Infinite reduction of couplings in non-renormalizable quantum field theory

Abstract

I study the problem of renormalizing a non-renormalizable theory with a reduced, eventually finite, set of independent couplings. The idea is to look for special relations that express the coefficients of the non-renormalizable terms as unique functions of a reduced set of independent couplings λ, such that the divergences are removed by means of field redefinitions plus renormalization constants for the λs. I consider non-renormalizable theories whose renormalizable subsector R is interacting. The "infinite" reduction is determined by i) perturbative meromorphy around the free-field limit of R, or ii) analyticity around the interacting fixed point of R. In general, prescriptions i) and ii) mutually exclude each other.When the reduction is formulated using i), the number of independent couplings remains finite or slowly grows together with the order of the expansion. The growth is slow in the sense that a reasonably small set of parameters is sufficient to make predictions up to very high orders. Instead, in case ii) the number of couplings generically remains finite. The infinite reduction is a tool to classify the non-renormalizable interactions and address the problem of their physical selection.

The Asymptotic Safety Scenario in Quantum Gravity

“…In a QFT with infinitely many relevant couplings (QCD in a lightfront formulation) the reduction principle was used by Perry-Wilson [174]. A general study of an ‘infinite reduction’ of couplings has been performed in [11]. …”

“…One reason lies in the fact that the number of independent relevant directions connected to the fixed point might not be known. Hidden dependencies would then allow for a (genuine or effective) reduction of couplings [236, 160, 174, 11, 16]. For quantum gravity the situation is further complicated by the fact that generic physical quantities are likely to be related only nonlocally and nonlinearly to the metric.…”

Dimensionally continued infinite reduction of couplings

“…The solution f 0,n (α) can be worked out in power series of α up to the order α r−1 , while the coefficient of α r is ill-defined: the problem is avoided introducing a new independent coupling λ nℓ 's with i > 0 belong to the evanescent sector, so they do not affect the physical quantities. These properties ensure that the violations of the invertibility conditions are less harmful than they appear at first sight: certainly they cause the introduction of new parameters, possibly infinitely many, but in general τ n grows with n and the physical new couplings appear at higher and higher orders, thus permitting low-order predictions with a relatively small number of independent couplings [3].…”

“…In the absence of three-leg vertices, the invertibility conditions found in [4,3] read in the notation of this paper (recall that here γ nℓ denotes the anomalous dimensions of the operators g 2pn O n (ϕ)) r n ≡ τ n + n − 1 + p n / ∈ N. (5.9)…”

Renormalization of a class of non-renormalizable theories

“…The arguments of this section apply to the theories of the form (2.3), but immediate generalizations work also with the most general non-renormalizable theory (see [7] for details). For definiteness, I work in even d spacetime dimensions.…”