Higher-order Renormalization Research Articles

Sequences of trees and higher-order renormalization group equations

We define a notion of higher-order renormalization group equations and investigate when a sequence of trees satisfies such an equation. In the strongest sense, the sequence of trees satisfies a kth-order renormalization group equation when applying any choice of Feynman rules results in a Green function satisfying a kth-order renormalization group equation, and we characterize all such sequences of trees. We also make some comments on sequences of trees, which require special choices of Feynman rules in order to satisfy a higher-order renormalization group equation.

Effective field theory, three-loop perturbative expansion, and their experimental implications in graphene many-body effects

Many-body electron-electron interaction effects are theoretically considered in monolayer graphene from a continuum effective field-theoretic perspective by going beyond the standard leading-order perturbative renormalization group (RG) analysis. Given that the bare fine structure constant in graphene is of order unity, which is neither small to justify a perturbative expansion nor large enough for strong-coupling theories to be applicable, the problem is a difficult one, with some similarity to 2+1-dimensional strong-coupling quantum electrodynamics (QED). In this work, we take a systematic and comprehensive analytical approach, working primarily at the Dirac point (intrinsic graphene), by going up to three loops in the diagrammatic expansion to both ascertain the validity of perturbation theory and to estimate quantitatively higher-order renormalization effects. While no direct signatures for non-Fermi liquid behavior at the Dirac point have yet been observed experimentally, there is ample evidence for the interaction-induced renormalization of the graphene velocity as the carrier density approaches zero. We provide a critical comparison between theory and experiment, using both bare- and screened-interaction (RPA) calculations. We find that while the one-loop RG analysis gives reasonable agreement with the experimental data, especially when screening and finite-density effects are included through the RPA, the two-loop analysis reveals a strong-coupling critical point in suspended graphene, signifying either a quantum phase transition or a breakdown of the weak-coupling RG approach. We show that the latter is more likely by adapting Dyson's argument for the breakdown of perturbative QED to the case of graphene. We propose future experiments and theoretical directions to make further progress on this important and difficult problem.

Higher-order renormalization of graphene many-body theory

We study the many-body theory of graphene Dirac quasiparticles interacting via the long-range Coulomb potential, taking as a starting point the ladder approximation to different vertex functions. We test in this way the low-energy behavior of the electron system beyond the simple logarithmic dependence of electronic correlators on the high-energy cutoff, which is characteristic of the large-N approximation. We show that the graphene many-body theory is perfectly renormalizable in the ladder approximation, as all higher powers in the cutoff dependence can be absorbed into the redefinition of a finite number of parameters (namely, the Fermi velocity and the weight of the fields) that remain free of infrared divergences even at the charge neutrality point. We illustrate this fact in the case of the vertex for the current density, where a complete cancellation between the cutoff dependences of vertex and electron self-energy corrections becomes crucial for the preservation of the gauge invariance of the theory. The other potentially divergent vertex corresponds to the staggered (sublattice odd) charge density, which is made cutoff independent by a redefinition in the scale of the density operator. This allows to compute a well-defined, scale invariant anomalous dimension to all orders in the ladder series, which becomes singular at a value of the interaction strength marking the onset of chiral symmetry breaking (and gap opening) in the Dirac field theory. The critical coupling we obtain in this way matches with great accuracy the value found with a quite different method, based on the resolution of the gap equation, thus reassuring the predictability of our renormalization approach.

Renormalizable 4D Quantum Gravity as a Perturbed Theory from CFT

We study the renormalizable quantum gravity formulated as a perturbed theory from conformal field theory (CFT) on the basis of conformal gravity in four dimensions. The conformal mode in the metric field is managed non-perturbatively without introducing its own coupling constant so that conformal symmetry becomes exact quantum mechanically as a part of diffeomorphism invariance. The traceless tensor mode is handled in the perturbation with a dimensionless coupling constant indicating asymptotic freedom, which measures a degree of deviation from CFT. Higher order renormalization is carried out using dimensional regularization, in which the Wess-Zumino integrability condition is applied to reduce indefiniteness existing in higher-derivative actions. The effective action of quantum gravity improved by renormalization group is obtained. We then make clear that conformal anomalies are indispensable quantities to preserve diffeomorphism invariance. Anomalous scaling dimensions of the cosmological constant and the Planck mass are calculated. The effective cosmological constant is obtained in the large number limit of matter fields.

Resummation and Higher Order Renormalization in 4D Quantum Gravity

Higher order renormalization in 4D quantum gravity is carried out using dimensional regularization with great care concerning the conformal-mode dependence. In this regularization, resummation can be automatically carried out without making an assumption like that of David, Distler and Kawai. In this paper we consider a model of 4D quantum gravity coupled to QED. Resummation inevitably implies a four-derivative quantum gravity. The renormalizability is directly checked up to O(er6) and O(tr2), where er and tr are the running coupling constants of QED and the traceless gravitational mode. There is no other running coupling constant in our model. The conformal mode is treated exactly, which means it is unrenormalized. It is found that Hathrell's results are included in our results. As a by-product, it is found that a higher-order gravitational correction to the beta function of QED is negative. An advantage of our model is that in the very high-energy regime, it closely resembles exactly solvable 2D quantum gravity. Thus, we can study physical states of 4D quantum gravity in this regime in parallel to those of 2D quantum gravity, which can be described with diffeomorphism invariant composite fields.

Resummation and Higher-order Renormalization in 4D Quantum Gravity

Resummation and Higher-order Renormalization in 4D Quantum Gravity

William Edward Caswell

William Edward Caswell

PHASE STRUCTURE OF THE THREE-DIMENSIONAL $g (\vec {\bf \phi}^2)^2$ FIELD THEORY

The phase structure of the three-dimensional superrenormalizable scalar-field model with the O (N)-invariant interaction g(ϕ2)2 is considered. By the method of canonical transformations, it is shown that this model describes the O (N)-symmetric system at both the small and the large coupling constant g. At the same time, phase transitions accompanied by symmetry rearrangement are possible at intermediate values of g. Comparison with the case of two dimensions shows a crucial influence of higher-order renormalization on the phase structure of the model.

Radiatively generated parton distributions for high energy collisions

The gluon and antiquark distributions of the nucleon are generated radiatively using the assumption that at some low resolution scale the nucleon consists entirely of valence quarks and valence-like gluons. The agreement between the uniquely predicted gluon and sea distributions and the available data on deep inelastic structure functions (including also recent low-Q2 measurements) and direct-photon production is demonstrated and discussed in detail. Simple parametrizations of the resulting (positive definite!) parton distributions are presented in the range 10−4≲x≦1 and 0.2≲Q2≲106 GeV2 as obtained according to the leading- and higher-order renormalization group evolution equations.

Renormalization of an SU(3) linearσmodel with mesons and baryons in the one-loop approximation

A linear SU(3) sigma model with mesons and baryons is demonstrated to be renormalizable in the one-loop approximation. The mesons and baryons are assigned to the (3,3*) direct-sum (3*,3) and the ((3,3*), (3*,3)) representations of chiral SU(3) x SU(3), respectively. The model incorporates both spontaneous and explicit symmetry breaking. The baryon symmetry-breaking terms are chosen to allow the model to describe the N-..xi.. mass difference. Higher-order renormalization is also considered.