QUANTUM FIELD THEORY IN THE LIMIT x≪1

Cornell University - arXiv, 2020

A formal expansion for the Green's functions of an interacting quantum field theory in a parameter that somehow encodes its "distance" from the corresponding non-interacting one was introduced more than thirty years ago, and has been recently reconsidered in connection with its possible application to the renormalization of non-hermitian theories. Besides this new and interesting application, this expansion has special properties already when applied to ordinary (i.e. hermitian) theories, and in order to disentangle the peculiarities of the expansion itself from those of non-hermitian theories, it is worth to push further the investigation limiting first the analysis to ordinary theories. In the present work we study some aspects related to the renormalization of a scalar theory within the framework of such an expansion. Due to its peculiar properties, it turns out that at any finite order in the expansion parameter the theory looks as non-interacting. We show that when diagrams of appropriate classes are resummed, this apparent drawback disappears and the theory recovers its interacting character. In particular we have seen that with a certain class of diagrams, the weak-coupling expansion results are recovered, thus establishing a bridge between the two expansions.

Physics Letters B, 1997

The asymptotic behaviour of cubic field theories is investigated in the Regge limit using the techniques of environmentally friendly renormalization, environmentally friendly in the present context meaning asymmetric in its momentum dependence. In particular we consider the crossover between large and small energies at fixed momentum transfer for a model scalar theory of the type φ 2 ψ. The asymptotic forms of the crossover scaling functions are exhibited for all two particle scattering processes in this channel to one loop in a renormalization group improved perturbation theory.

Physics Letters B, 1995

Direct verification of the existence of an infinite set of multicritical non-perturbative FPs (Fixed Points) for a single scalar field in two dimensions, is in practice well outside the capabilities of the present standard approximate non-perturbative methods. We apply a derivative expansion of the exact RG (Renormalization Group) equations in a form which allows the corresponding FP equations to appear as non-linear eigenvalue equations for the anomalous scaling dimension η. At zeroth order, only continuum limits based on critical sine-Gordon models, are accessible. At second order in derivatives, we perform a general search over all η ≥ .02, finding the expected first ten FPs, and only these. For each of these we verify the correct relevant qualitative behaviour, and compute critical exponents, and the dimensions of up to the first ten lowest dimension operators. Depending on the quantity, our lowest order approximate description agrees with CFT (Conformal Field Theory) with an accuracy between 0.2% and 33%; this requires however that certain irrelevant operators that are total derivatives in the CFT are associated with ones that are not total derivatives in the scalar field theory. CERN-TH.7403/94 SHEP 94/95-04 hep-th/9410141 October, 1994. * On Leave from Southampton University, U.K. (Address after 1/10/94). Circumstantial evidence strongly suggests that there exists an infinite set of multicritical non-perturbative FPs for a single scalar field in two dimensions, corresponding to the universality classes of multicritical Ising models, equivalently to the diagonal invariants of the unitary minimal (p, p + 1) conformal models with p = 3, 4, · · · [1] [2], however direct verification of these facts is in practice well outside the capabilities of the standard approximate non-perturbative methods: lattice Monte Carlo, resummations of weak or strong coupling perturbation theory and the epsilon expansion. (The impracticableness of the epsilon expansion for higher p is covered in ref.[3], implying similar difficulties in weak coupling perturbation theory, while lattice methods suffer from difficulties of locating and accurately computing the multicritical points in the at least p − 2 dimensional bare coupling constant space)

Annals of Physics, 1977

A new renormalization-group method for treating Hamiltonian quantum field theories is devised. The method involves integration over energy fluctuations of successively smaller size, and the computation of effective Hamiltonians which reflect the effects of the fluctuations which have been integrated out. The method is then used in qualitative analysis of highenergy small-angle scattering. Model field theories based on scalar 4" theory and quantum electrodynamics (QED) are described, and details of the integration procedure are discussed in the context of these model systems. Simple recursion relations which sum the logarithms associated with ladder graphs in Q" theory and with tower graphs in QED are demonstrated. The recursion relations for QED are shown to generate an infinite set of marginal interactions, all of which accumulate logarithms; the existence of these marginal interactions accounts for the complexity of previous analyses of high-energy scattering.

数理解析研究所講究録, 2006

We study a scaling limit for the generarized spin-boson model and a generalization of the Nelson model. Applying it to a model for the field of the nuclear force with isospin, we obtain an effective potential of the interaction between nucleons. Also, we get some applications to condensed matter physics.

Annals of Physics, 2012

We analyze how corrections linear in the effective range, r 0 , affect quantities in the three-body sector within an effective field theory with short-range interactions. We demonstrate that relevant observables can be straightforwardly obtained using a perturbative expansion in powers of r 0 .

2020

As applied to quantum theories, the program of renormalization is successful for ‘renormalizable models’ but fails for ‘nonrenormalizable models’. After some conceptual discussion and analysis, an enhanced program of renormalization is proposed that is designed to bring the ‘nonrenormalizable models’ under control as well. The new principles are developed by studying several, carefully chosen, soluble examples, and include a recognition of a ‘hard-core’ behavior of the interaction and, in special cases, an extremely elementary procedure to remove the source of all divergences. Our discussion provides the background for a recent proposal for a nontrivial quantization of nonrenormalizable scalar quantum field models, which is briefly summarized as well. Dedication: It is a pleasure to dedicate this article to the memory of Prof. Alladi Ramakrishnan who, besides his own important contributions to science, played a crucial role in the development of modern scientific research and educat...

Annals of Physics, 1977

It is proposed that field theories quantized in a curved space-time manifold can be conveniently regularized and renormalized with the aid of Pauli-Villars regulator fields. The method avoids the conceptual difficulties of covariant point-separation approaches, by always starting from a manifestly generally covariant action, and the technical limitations of the dimensional regularization approach, which requires solution of the theory in arbitrary dimension in order to go beyond a weak-field expansion. An action is constructed which renormalizes the weak-field perturbation theory of a massive scalar field in two space-time dimensions, it is shown that the trace anomaly previously found in dimensional regularization and some point-separation calculations also arises in perturbation theory when the theory is Pauli-Villars regulated. We then study a specific solvable two-dimensional model of a massive scalar field in a Robertson-Walker asymptotically flat universe. It is shown that the action previously considered leads, in this model, to a well-defined finite expectation value for the stress-energy tensor. The particle production (<0 in / @(x, t)l 0 in) for t-+ co) is computed explicitly. Finally, the validity of weak-field perturbation theory (in the appropriate range of parameters) is checked directly in the solvable model, and the trace anomaIy computed in the asymptotic regions t + i a independently of any weakfield approximation. The extension of our model to higher dimensions and the renormalization of interacting (scalar) field theories are briefly discussed.

arXiv (Cornell University), 2023

We focus on the behavior of (2+1)d λϕ 4 and (5+1)d λϕ 3 or λ|ϕ| 3 theories in different regimes and compare the results obtained from the adaptive perturbation method with those obtained from lattice simulation. These theories are simple models that exhibit asymptotic freedom, which is a property that is also observed in more complex theories such as QCD, which describes the strong interaction between quarks and gluons. Asymptotic freedom is an important feature of these theories because it allows for a perturbative treatment of interactions at high energies. However, the standard perturbation scheme breaks down in the presence of strong interactions, and the adaptive perturbation method, which involves resuming the Feynman diagrams, is more suitable for studying these interactions. Our research involves comparing the perturbation result to lattice simulation. In the case of the ϕ 3 theory, there is no stable vacuum, so we explore evidence from the |ϕ| 3 theory instead. Our results appear to show that resummation improves the strong coupling result for both the λϕ 4 and λ|ϕ| 3 theories. Additionally, we improve the resummation method for the three-point coupling vertex and study the RG flow to analyze the resummation contribution and theoretical properties.

Physical Review B, 2015

We study the renormalization of the Fermi surface coupled to a massless boson near three spatial dimensions. For this, we set up a Wilsonian RG with independent decimation procedures for bosons and fermions, where the four-fermion interaction "Landau parameters" run already at tree level. Our explicit one loop analysis resolves previously found obstacles in the renormalization of finite density field theory, including logarithmic divergences in non-local interactions and the appearance of multilogarithms. The key aspects of the RG are the above tree level running, and a UV-IR mixing between virtual bosons and fermions at the quantum level, which is responsible for the renormalization of the Fermi velocity. We apply this approach to the renormalization of 2k F singularities, and a companion paper considers the RG for Fermi surface instabilities. We end with some comments on the renormalization of finite density field theory with the inclusion of Landau damping of the boson.

Physical Review D, 1981

%'e study two continuum methods of regulating the formal strong-coupling expansion of the Green's functions, obtained by expanding the path integral in powers of the kinetic energy {inverse free propagator). Our continuum regulations amount to introducing either a hard (0 function) or soft (Gaussian) cutoff A in momentum space. The cutoA' takes the place of the usual spatial cutoff, the lattice spacing, which arises when the path integral is defined as the continuum limit of,ordinary integrals on a Euclidean space-time lattice. We find, by investigating free field theory and g$4 field theory in one dimension, that the 8-function regulation is more accurate than the Gaussian and, unlike the Gaussian, preserves certain continuum Green s-function identities. The extension to field theories with fermions is trivial and we give the strong-coupling graphical rules for an arbitrary field theory with fermions and bosons in d dimensions.

Il Nuovo Cimento A Series 10, 1969

ysis c'm be applied, with minor forinal modifications, to any renormalizable theory, in particular to quantum eleetrodynamies, a subject of wider physical interest. Section 2 contains a short review of results obtained in previous investigations, in order to introduce the principal tools for our later analysis. One of the main features of the renormalization method by finite-part integration rules is the lack of uniqueness of the regularized theory. In fact as one can see from some specific examples and also as it is quite clear from the axiomatic definition given in Sect. 4 of ref. (~), an infinite number of finitepart integrals does in general exist. With any of these finite-part integrals we can associate a regularized theory, the propagators of which will satisfy branching equations involving the particular finite-part integration rule. Since the physical properties of the system described by the theory do not depend upon the rule selected particularly for this purpose, it is possible to introduce the so-called <~ renormalization group ,~, defined as the group of all transformations among different regularized theories. The physical content of the theory is invariant with respect to this group. The renormalization group, arising from the regularization methods by finite-part integration, appears, in its structure, as an enlargement of the group studied by other authors ('~). This contains the latter as a particular case, when the finite-part integrals are restricted to vary in a special subclass. A first implicit characterization of the group follows easily from results of previous works (~-~) and is given in Secl. 3. On the other hand, tile successes (~) of the analysis of the group according to the interpretation of the above-mentioned authors leads one to believe that very useful information on the functional dependence of the propagators on the field-theoretical parameters could be gained from a deep knowledge of the group structure according' to our interpretation. it should for instance be possible to <(improve ,) a perturbative approximation by considering the fact that the theory in its full content is invariant under the group even though its ~pproximations are not. The present work provides a complete cht~racterization of the transformations among different regularized theories. This marks therefore a first neces-(5) E. C. G. ST[ECKELBFR(~ and A. I'ETERMANN:

Physical Review D, 1978

2010

We study the convergence of a functional renormalisation group technique by looking at the ratio between the fermion-fermion scattering length and the dimer-dimer scattering length for a system of nonrelativistic fermions. We find that in a systematic expansion in powers of the fields there is a rapid convergence of the result that agrees with know exact results.

1999

The Schrodinger equation with a two-dimensional delta-function potential is a simple example of an asymptotically free theory that undergoes dimensional transmutation. Renormalization requires the introduction of a mass scale, which can be lowered perturbatively until an infrared cutoff produced by non-perturbative effects such as bound state formation is encountered. We outline the effective field theory and similarity renormalization group techniques