Power Counting Theorem Research Articles

Perturbative analysis of the Wess-Zumino flow

We investigate an interacting supersymmetric gradient flow in the Wess-Zumino model. Thanks to the nonrenormalization theorem and an appropriate initial condition, we find that any correlator of flowed fields is ultraviolet finite. This is shown at all orders of the perturbation theory using the power counting theorem for one-particle irreducible supergraphs. Since the model does not have the gauge symmetry, the mechanism of realizing the ultraviolet finiteness is quite different from that of the Yang-Mills flow, and this could provide further understanding of the gradient flow approach.

On the Power-counting Renormalizability of a Lifshitz-type QFT in Configuration Space

Recently, Hořava (Phys. Rev. D. 79, 084008, 2009) proposed a theory of gravity in 3+1 dimensions with anisotropic scaling using the traditional framework of quantum field theory (QFT). Such an anisotropic theory of gravity, characterized by a dynamical critical exponent z, has proven to be power-counting renormalizable at a z=3 Lifshitz Point. In the present article, we develop a mathematically precise version of power-counting theorem in Lorentz violating theories and apply this to the Hořava-Lifshitz (scalar field) models in configuration space. The analysis is performed under the light of the systematic use of the concept of extension of homogeneous distributions, a concept tailor-made to address the problem of the ultraviolet renormalization in QFT. This becomes particularly transparent in a Lifshitz-type QFT. In the specific case of the \({\phi _{4}^{4}}\)-theory, we show that is sufficient to take z=3 in order to reach the ultraviolet finiteness of the S-matrix in all orders.

A Renormalizable 4-Dimensional Tensor Field Theory

We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on U(1)4 is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the $${\phi^6}$$ rather than of the $${\phi^4}$$ type, since two different $${\phi^6}$$ -type interactions are log-divergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new locality principle is established for this category of graphs which allows to renormalize their divergences through counterterms of the form of the bare Lagrangian interactions. The model also has an unexpected anomalous log-divergent $${(\int \phi^2)^2}$$ term, which can be interpreted as the generation of a scalar matter field out of pure gravity.

Linearized group field theory and power-counting theorems

We introduce a linearized version of group field theory. It can be viewed either as a group field theory over the additive group of a vector space or as an asymptotic expansion of any group field theory around the unit group element. We prove exact power-counting theorems for any graph of such models. For linearized colored models the power counting of any amplitude is further computed in terms of the homology of the graph. An exact power-counting theorem is also established for a particular class of graphs of the nonlinearized models, which satisfy a planarity condition. Examples and connections with previous results are discussed.

Power-counting theorem for staggered fermions

Lattice power-counting is extended to QCD with staggered fermions. As preparation, the difficulties encountered by Reisz's original formulation of the lattice power-counting theorem are illustrated. One of the assumptions that is used in his proof does not hold for staggered fermions, as was pointed out long ago by Lüscher. Finally, I generalize the power-counting theorem, and the methods of Reisz's proof, such that the difficulties posed by staggered fermions are overcome.

Weak Power-Counting Theorem for the Renormalization of the Nonlinear Sigma Model in Four Dimensions

The formulation of the non-linear sigma model in terms of flat connection allows the construction of a perturbative solution of a local functional equation encoding the underlying gauge symmetry. In this paper we discuss some properties of the solution at the one-loop level in D=4. We prove the validity of a weak power-counting theorem in the following form: although the number of divergent amplitudes is infinite only a finite number of divergent amplitudes have to be renormalized. The proof uses the linearized functional equation of which we provide the general solution in terms of local functionals. The counteterms are given in terms of linear combinations of these invariants and the coefficients are fixed by a finite number of divergent amplitudes. The latter contain only insertions of the composite operators $\phi_0$ (the constraint of the non-linear sigma model) and $F_\mu$ (the flat connection). These amplitudes are at the top of a hierarchy implicit in the functional equation. As an example we derive the counterterms for the four-point amplitudes.

A local formulation of lattice Wess-Zumino model with exact U(1)Rsymmetry

A lattice Wess-Zumino model is formulated on the basis of Ginsparg-Wilson fermions. In perturbation theory, our formulation is equivalent to the formulation by Fujikawa and Ishibashi and by Fujikawa. Our formulation is, however, free from a singular nature of the latter formulation due to an additional auxiliary chiral supermultiplet on a lattice. The model posssesses an exact U(1)R symmetry as a supersymmetric counterpart of the L?scher lattice chiral U(1) symmetry. A restration of the supersymmetric Ward-Takahashi identity in the continuum limit is analyzed in renormalized perturbation theory. In the one-loop level, a supersymmetric continuum limit is ensured by suitably adjusting a coefficient of a single local term *. The non-renormalization theorem holds to this order of perturbation theory. In higher orders, on the other hand, coefficents of local terms with dimension ? 4 that are consistent with the U(1)R symmetry have to be adjusted for a supersymmetric continuum limit. The origin of this complexicity in higher-order loops is clarified on the basis of the Reisz power counting theorem. Therefore, from a view point of supersymmetry, the present formulation is not quite better than a lattice Wess-Zumino model formulated by using Wilson fermions, although a number of coefficients which require adjustment is much less due to the exact U(1)R symmetry. We also comment on an exact non-linear fermionic symmetry which corresponds to the one studied by Bonini and Feo; an existence of this exact symmetry itself does not imply a restoration of supersymmetry in the continuum limit without any adjustment of parameters.

Renormalisation of ?4-Theory on Non-Commutative $$\mathbb{R}^{4}$$ to All Orders

We present the main ideas and techniques of the proof that the duality-covariant four-dimensional non-commutative Φ4-model is renormalisable to all orders. This includes the reformulation as a dynamical matrix model, the solution of the free theory by orthogonal polynomials as well as the renormalisation byflow equations involving power-counting theorems for ribbon graphs drawn on Riemann surfaces

Power-Counting Theorem for Non-Local Matrix Models and Renormalisation

Solving the exact renormalisation group equation à la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties—typically arising from orthogonal polynomials—which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative ℝ D in matrix formulation.

NONCOMMUTATIVE U(1) SUPER-YANG–MILLS THEORY: PERTURBATIVE SELF-ENERGY CORRECTIONS

The quantization of the noncommutative [Formula: see text], U(1) super-Yang–Mills action is performed in the superfield formalism. We calculate the one-loop corrections to the self-energy of the vector superfield. Although the power-counting theorem predicts quadratic ultraviolet and infrared divergences, there are actually only logarithmic UV and IR divergences, which is a crucial feature of noncommutative supersymmetric field theories.

Sine-Gordon breather form factors and quantum field equations

Using the results of previous investigations on sine-Gordon form factors exact expressions of all breather matrix elements are obtained for several operators: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. Formulae for the asymptotic behavior of bosonic form factors are presented which are motivated by Weinberg's power counting theorem in perturbation theory. It is found that the quantum sine-Gordon field equation holds and an exact relation between the ``bare'' mass and the renormalized mass is obtained. Also a quantum version of a classical relation for the trace of the energy momentum is proven. The eigenvalue problem for all higher conserved charges is solved. All results are compared with perturbative Feynman graph expansions and full agreement is found.

Universality of the axial anomaly in lattice QCD

We prove that lattice QCD generates the axial anomaly in the continuum limit under very general conditions on the lattice action, which includes the case of Ginsparg-Wilson fermions. The ingredients going into the proof are gauge invariance, locality of the Dirac operator, absence of fermion doubling, the general form of the lattice Ward identity, and the power counting theorem of Reisz. The results generalize in an obvious way to SU(N) lattice gauge theories.

Chiral symmetry restoration and axial vector renormalization for Wilson fermions

Lattice gauge theories with Wilson fermions break chiral symmetry. In the U(1) axial vector current this manifests itself in the anomaly. On the other hand it is generally expected that the axial vector flavour mixing current is non-anomalous. We give a short, but strict proof of this to all orders of perturbation theory, and show that chiral symmetry restauration implies a unique multiplicative renormalization constant for the current. This constant is determined entirely from an irrelevant operator in the Ward identity. The basic ingredients going into the proof are the lattice Ward identity, charge conjugation symmetry and the power counting theorem. We compute the renormalization constant to one loop order. It is largely independent of the particular lattice realization of the current.

Lattice QED and universality of the axial anomaly

We give a perturbative proof that U(1) lattice gauge theories generate the axial anomaly in the continuum limit under very general conditions on the lattice Dirac operator. These conditions are locality, gauge covariance and the absence of species doubling. They hold for Wilson fermions as well as for realizations of the Dirac operator that satisfy the Ginsparg-Wilson relation. The proof is based on the lattice power counting theorem. The results generalize to non-abelian gauge theories.

The axial anomaly in lattice QED. A universal point of view

We give a perturbative proof that U(1) lattice gauge theories generate the axial anomaly in the continuum limit under very general conditions on the lattice Dirac operator. These conditions are locality, gauge covariance and the absence of species doubling. They hold for Wilson fermions as well as for realizations of the Dirac operator that satisfy the Ginsparg–Wilson relation. The proof is based on the lattice power counting theorem.

Relating inclusive e+e− annihilation to electroproduction sum rules in quantum chromodynamics

The Broadhurst-Kataev conjecture, that the “discrepancy” in the connection with the π 0 → γγ anomaly equals the beta function β( α ) times a power series in the effective coupling α , is proven to all orders of perturbative quantum chromodynamics. The use of nested short-distance expansions is justified via Weinberg's power-counting theorem.

Lattice gauge theory: Renormalization to all orders in the loop expansion

The renormalization of SU( N) lattice gauge theories is discussed. The program is based on lattice power counting theorems. By choice of an admissible gauge fixing, e.g. the Lorentz gauge, the theory exhibits a BRS-symmetry. It is proven that the existence of such a symmetry insures finiteness of the continuum limit by simply renormalizing the coupling constant and field strentghs, to every order in perturbation theory.

A power counting theorem for Feynman integrals on the lattice

A convergence theorem is proved, which states sufficient conditions for the existence of the continuum limit for a wide class of Feynman integrals on a space-time lattice. A new kind of a UV-divergence degree is introcduced, which allows the formulation of the theorem in terms of power counting conditions.

Heavy-mass power-counting theorem and a proof of decoupling in the momentum-subtraction scheme

We establish a heavy-mass power-counting prescription describing the effects of virtual heavy particles at low energy in an arbitrary Feynman diagram using momentum subtraction to remove divergences from its corresponding Feynman integral. Furthermore, we show that in the absence of heavy-mass-dependent vertices, the virtual heavy-mass effects are suppressed in the low-energy regime by an inverse power of the heavy mass given by our power-counting prescription. As a result, we obtain a simple proof of the Appelquist-Carazzone decoupling theorem.

Erratum: Convergence of the generalized and improved Dyson-Salam renormalization scheme

A direct convergence proof of the generalized and improved Dyson-Salam renormalization scheme, introduced in our earlier work as a consistent renormalization scheme in momentum space for renormalizable as well as nonrenormalizable theories, is readily given by expressing the renormalized Feynman integrand in a form to which the power-counting theorem is immediately applicable.