I introduce an approximation scheme that allows to deduce differential equations for the renormalization group $β$-function from a Schwinger--Dyson equation for the propagator. This approximation is proven to give the dominant asymptotic behavior of the perturbative solution. In the supersymmetric Wess--Zumino model and a $ϕ^3_6$ scalar model which do not have divergent vertex functions, this simple Schwinger--Dyson equation for the propagator captures the main quantum corrections.
A non-subtractive recipe of Casimir energy renormalization efficient in the presence of logarithmically divergent terms is proposed. It is demonstrated that it can be applied even then, when energy levels can be obtained only numerically and neither their asymptotical behavior, nor the analytic form of spectral equation is known. The results of calculations performed with this method are compared to those obtained by means of explicit subtraction of divergent terms from energy.
A stochastic process with self-interaction as a model of quantum field theory is studied. We consider an Ornstein-Uhlenbeck stochastic process x(t) with interaction of the form x^{(α)}(t)^4, where $α$ indicates the fractional derivative. Using Bogoliubov's R-operation we investigate ultraviolet divergencies for the various parameters $α$. Ultraviolet properties of this one-dimensional model in the case $α=3/4$ are similar to those in the $ϕ^4_4$ theory but there are extra counterterms. It is shown that the model is two-loops renormalizable. For $5/8\leq α< 3/4$ the model has a finite number of…
The structure of the UV divergencies in higher dimensional nonrenormalizable theories is analysed. Based on renormalization operation and renormalization group theory it is shown that even in this case the leading divergencies (asymptotics) are governed by the one-loop diagrams the number of which, however, is infinite. Explicit expression for the one-loop counter term in an arbitrary D-dimensional quantum field theory without derivatives is suggested. This allows one to sum up the leading asymptotics which are independent of the arbitrariness in subtraction of higher order operators. Diagramm…
The infinitesimal unitary transformation, introduced recently by F.Wegner, to bring the Hamiltonian to diagonal (or band diagonal) form, is applied to the Hamiltonian theory as an exact renormalization scheme. We consider QED on the light front to illustrate the method. The low-energy generated interaction, induced in the renormalized Hamiltonian to the order α, is shown to be negative to insure together with instantaneous term and perturbative photon exchange the bound states for positronium. It is possible to perform the complete complete elimination of the eeγ-vertex in the instant form fra…
The effective equations of motion for a point charged particle taking account of radiation reaction are considered in various space-time dimensions. The divergencies steaming from the pointness of the particle are studied and the effective renormalization procedure is proposed encompassing uniformly the cases of all even dimensions. It is shown that in any dimension the classical electrodynamics is a renormalizable theory if not multiplicatively beyond d=4. For the cases of three and six dimensions the covariant analogs of the Lorentz-Dirac equation are explicitly derived.
A new formalism for the perturbative construction of algebraic quantum field theory is developed. The formalism allows the treatment of low dimensional theories and of non-polynomial interactions. We discuss the connection between the Stueckelberg-Petermann renormalization group which describes the freedom in the perturbative construction with the Wilsonian idea of theories at different scales . In particular we relate the approach to renormalization in terms of Polchinski's Flow Equation to the Epstein-Glaser method. We also show that the renormalization group in the sense of Gell-Mann-Low (w…
We analyze divergencies in 2-point and 3-point functions for noncommutative $θ$-expanded SU(2)-gauge theory with massless fermions. We show that, after field redefinition and renormalization of couplings, one divergent term remains.
In this paper we construct the 2 dimensional Euclidean $ϕ^4$ quantum field theory using the method of loop vertex expansion. We reproduce the results of standard constructive theory, for example the Borel summability of the Schwinger functions in the coupling constant. Our method should be also suitable for the future construction of Grosse-Wulkenhaar models on non-commutative space-time.
The loop vertex expansion (LVE) is a constructive technique which uses only canonical combinatorial tools and no space-time dependent lattices. It works for quantum field theories without renormalization. Renormalization requires scale analysis. In this paper we provide an enlarged formalism which we call the multiscale loop vertex expansion (MLVE). We test it on what is probably the simplest quantum field theory which requires some kind of renormalization, namely a combinatorial model of the vector type with quartic interaction and a propagator which mimicks the power counting of $ϕ^4_2$. An …
In this paper we construct the noncommutative Grosse-Wulkenhaar model on 2-dimensional Moyal plane with the method of loop vertex expansion. We treat renormalization with this new tool, adapt Nelson's argument and prove Borel summability of the perturbation series. This is the first non-commutative quantum field theory model to be built in a non-perturbative sense.
In this paper we continue our program of non-pertubative constructions of tensorial group field theories (TGFT). We prove analyticity and Borel summability in a suitable domain of the coupling constant of the simplest super-renormalizable TGFT which contains some ultraviolet divergencies, namely the color-symmetric quartic melonic rank-four model with Abelian $U(1)$ gauge invariance, nicknamed $U(1)-T^4_4$. We use a multiscale loop vertex expansion. It is an extension of the loop vertex expansion (the basic constructive technique for non-local theories) which is required for theories that invo…
Based upon the intrinsic relation between the divergent lower point functions and the convergent higher point ones in the renormalizable quantum field theories, we propose a new method for regularization and renormalization in QFT. As an example, we renormalize the $ϕ^{4}$ theory at the one loop order by means of this method.
We investigate the possibility of generalizing differential renormalization of D.Z.Freedman, K.Johnson and J.I.Latorre in an invariant fashion to theories with infrared divergencies via an infrared $\tilde{R}$ operation. Two-dimensional $σ$ models and the four-dimensional $ϕ^4$ theory diagrams with exceptional momenta are used as examples, while dimensional renormalization serves as a test scheme for comparison. We write the basic differential identities of the method simultaneously in co-ordinate and momentum space, introducing two scales which remove ultraviolet and infrared singularities. T…
We study the theory of noncommutative U(N) Yang-Mills field interacting with scalar and spinor fields in the fundamental and the adjoint representations. We include in the action both the terms describing interaction between the gauge and the matter fields and the terms which describe interaction among the matter fields only. Some of these interaction terms have not been considered previously in the context of noncommutative field theory. We find all counterterms for the theory to be finite in the one-loop approximation. It is shown that these counterterms allow to absorb all the divergencies …
We investigate the renormalization group theory of generalized multi-vertex sine-Gordon model by employing the dimensional regularization method and also the Wilson renormalization group method. The vertex interaction is given by $\cos(k_j\cdot ϕ)$ where $k_j$ ($j=1,2,\cdots,M$) are momentum vectors and $ϕ$ is an $N$-component scalar field. The beta functions are calculated for the sine-Gordon model with multi cosine interactions. The second-order correction in the renormalization procedure is given by the two-point scattering amplitude for tachyon scattering. We show that new vertex interacti…
The non-renormalization theorems of chiral vertex functions are derived on the basis of an algebraic analysis. The property, that the interaction vertex is a second supersymmetry variation of a lower dimensional field monomial, is used to relate chiral Green functions to superficially convergent Green functions by extracting the two supersymmetry variations from an internal vertex and transforming them to derivatives acting on external legs. The analysis is valid in the massive as well as in the massless model and can be performed irrespective of properties of the superpotential at vanishing m…
We use the Schwinger-Dyson equations as a starting point to derive renormalization flow equations. We show that Katanin's scheme arises as a simple truncation of these equations. We then give the full renormalization group equations up to third order in the irreducible vertex. Furthermore, we show that to the fifth order, there exists a functional of the self-energy and the irreducible four-point vertex whose saddle point is the solution of Schwinger-Dyson equations.
Nonperturbative renormalization group beyond melonic sector: The Effective Vertex Expansion method for group fields theories
Commissariat à l’Énergie Atomique (CEA, LIST)（法国原子能和替代能源委员会）；Faculté des Sciences et Techniques (ICMPA-UNESCO Chair), Université d’Abomey- Calavi（阿博米-卡拉维大学科学与技术学院）
We study Landau-gauge Yang-Mills theory by means of a nonperturbative vertex expansion of the quantum effective action. Using an exact renormalization group equation, we compute the fully dressed gluon and ghost propagators to lowest nontrivial order in the vertex expansion. In the mid-momentum regime, $p^2\sim\mathcal{O}(1)\text{GeV}^2$, we probe the propagator flow with various {\em ansätze} for the three- and four-point correlations. We analyze the potential of these truncation schemes to generate a nonperturbative scale. We find universal infrared behavior of the propagators, if the gluon …
We calculate the two-loop renormalization group (RG) beta-function of a massless scalar field theory from the irreducible version of Polchinski's exact RG flow equation. To obtain the correct two-loop result within this method, it is necessary to take the full momentum-dependence of the irreducible four-point vertex and the six-point vertex into account. Although the same calculation within the orthodox field theory method is less tedious, the flow equation method makes no assumptions about the renormalizability of the theory, and promises to be useful for performing two-loop calculations for …
In this work we study the renormalization of the electrodynamics of spin 1/2 fermions in the Poincaré projector formalism which is second order in the derivatives of the fields. We analyze the superficial degree of divergence of the vertex functions of this theory, calculate at one-loop level the vacuum polarization, fermion self-energy and γ-fermion-fermion vertex function and the divergent piece of the one-loop contributions to the γ-γ-fermion-fermion vertex function. It is shown that these functions are renormalizable independently of the value of the gyromagnetic factor g which is a free p…
In this paper we consider the possibility of chiral (charge or spin density wave) symmetry breaking in graphene due to long-range Coulomb interaction by comparing the results of the Bethe-Salpeter and functional renormalization-group approaches. The former approach performs summation of ladder diagrams in the particle-hole channel, and reproduces the results of the Schwinger-Dyson approach for the critical interaction strength of the quantum phase transition. The renormalization-group approach combines the effect of different channels and allows to study the role of vertex corrections. The cri…
We explore the properties of a recently proposed background independent exact renormalization group approach to gauge theories and gravity. In the process we also develop the machinery needed to study it rigorously. The proposal comes with some advantages. It preserves gauge invariance manifestly, avoids introducing unphysical fields, such as ghosts and Pauli-Villars fields, and does not require gauge-fixing. However, we show that in the simple case of SU(N) Yang-Mills it does not completely regularise the longitudinal part of vertex functions already at one loop, invalidating certain methods …
We present a novel analysis of the gluon gap equation, where its full nonlinear structure is duly taken into account. In particular, while in previous treatments the linearization of this homogeneous integral equation introduced an indeterminacy in the scale of the corresponding mass, the current approach determines it uniquely, once the value of the gauge coupling at a given renormalization point is used as input. A crucial ingredient for this construction is the "kinetic term" of the gluon propagator, whose form is not obtained from the complicated equation governing its evolution, but is ra…
We investigate the non-perturbative renormalization group flow of the scalar Galileon model in flat space. We discuss different expansion schemes of the Galileon truncation, including a heat-kernel based derivative expansion, a vertex expansion in momentum space and a curvature expansion in terms of a covariant geometric formulation. We find that the Galileon symmetry prevents a quantum induced renormalization group running of the Galileon couplings. Consequently, the Galileon truncation only features a trivial Gaussian fixed point.
In recent years, some interesting investigations of the non-perturbative renormalization group equations for tensorial group field theories have been done in the truncation method, and completely discarding the Ward identities from their analysis. In this letter, in continuation of our recent series of papers, we present a new framework of investigation, namely the effective vertex expansion, allowing to consider infinite sectors rather than finite dimensional subspaces of the full theory space. We focus on the ultraviolet behavior, and provide a new and complete description of the renormaliza…
We study renormalized quenched strong-coupling QED in four dimensions in arbitrary covariant gauge, in the Dyson-Schwinger equation formalism. Above the chiral critical coupling, we show that there is no finite chiral limit. This behaviour is found to be independent of the detailed choice of proper vertex, provided that the vertex is consistent with the Ward-Takahashi identity and multiplicative renormalizability. The finite solutions previously reported lie in an unphysical regime of the theory with multiple solutions and oscillating mass functions. This study is consistent with the assertion…
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 …
The renormalization group flow in the theory space of a BRST invariant string $σ$-model is investigated. For the open bosonic string the non-perturbative off-shell effective action and its gauge symmetry properties are determined from $β$-functions defined by the local Weyl anomaly. The interactions are shown to explicitly break the free theory BRST invariance generating new non-linear gauge symmetries of the type present in Witten's string field theory. In the Feynman-Siegel gauge the $σ$-model is shown to generate Witten's structure of vertex couplings.
Given a Hamiltonian with a continuous symmetry one can generally factorize that symmetry and consider the dynamics on invariant Hilbert Spaces. In Statistical Mechanics this procedure is known as the vertex-IRF map, and in certain cases, like rotational invariant Hamiltonians, can be implemented via group theoretical techniques. Using this map we translate the DMRG method, which applies to 1d vertex Hamiltonians, into a formulation adequate to study IRF Hamiltonians. The advantage of the IRF formulation of the DMRG method ( we name it IRF-DMRG), is that the dimensions of the Hilbert Spaces inv…
The two-loop three-gluon vertex is calculated in an arbitrary covariant gauge, in the limit when one of the external momenta vanishes. The differential Ward-Slavnov-Taylor (WST) identity related to this limit is discussed, and the relevant results for the ghost-gluon vertex and two-point functions are obtained. Together with the differential WST identity, they provide another independent way for calculating the three-gluon vertex. The renormalization of the results obtained is also presented.
This paper, in addition to our recent works, intends to explore the behavior of the Wetterich flow equations in the portion of the theory space spanned by non-branching melons constrained with Ward-identities. We focus on a rank-5 just-renormalizable tensorial group field theory and consider a non-trivial extension of the local potential approximation namely effective vertex expansion for just-renormalizable quartic melonic interactions, disregarding effects coming from disconnected interactions. Investigating the dynamically constrained flow, we show explicitly that results weakly rely on the…
The 4PI effective action provides a a hierarchy of integral equations which have the form of Bethe-Salpeter equations. The vertex functions obtained from these equations can be used to truncate the exact renormalization group flow equations. This truncation has the property that the flow is a total derivative with respect to the flow parameter and is equivalent to solving the nPI equations of motion. This result establishes a direct connection between two non-perturbative methods.
By exploiting the convexity of the two-particle-irreducible (2PI) effective action, we describe a procedure for extracting n-point vertex functions. This procedure is developed within the context of a zero-dimensional "quantum field theory" and subsequently extended to higher dimensions. These results extend the practicability and utility of a recent, alternative approach to the functional renormalization group programme [see arXiv:1907.06503; arXiv:2107.12914], and clarify the relationship between the flow equations for coupling parameters and vertices.
As far as the renormalization in perturbative QCD is concerned the n-point one particle irreducible (1PI) proper vertex function is the basic building block where the ultra-violet (UV) divergence occurs when the loop momentum integration limit goes to infinity. In this paper we express the S-matrix element for the $gg \rightarrow ggg$ scattering process at all orders in coupling constant in terms of 5-point, 4-point, 3-point 1PI proper vertex functions and the (full) propagator by using the path integral formulation of QCD.
We study response of liquid to a scale transformation, which generates a change of the liquid density, and obtain a set of differential equations for correlation functions. The set of equations, which we call density renormalization group equations (DRGEs), is similar to the BBGKY hierarchy as it relates different multiple-point correlation functions. In particular, we derive DRGEs for one-particle irreducible vertex functions of liquid by performing Legendre transformations, which enables us to calculate properties of liquid at higher density in terms of correlation functions at lower density…
The QED renormalization is restudied by using a mass-dependent subtraction which is performed at a time-like renormalization point. The subtraction exactly respects necessary physical and mathematical requirements such as the gauge symmetry, the Lorentz- invariance and the mathematical convergence. Therefore, the renormalized results derived in the subtraction scheme are faithful and have no ambiguity. Especially, it is proved that the solution of the renormalization group equation satisfied by a renormalized wave function, propagator or vertex can be fixed by applying the renormalization boun…
We investigate the quark-gluon three-point correlation function within a one-loop computation performed in the Curci-Ferrari massive extension of the Faddeev-Popov gauge-fixed action. The mass term is used as a minimal way for taking into account the influence of the Gribov ambiguity. Our results, with renormalization-group improvement, are compared with lattice data. We show that the comparison is in general very satisfactory for the functions which are compatible with chiral symmetry, except for one. We argue that this may be due to large systematic errors {when extracting this function from…
The conformal symmetry of the QCD Lagrangian for massless quarks is broken both by renormalization effects and the gauge fixing procedure. Renormalized primitive divergent amplitudes have the property that their form away from the overall coincident point singularity is fully determined by the bare Lagrangian, and scale dependence is restricted to $δ$-functions at the singularity. If gauge fixing could be ignored, one would expect these amplitudes to be conformal invariant for non-coincident points. We find that the one-loop three-gluon vertex function $Γ_{μνρ}(x,y,z)$ is conformal invariant i…
We present a unified framework for the renormalisation of the Hamiltonian and eigenbasis of a system of correlated electrons, unveiling thereby the interplay between electronic correlations and many-particle entanglement. For this, we extend substantially the unitary renormalization group (URG) scheme introduced in Refs.\cite{anirbanmotti,anirbanmott2,mukherjee2020}. We recast the RG as a discrete flow of the Hamiltonian tensor network, i.e., the collection of various $2n$-point scattering vertex tensors comprising the Hamiltonian. The renormalisation progresses via unitary transformations tha…
In this paper we present an improved RI-type prescription appropriate for the non-perturbative renormalization of gauge invariant nonlocal operators. In this prescription, the non-perturbative vertex function is improved by subtracting unwanted finite lattice spacing ($a$) effects, calculated in lattice perturbation theory. The method is versatile and can be applied to a wide range of fermion and gluon actions, as well as types of nonlocal operators. The presence of operator mixing can also be accommodated. Compared to the standard RI' prescription, this variant can be recast as a supplementar…
We find a geometrical description from a field theoretical setup based on Wilson's renormalization group in real space. We show that renormalization group equations of coupling parameters encode the metric structure of an emergent curved space, regarded to be an Einstein equation for the emergent gravity. Self-consistent equations of local order-parameter fields with an emergent metric turn out to describe low energy dynamics of a strongly coupled field theory, analogous to the Maxwell equation of the Einstein-Maxwell theory in the AdS$_{d+2}$/CFT$_{d+1}$ duality conjecture. We claim that the …
With the introduction of a spectral representation, the Schwinger-Dyson equation (SDE) for the fermion propagator is formulated in Minkowski space in QED. After imposing the on-shell renormalization conditions, analytic solutions for the fermion propagator spectral functions are obtained in four dimensions with a renormalizable version of the Gauge Technique anzatz for the fermion-photon vertex in the quenched approximation in the Landau gauge. Despite the limitations of this model, having an explicit solution provides a guiding example of the fermion propagator with the correct analytic struc…
The DC electrical conductivity of an ultrarelativistic QED plasma is studied in real time by implementing the dynamical renormalization group. The conductivity is obtained from the realtime dependence of a dissipative kernel related to the retarded photon polarization. Pinch singularities in the imaginary part of the polarization are manifest as growing secular terms that in the perturbative expansion of this kernel. The leading secular terms are studied explicitly and it is shown that they are insensitive to the anomalous damping of hard fermions as a result of a cancellation between self-ene…
Fermionic functional renormalization group (f-FRG) is applied to describe Bose-Einstein condensation (BEC) of dimers for a two-component fermionic system with attractive contact interaction. In order to describe the system of dimers without introducing auxiliary bosonic fields (bosonization), we propose a new exact evolution-equation of the effective action in f-FRG with an infrared regulator for the fermion vertex. Then we analyze its basic properties in details. We show explicitly that the critical temperature of the free Bose gas is obtained naturally by this method without bosonization. Me…
