Super-renormalizable Theory Research Articles

Introduction to Nonlocal Field Theory Including Gravity

We give minireview of nonlocal field theory (infinite derivative field theory). We start with the discussion of the main peculiarities of nonlocal field theory on the example of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d = 4$$\\end{document} scalar \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\phi }^{4}}$$\\end{document}-model. The nonlocal \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\phi }^{4}}$$\\end{document}-model is ultraviolet finite, unitary, and macrocausal. One of the problems of nonlocal field theory is that the formfactor is an arbitrary entire function that makes the predictions extremely weak. We propose some additional principle that allows to fix the formfactor. Also we review the main results obtained in nonlocal quantum gravity, namely the nonlocal generalization of Einstein gravity leads to the superrenormalizable theory.

Hidden non-locality and self-superrenormalization of quantum gravity

We show that the formation/evaporation of Black Holes (BH) unitarizes quantum gravity at all the orders of the perturbation theory. Non-perturbative quantum effects save the scattering amplitudes from any polynomial divergences. Such a phenomena is intimately related to the dynamical emergence of an effective non-locality as well as emergent modifications of the Heisenberg’s uncertainty principle. The BH production delocalizes quantum gravity vertices and propagators as a consequence of its holographically stored entropy. In this sense, quantum gravity is a superrenormalizable theory, although non-locality is hidden in its action.

A new renormalon in two dimensions

According to standard lore, perturbative series of super-renormalizable theories have only instanton singularities. In this paper we show that two-dimensional scalar theories with a spontaneously broken O(N ) symmetry at the classical level, which are super-renormalizable, have an IR renormalon singularity at large N . Since perturbative expansions in these theories are made around the “false vacuum” in which the global symmetry is broken, this singularity can be regarded as a manifestation of the non-perturbative absence of Goldstone bosons. We conjecture that the Borel singularity in the ground state energy of the Lieb-Liniger model is a non-relativistic manifestation of this phenomenon. We also provide en passant a detailed perturbative calculation of the Lieb-Liniger energy up to two-loops, and we check that it agrees with the prediction of the Bethe ansatz.

Finite entanglement entropy of black holes

We compute the area term contribution to black holes’ entanglement entropy (using the conical technique) for a class of local or weakly non-local super-renormalizable gravitational theories coupled to matter. For the first time, we explicitly prove that all the beta functions in the proposed theory, except for the cosmological constant, are identically zero in cut-off regularization scheme and not only in dimensional regularization scheme. In particular, we show that there is no divergence quadratic in cut-off and hence there is no contribution to the beta function of the Newton constant. As a consequence of this result, we argue that in these theories of gravity conical entropy is a sensible definition of physical entropy, in particular, it is positive-definite and gauge independent. On top of this the conical entropy, being expressed only in terms of the classical Newton constant, turns out to be finite and naturally coincides with Bekenstein-Hawking entropy. Finally, we propose a theory in which the renormalization of the Newton constant is entirely due to the Standard Model matter, arguing that such a contribution does not give the usual interpretational problems of conical entropy discussed in the literature.

Finite quantum gauge theories

We explicitly compute the one-loop exact beta function for a nonlocal extension of the standard gauge theory, in particular Yang-Mills and QED. The theory, made of a weakly nonlocal kinetic term and a local potential of the gauge field, is unitary (ghost-free) and perturbatively super-renormalizable. Moreover, in the action we can always choose the potential (consisting of one "killer operator") to make zero the beta function of running gauge coupling constant. The outcome is "a UV finite theory for any gauge interaction". Our calculations are done in D=4, but the results can be generalized to even or odd spacetime dimensions. We compute the contribution to the beta function from two different killer operators by using two independent techniques, namely the Feynman diagrams and the Barvinsky-Vilkovisky traces. By making the theories finite we are able to solve also the Landau pole problems, in particular in QED. Without any potential the beta function of the one-loop super-renormalizable theory shows a universal Landau pole in the running coupling constant in the ultraviolet regime (UV), regardless of the specific higher-derivative structure. However, the dressed propagator shows neither the Landau pole in the UV, nor the singularities in the infrared regime (IR).

Emergence of Lowenstein–Zimmermann mass terms for QED3

In this paper we consider a super-renormalizable theory of massless QED in (2 + 1) dimensions and discuss their BRST symmetry transformation. By extending the BRST transformation, we derive the Nielsen identities for the theory. Further, we compute the generalized BRST (so-called FFBRST) transformation by making the transformation parameter field dependent. Remarkably, we observe that the Lowenstein–Zimmermann mass terms, containing Lowenstein–Zimmermann parameter which plays an important role in the BPHZL renormalization program, along with the external sources coupled to the nonlinear BRST variations appear naturally in the theory through an FFBRST transformation.

Standard model and graviweak unification with (super)renormalizable gravity. Part I: Visible and invisible sectors of the universe

We develop a self-consistent Spin (4, 4)-invariant model of the unification of gravity with weak SU(2) gauge and Higgs fields in the visible and invisible sectors of our universe. We consider a general case of the graviweak unification, including the higher-derivative super-renormalizable theory of gravity, which is a unitary, asymptotically-free and perturbatively consistent theory of the quantum gravity.

Do we have unitary and (super)renormalizable quantum gravity below the Planck scale?

We explore how the stability of metric perturbations in higher derivative theories of gravity depends on the energy scale of initial seeds of such perturbations and on a typical energy scale of the gravitational vacuum background. It is shown that, at least in the cases of specific cosmological backgrounds, that the unphysical massive ghost which is present in the spectrum of such theories is not growing up as a physical excitation and remains in the vacuum state, until the initial frequency of the perturbation is close to the Planck order of magnitude. In this situation the existing versions of renormalizable and superrenormalizable theories can be seen as very satisfactory effective theories of Quantum Gravity below the Planck scale.

Hidden quantum groups symmetry of super-renormalizable gravity

In this paper, we consider the relation between the super-renormalizable theories of quantum gravity studied by Biswas, Gerwick, Koivisto, and Mazumdar [Phys. Rev. Lett. 108, 031101 (2012)] and Modesto [arXiv:1107.2403; arXiv:1202.0008] and an underlying noncommutativity of space-time. For one particular super-renormalizable theory, we show that at the linear level (quadratic in the Lagrangian) the propagator of the theory is the same one we obtain starting from a theory of gravity endowed with $\ensuremath{\theta}$-Poincar\'e quantum groups of symmetry. Such a theory is over the so-called $\ensuremath{\theta}$-Minkowski noncommutative space-time. We shed new light on this link and show that, among the theories considered in these references, there exists only one nonlocal and Lorentz invariant super-renormalizable theory of quantum gravity that can be described in terms of a quantum-group symmetry structure. We also emphasize contact with preexistent works in the literature and discuss preservation of the equivalence principle in our framework.

Determination of the propagator in QED 3 by spectral function

To study the infrared behaviour of the propagator, exponentiation of the lowest order spectral function has been known.We show this method is helpful in super renormalizable theory with dimension-full coupling constant.In the 1/N approximation anomalous dimension is independent of $N$,which plays an important role for confinement and pair condensation.

Dynamical Lorentz symmetry breaking from a(3+1)-dimensional axion-Wess-Zumino model

We study renormalizable Abelian vector-field models in the presence of the Wess-Zumino interaction with pseudoscalar matter. Renormalizability is achieved by supplementing the standard kinetic term of vector fields with higher derivatives. The appearance of a fourth power of momentum in the vector-field propagator leads to the superrenormalizable theory in which the \ensuremath{\beta} function, the vector-field renormalization constant, and the anomalous mass dimension are calculated exactly. It is shown that this model has the infrared stable fixed point and its low-energy limit is nontrivial. The modified effective potential for the pseudoscalar matter leads to the possible occurrence of dynamical breaking of Lorentz symmetry. This phenomenon is related to the modification of electrodynamics by means of the Chern-Simons (CS) interaction polarized along a constant CS vector. Its presence makes the vacuum that has been recently estimated from astrophysical data optically active. We examine two possibilities for the CS vector to be timelike or spacelike, under the assumption that it originates from VEV of some pseudoscalar matter and show that only the latter one naturally arises in the framework of the present model.

Reduction of the N-component scalar model at the two-loop level.

Dimensional reduction of high temperature field theories improves IR features of their perturbative treatment. A crucial question is, what three-dimensional theory is representing the full system the most faithful way. Careful investigation of the induced 3-dimensional counterterm structure of the finite temperature 4D O(N) symmetric scalar theory at 2-loop level leads to proposing the presence of non-local operators in the effective theory. On scales beyond ${\cal O}(T^{-1})$, the scaling behavior of the couplings, consequently, deviates from the usual three-dimensional scaling characteristic for superrenormalizable theories.

CONFORMAL SECTOR IN D=6 QUANTUM GRAVITY

We discuss the conformal factor dynamics in D=6. Accepting the proposal that higher derivative dimensionless terms in the anomaly-induced effective action may be dropped, we obtain a super-renormalizable (like in D=4) effective theory for the conformal factor. The one-loop analysis of this theory gives the anomalous scaling dimension for the conformal factor and provides a natural mechanism to solve the cosmological constant problem.

Renormalization group and the ultraviolet problem in the Luttinger model

The Luttinger model describes a non-local interacting relativistic theory for spinless and massless fermions. Albeit the exact solution is already known, the perturbative approach to the model via the renormalization group is useful on account of the connection to the study of more realistic models' behaviour near the Fermi surface. In this work we show that the effective potential describing the interaction on the physical scalep 0 −1 is analytical in the coupling constants, and has an exponential decay on that scale. Besides the physical motivation of this approach, the problem is also technically interesting, since it is an example of a trivially superrenormalizable theory, as far as the ultraviolet region is concerned; nevertheless the proof is quite delicate, as the convergence of the perturbative series does not follow from the superficial bounds (which would give logarithmic and linear divergences), but is due to accidental compensations furnished by the particular symmetry properties of the model.

LEADING INFRARED LOGARITHMS AND VACUUM STRUCTURE OF QCD3

QCD 3 is a superrenormalizable massless theory; therefore off-mass-shell infrared divergences appear in the loop expansion. We show how certain infrared divergences can be subtracted by changing the boundary conditions in the functional integral, letting the vector potentials approach non-zero constant values at infinity. Infrared divergences, in the Green's functions, come together with powers of logarithms of the external momenta, and among the infrared divergences we deal with, there are those that give rise to the leading and first subleading logarithms. We show how for two-point functions it is possible to sum the leading and first subleading logarithms to all orders. This procedure defines a nonperturbative approximation for QCD 3. We find that in the ultraviolet region these summations are well defined, while in the infrared region, some additional prescription is needed to make sense out of them. From the construction developed to cancel infrared divergences, we derive the presence of arbitrarily large field strengths in the vacuum. We respect Lorentz invariance, because in this construction, one averages all directions and strengths of these fields. Possible applications to the infrared problem of QCD 4 are indicated.

SU(2) Lattice gauge theory in (2+1)D

Cluster expansion methods are applied to theSU(2) lattice gauge model in (2+1) dimensions. Strong-coupling series are calculated for the vacuum energy per site, the axial string tension, and the scalar mass gap; while ELCE approximants are used to estimate the string tension beyond its roughening transition. The simple scaling behaviour expected of this super-renormalizable theory is clearly seen, and we estimate that in the continuum limit the string tension σ∼(0.14±0.01)g4, while the mass gapM s ∼(2.2±0.25)g2. More accurate Monte Carlo simulations are needed to check the universality between the Hamiltonian and Euclidean versions of this model.

Nonanalyticity of the perturbative expansion for super-renormalizable massless field theories

The complete infrared expansion of Feynman amplitudes is established at any dimension d. The so called infrared finite parts develop poles at rational d. We prove a conjecture by Parisi by constructing an infrared subtraction procedure which defines finite amplitudes in such dimensions. The corresponding counterterms are associated to nonlocal operators and are generated in a nonperturbative way for super-renormalizable theories. We determine at all orders the perturbative expansion which contains powers and logarithms of the coupling constant.

Scale limit and low momentum behavior of Euclidean fields. II

The problem of the limit behavior of a rescaled random field εαφ (x/ε) as ε→0 is considered as an analog for random fields of the classical limit theorems for sums of random variables. If the random field φ does not depend on the scale ε, the limit is Gaussian in Euclidean field theory with a mass gap. If φ itself is a function of ε (giving a double sequence scheme), any infinitely divisible distribution can appear as the limit. In this formulation the limit of ε−dφ (x/ε) as ε→0 is discussed. It is proved that the limit field (if it exists) is a random field with values independent at every point. A correspondence between this scale limit and low momentum behavior is revealed. It is shown also that the scale limit may be considered as a limit of field theory when m→∞ and the coupling constant g→∞ in superrenormalizable theories and m→∞, g→0 for nonrenormalizable interactions. The scale limit of some ultraviolet regularized Euclidean fields is calculated. As an application some functional integrals are evaluated in the Ginzburg–Landau model in the low momentum limit.

Superrenormalizable theories with higher derivatives

Superrenormalizable theories with higher derivatives

Singular potentials and the divergence problem in quantum field theory

The classification of quantum field theories in superrenormalizable, renormalizable and nonrenormalizable theories corresponds in potential theory to a classification of potentials according to their singular behaviour at the origin. For potentials which are equally or more singular than the «kinetic repulsion» arising from quantum motion, the wave functions obey boundary conditions at the origin different from those of the free solutions. A perturbative approach, therefore, in general, fails which is formally reflected in the occurrence of divergences similar to those in the field-theoretical case. The present investigation seems to indicate that this formal resemblance actually arises from a common origin. On the basis of this apparent analogy, the knowledge of the exact solutions in potential theory suggests possible nonperturbative methods which may avoid the divergences in renormalizable field theories.