Renormalization of a class of non-renormalizable theories

Certain power-counting non-renormalizable theories, including the most general self-interacting scalar fields in four and three dimensions and fermions in two dimensions, have a simplified renormalization structure. For example, in four-dimensional scalar theories, 2n derivatives of the fields, n>1, do not appear before the nth loop. A new kind of expansion can be defined to treat functions of the fields (but not of their derivatives) non-perturbatively. I study the conditions under which these theories can be consistently renormalized with a reduced, eventually finite, set of independent couplings. I find that in common models the number of couplings sporadically grows together with the order of the expansion, but the growth is slow and a reasonably small number of couplings is sufficient to make predictions up to very high orders. Various examples are solved explicitly at one and two loops.Comment: 38 pages, 1 figure; v2: more explanatory comments and references; appeared in JHE

A note on the improvement ambiguity of the stress tensor and the critical limits of correlation functions

I study various properties of the critical limits of correlators containing insertions of conserved and anomalous currents. In particular, I show that the improvement term of the stress tensor can be fixed unambiguously, studying the RG interpolation between the UV and IR limits. The removal of the improvement ambiguity is encoded in a variational principle, which makes use of sum rules for the trace anomalies a and a'. Compatible results follow from the analysis of the RG equations. I perform a number of self-consistency checks and discuss the issues in a large set of theories.Comment: 15 page

Infinite reduction of couplings in non-renormalizable quantum field theory

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 irrelevant terms as unique functions of a reduced set of independent couplings lambda, such that the divergences are removed by means of field redefinitions plus renormalization constants for the lambda's. I consider non-renormalizable theories whose renormalizable subsector R is interacting and does not contain relevant parameters. 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 irrelevant interactions and address the problem of their physical selection.Comment: 40 pages; v2: more explanatory comments; appeared in JHE

Higher-spin current multiplets in operator-product expansions

Various formulas for currents with arbitrary spin are worked out in general space-time dimension, in the free field limit and, at the bare level, in presence of interactions. As the n-dimensional generalization of the (conformal) vector field, the (n/2-1)-form is used. The two-point functions and the higher-spin central charges are evaluated at one loop. As an application, the higher-spin hierarchies generated by the stress-tensor operator-product expansion are computed in supersymmetric theories. The results exhibit an interesting universality.Comment: 19 pages. Introductory paragraph, misprint corrected and updated references. CQG in pres

Consistent irrelevant deformations of interacting conformal field theories

I show that under certain conditions it is possible to define consistent irrelevant deformations of interacting conformal field theories. The deformations are finite or have a unique running scale ("quasi-finite"). They are made of an infinite number of lagrangian terms and a finite number of independent parameters that renormalize coherently. The coefficients of the irrelevant terms are determined imposing that the beta functions of the dimensionless combinations of couplings vanish ("quasi-finiteness equations"). The expansion in powers of the energy is meaningful for energies much smaller than an effective Planck mass. Multiple deformations can be considered also. I study the general conditions to have non-trivial solutions. As an example, I construct the Pauli deformation of the IR fixed point of massless non-Abelian Yang-Mills theory with N_c colors and N_f <~ 11N_c/2 flavors and compute the couplings of the term F^3 and the four-fermion vertices. Another interesting application is the construction of finite chiral irrelevant deformations of N=2 and N=4 superconformal field theories. The results of this paper suggest that power-counting non-renormalizable theories might play a role in the description of fundamental physics.Comment: 23 pages, 5 figures; reference updated - JHE

Inequalities for trace anomalies, length of the RG flow, distance between the fixed points and irreversibility

I discuss several issues about the irreversibility of the RG flow and the trace anomalies c, a and a'. First I argue that in quantum field theory: i) the scheme-invariant area Delta(a') of the graph of the effective beta function between the fixed points defines the length of the RG flow; ii) the minimum of Delta(a') in the space of flows connecting the same UV and IR fixed points defines the (oriented) distance between the fixed points; iii) in even dimensions, the distance between the fixed points is equal to Delta(a)=a_UV-a_IR. In even dimensions, these statements imply the inequalities 0 =< Delta(a)=< Delta(a') and therefore the irreversibility of the RG flow. Another consequence is the inequality a =< c for free scalars and fermions (but not vectors), which can be checked explicitly. Secondly, I elaborate a more general axiomatic set-up where irreversibility is defined as the statement that there exist no pairs of non-trivial flows connecting interchanged UV and IR fixed points. The axioms, based on the notions of length of the flow, oriented distance between the fixed points and certain "oriented-triangle inequalities", imply the irreversibility of the RG flow without a global a function. I conjecture that the RG flow is irreversible also in odd dimensions (without a global a function). In support of this, I check the axioms of irreversibility in a class of d=3 theories where the RG flow is integrable at each order of the large N expansion.Comment: 24 pages, 3 figures; expanded intro, improved presentation, references added - CQ

Low-energy Phenomenology Of Scalarless Standard-Model Extensions With High-Energy Lorentz Violation

We consider renormalizable Standard-Model extensions that violate Lorentz symmetry at high energies, but preserve CPT, and do not contain elementary scalar fields. A Nambu--Jona-Lasinio mechanism gives masses to fermions and gauge bosons, and generates composite Higgs fields at low energies. We study the effective potential at the leading order of the large-N_{c} expansion, prove that there exists a broken phase and study the phase space. In general, the minimum may break invariance under boosts, rotations and CPT, but we give evidence that there exists a Lorentz invariant phase. We study the spectrum of composite bosons and the low-energy theory in the Lorentz phase. Our approach predicts relations among the parameters of the low-energy theory. We find that such relations are compatible with the experimental data, within theoretical errors. We also study the mixing among generations, the emergence of the CKM matrix and neutrino oscillations.Comment: 32 pages; v2: typos corrected, more references, some more comments - PR

Tailoring Dielectric Properties of Multilayer Composites Using Spark Plasma Sintering

A straightforward and simple way to produce well-densified ferroelectric ceramic composites with a full control of both architecture and properties using spark plasma sintering (SPS) is proposed. SPS main outcome is indeed to obtain high densification at relatively low temperatures and short treatment times thus limiting interdiffusion in multimaterials. Ferroelectric/dielectric (BST64/MgO/BST64) multilayer ceramic densified at 97% was obtained, with unmodified Curie temperature, a stack dielectric constant reaching 600, and dielectric losses dropping down to 0.5%, at room-temperature. This result ascertains SPS as a relevant tool for the design of functional materials with tailored properties

HyperK\"ahler quotients and N=4 gauge theories in D=2

We consider certain N=4 supersymmetric gauge theories in D=2 coupled to quaternionic matter multiplets in a minimal way. These theories admit as effective theories sigma-models on non-trivial HyperK\"ahler manifolds obtained as HyperK\"ahler quotients. The example of ALE manifolds is discussed. (Based on a talk given by P. Fr\'e at the F. Gursey Memorial Conference, Istanbul, June 1994).Comment: 22 pages, Latex, no figure

Renormalizable acausal theories of classical gravity coupled with interacting quantum fields

We prove the renormalizability of various theories of classical gravity coupled with interacting quantum fields. The models contain vertices with dimensionality greater than four, a finite number of matter operators and a finite or reduced number of independent couplings. An interesting class of models is obtained from ordinary power-counting renormalizable theories, letting the couplings depend on the scalar curvature R of spacetime. The divergences are removed without introducing higher-derivative kinetic terms in the gravitational sector. The metric tensor has a non-trivial running, even if it is not quantized. The results are proved applying a certain map that converts classical instabilities, due to higher derivatives, into classical violations of causality, whose effects become observable at sufficiently high energies. We study acausal Einstein-Yang-Mills theory with an R-dependent gauge coupling in detail. We derive all-order formulas for the beta functions of the dimensionality-six gravitational vertices induced by renormalization. Such beta functions are related to the trace-anomaly coefficients of the matter subsector.Comment: 36 pages; v2: CQG proof-corrected versio