract

A general regularization/renormalization scheme based on intrinsic properties of quantum fields as operator-valued distributions with adequate test functions is presented. The paracompactness property of the Minkowskian or Euclidean manifolds permits a unique definition of fields through integrals over the manifold based on test functions which are partition of unity (PU). These test functions turn out to provide a direct Lorentz-invariant scheme to the extension procedure of singular distributions and possess the unique property of being equal to their Taylor remainder of any order. When expressed through Lagrange’s formulae, this remainder leads to specific procedures of extension in the UV and IR domains. These results, directly obtainable at the physical dimension D = 4, are found to depend on an arbitrary scale present in the definition of any PU test functions and relevant to the final RG-analysis of physical amplitudes. The method is of general character in that it comprises the well-known symmetry-preserving procedures of Bogoliubov–Parasiuk–Hepp–Zimmermann, Pauli–Villars subtractions at the level of propagators and dispersion relations. Symmetry preservation is indeed verified explicitly in simple QED/QCD gauge-boson contributions and in a covariant light-front dynamics treatment of the Yukawa model.

Aspects of fine-tuning of the Higgs mass within finite field theories

“…This has been known for a long time. However, its full significance for practical calculations was not fully recognized until recently [4,10,11,[13][14][15].…”

“…[4], the test function f should have peculiar properties. It is chosen as a super regular partition of unity (PU, see [4] for more details on PUs), i.e. a function of finite support which is 1 everywhere except at the boundaries.…”

“…1, it is given by [4] T > ðXÞ ðÀXÞ k a k k! @ kþ1 X ½XT > ðXÞ Z 2 a dt t ða À tÞ k : (9) The value of k in (9) corresponds to the order of singularity of the original distribution T > ðXÞ [4]. In practice, it can be chosen as the smallest integer, positive or null, which leads to a nonsingular extensionT > ðXÞ.…”

“…This scale is the only remnant of the presence of the test function. The extension of singular distributions in the IR domain can be done similarly [4,15]. For an homogeneous distribution in one dimension, with T < ½X=t ¼ t kþ1 T < ðXÞ, the extension of the distribution in the IR domain is given bỹ…”

“…Doing this, the extensionT < ðXÞ is nothing else than the pseudofunction (Pf) of 1=X kþ1 [4,12,15] T < ðXÞ ¼ Pf 1…”