Three-point Vertex Research Articles

Extended iterative scheme for QCD: three-point vertices

In the framework of a generalized iterative scheme introduced previously to account for the non-analytic coupling dependence associated with the renormalization-group invariant mass scale Lambda, we establish the self-consistency equations of the extended Feynman rules (Lambda-modified vertices of zeroth perturbative order) for the three-gluon vertex, the two ghost vertices, and the two vertices of massless quarks. Calculations are performed to one-loop-order, in Landau gauge, and at the lowest approximation level (r=1) of interest for QCD. We discuss the phenomenon of compensating poles inherent in these equations, by which the formalism automatically cancels unphysical poles on internal lines, and the role of composite-operator information in the form of equation-of-motion condensate conditions. The observed near decoupling of the four-gluon conditions permits a solution to the 2-and-3-point conditions within an effective one-parameter freedom. There exists a parameter range in which one solution has all vertex coefficients real, as required for a physical solution, and a narrower range in which the transverse-gluon and massless-quark propagators both exhibit complex-conjugate pole pairs.

Easy way to solve two-loop vertex integrals

Negative dimensional integration is a step further dimensional regularization ideas. In this approach, based on the principle of analytic continuation, Feynman integrals are polynomial ones and for this reason very simple to handle, contrary to the usual parametric ones. The result of the integral worked out in $D<0$ must be analytically continued again --- of course --- to real physical world, $D>0$, and this step presents no difficulties. We consider four two-loop three-point vertex diagrams with arbitrary exponents of propagators and dimension. These original results give the correct well-known particular cases where the exponents of propagators are equal to unity.

Nonperturbative three-point vertex in massless quenched QED and perturbation theory constraints

Dong, Munczek, and Roberts have shown how the full 3-point vertex that appears in the Schwinger-Dyson equation for the fermion propagator can be expressed in terms of a constrained function ${W}_{1}$ in massless quenched QED. However, this analysis involved two key assumptions: that the fermion anomalous dimension vanishes in the Landau gauge and that the transverse vertex has a simplified dependence on momenta. Here we remove these assumptions and find the general form for a new constrained function ${U}_{1}$ that ensures the multiplicative renormalizability of the fermion propagator nonperturbatively. We then study the restriction imposed on ${U}_{1}$ by recent perturbative calculations of the vertex and compute its leading logarithmic expansion. Since ${U}_{1}$ should reduce to this expansion in the weak coupling regime, this should serve as a guide to its nonperturbative construction. We comment on the perturbative realization of the constraints on ${U}_{1}.$

A Study of Fermionic Three-Point Vertex with Gluon Condensate

We derive the dimension-four gluon condensate contributions to the fermionic three-point vertex with different forms of the gluon condensate component of the nonlocal two-gluon vacuum expectation value (vev). We find that the result with the non-covariant form of nonlocal two-gluon vev in the fixed-point gauge depends on whether the zero-point is set on the fermionic- or gluonic-line. But this uncertainty can be avoided by using the covariant form of the nonlocal two-gluon vev.

Zero-Point Problem in Fixed-Point Gauge of Nonperturbative QCD

We discuss the essence of the zero-point in fixed-point gauge of nonperturbative QCD and describe how to set the zero-point appropriately in deriving the nonperturbative quark–gluon three-point vertex functions including gluon condensates.

CHORD DIAGRAMS AND BPHZ SUBTRACTIONS

The combinatorics of the BPHZ subtraction scheme for a class of ladder graphs for the three-point vertex in ɸ3 theory is transcribed into certain connectivity relations for marked chord diagrams (knots with transversal intersections). The resolution of the singular crossings using the equivalence relations in these examples provides confirmation of a proposed fundamental relationship between knot theory and renormalization in perturbative quantum field theory.

Hidden symmetry, exact relations, and a small parameter in surface growth models with diffusion

An exact relation between the Green's function and the dressed three-point vertex function Γ was found for the Langevin equations describing surface growth with diffusion in 1 + d dimensions. This relation follows from a hidden symmetry of the problem with a gauge function depending on time only and turns out to be exactly the same as that found by Lebedev and L'vov for the Kardar-Parisi-Zhang (KPZ) equation of surface roughening. By a similar analysis as was done for the KPZ equation we conclude that in the region of strong coupling Γ − Γ ∼ 0.1 Γ 0, where Γ 0 is the bare value of the vertex Γ.

Background field method: Alternative way of deriving the pinch technique's results.

We show that the background field method (BFM) is a simple way of deriving the same gauge-invariant results which are obtained by the pinch technique (PT). For illustration we construct gauge-invariant self-energy and three-point vertices for gluons at one-loop level by BFM and demonstrate that we get the same results which were derived via PT. We also calculate the four-gluon vertex in BFM and show that this vertex obeys the same Ward identity that was found with PT.}

Ward identities for surface-growth models with diffusion.

Using the method of Hwa and Fisher [Phys. Rev. B 49, 3136 (1994)], we derive Ward identities relating the two- and three-point vertex functions in the dynamic functional theory for surface-growth models with diffusion. These identities result from the statistical transformation symmetries in the underlying Langevin equations for these models. An explicit scaling form for the full three-point vertex function is also obtained. As a result one can show that the scaling properties of the full three-point vertex function are the same as those of the ``bare'' vertex function obtained in the case of no stochastic noise. This implies that the coupling constant for the nonlinear term in the Langevin equation is not renormalized.

Covariant three-body equations in φ 3 field theory

We derive four-dimensional relativistic three-body equations for the case of a field theory with a three-point interaction vertex. These equations describe the coupled 2 → 2, 2 → 3, and 3 → 3 processes, and provide the means of calculating the kernel of the 2 → 2 Bethe-Salpeter equation. Our equations differ from all previous formulations in two essential ways. Firstly, we have overcome the overcounting problems inherent in earlier works. Secondly, we have retained all possible two-body forces when one particle is a spectator. In this respect, we show how it is necessary to also retain certain three-body forces as these can give rise to (previously overlooked) two-body forces when used in a 2 → 3 process. The revealing of such hidden two-body forces gives rise to a further novel feature of our equations, namely, to the appearance of a number of subtraction terms. In the case of the πNN system, for example, the NN potential involves a subtraction term where two pions, exchanged between the nucleons, interact with each other through the ππ t-matrix. The necessity of an input ππ interaction is surprising and contrasts markedly with the corresponding three-dimensional description of the πNN system where no such interaction explicitly appears. This illustrates the somewhat unexpected result that the four-dimensional equations differ from the three-dimensional ones even at the operator level.

Monte-Carlo simulations of a flux tube model

We study, using Monte-Carlo simulations, a lattice version of the Patel flux tube model with the three-point baryonic vertices. Employing finite size scaling analysis for the heat-capacity we show that for large quark masses the model undergoes the second order phase transition.

Crossing-symmetric two-particle reduction of four-point vertex.

We develop a nonperturbative reduction scheme for the four-point vertex [ital T] of a generic field theory with interactions among bosons, fermions, and antifermions. We exhibit integral equations which express, in a manifestly crossing-symmetric way, the full vertex in terms of its two-particle irreducible part, together with the dressed three-point vertices and two-point propagators. This scheme generalizes the usual summation of ladder or bubble diagrams, thus providing for the consistent summation of a larger subset of all diagrams.

An approach to the evaluation of three- and four-point ladder diagrams

An approach to the calculation of ladder graphs with three and four external lines is considered (in the case of massless internal particles and arbitrary external momenta). Simple formulae for reducing four-point diagrams to three-point vertices are derived. Exact results for diagrams up to the two-loop level are obtained in terms of polylogarithms.

Theory of strongly correlated electron systems on the basis of a diagrammatic technique for Hubbard operators.

The Hubbard model with infinite intrasite Coulomb interaction is studied in terms of the diagrammatic technique for Hubbard operators. Exact diagrammatic representations for single-particle electron Green's functions ${\mathit{scrG}}_{\mathrm{\ensuremath{\uparrow}}}$,${\mathit{scrG}}_{\mathrm{\ensuremath{\downarrow}}}$, as well as for the spin Green's function scrD are constructed. All three Green's functions are expressed through a unique system of four-point and three-point vertex parts, for which graphical expressions are given in terms of diagrammatic series. The general graphical representations are given for both the paramagnetic and the ferromagnetic phases. Evaluation of the four-point vertex parts for a three-dimensional system are performed within the generalized random-phase approximation (GRPA), which consists in summation of all loop-type diagrams. The GRPA expressions for the vertex parts are used to calculate the Green's functions. It is shown that the electron Green's functions ${\mathit{scrG}}_{\mathrm{\ensuremath{\uparrow}}}$ and ${\mathit{scrG}}_{\mathrm{\ensuremath{\downarrow}}}$ in the paramagnetic phase have a propagator character, whereas the spin Green's function scrD is more complex exhibiting existence of incoherent states.The propagator-type and incoherent-type contributions correspond to two terms in the expression for the dynamical susceptibility. One of them has the typical Pauli character for itinerant magnetism, and the other is of Curie form (\ensuremath{\propto}1/T), manifesting the appearance of localized magnetic moments. It has been shown that there is a critical electron concentration ${\mathit{n}}_{\mathit{c}}$, for which a change of the regime from the itinerant-type magnetism to the magnetic behavior typical for localized magnetic moments takes place. Simultaneously, at the same critical concentration the paramagnetic phase becomes unstable against the ferromagnetic ordering. In the simplest approximation for the electron Green's-function lines which enter the loop diagrams, it turns out that ${\mathit{n}}_{\mathit{c}}$=0.66 for the case of constant density of states of the unperturbed band. The exact diagrammatic expressions for all three Green's functions are also given in the ferromagnetic phase. It is shown that two of them, ${\mathit{scrG}}_{\mathrm{\ensuremath{\downarrow}}}$ and scrD, are of the propagator type, whereas ${\mathit{scrG}}_{\mathrm{\ensuremath{\uparrow}}}$ (for the spin aligned against the spontaneous magnetization) contains incoherent contributions. The presented theory can be further developed in many different directions.

A THEORY OF FERROMAGNETISM IN THE HUBBARD MODEL WITH INFINITE COULOMB INTERACTION

On the basis of a regular perturbation theory in the form of diagrammatic technique with X-operators, a ferromagnetic state in the Hubbard model with infinite on-site Coulomb interaction U is studied for a wide range of electron concentration n. Treating the kinetic energy of electrons as perturbation, we get the exact graphic representations for three Green’s functions [Formula: see text] and [Formula: see text] for electrons with spin ↑ and ↓ (spontaneous magnetic moment is in ↑ direction) and [Formula: see text] for spin waves. All of them are expressed through the unique system of four-point and three-point vertex parts for effective electron-magnon and electron-electron interactions. These vertex parts are calculated in two approximations: a low-density approximation for n≪1 or nh=1–n≫1, and the generalized random-phase approximation (GRPA), which was suggested earlier by us for the description of paramagnetic phase in this model. An important result in both cases is the understanding of essential difference of electron states with spin ↑ and ↓. The state for ↑ has coherent character in all region of electron concentration (n&gt;nc) where a ferromagnetic state exists, while the state for spin ↓ has mixed characters including both coherent and incoherent contribution. For the saturated ferromagnetism (n&gt;ns) when nh≪1, [Formula: see text] is an entirely incoherent (nonquasiparticle) state. Appearance of incoherent state is probably a general property of strongly correlated systems distinguishing its behavior from the Fermi liquid. We show also that the Hubbard-I approximation has no region of applicability for the electron with spin ↓ in a ferromagnet and we found a new factor describing the correlation-narrowing of ↓ spin electron band. Green’s function [Formula: see text] calculated in two different limits n≪1 and nh≪1, are jointed together to allow us to calculate ns which lies in the intermediate concentration regime. For simple cubic lattice we found ns=0.87. In the limit nh≪1 our results reduce to the earlier known ones, including Nagaoka’s result for the spin wave energy. We point out the way to construct a rigorous theory for ferromagnetic state in the intermediate range of electron concentration. In the conclusion several important problems are discussed in connection to the continuation of the present work.

Truncating the Schwinger-Dyson equations: How multiplicative renormalizability and the Ward identity restrict the three-point vertex in QED.

Nonperturbative studies of field theory require the Schwinger-Dyson equations to be truncated to make them tractable. Thus, when investigating the behavior of the fermion propagator, for example, an Ansatz for the three-point vertex has to be made. While the well-known Ward identity determines the longitudinal part of this vertex in terms of the fermion propagator as shown by Ball and Chiu, it leaves the transverse part unconstrained. However, Brown and Dorey have recently emphasized that the requirement of multiplicative renormalizability is not satisfied by arbitrary Ansa$iuml---tze for the vertex. We show how this requirement restricts the form of the transverse part. By considering the example of QED in the quenched approximation, we present a form for the vertex that not only satisfies the Ward identity but is multiplicatively renormalizable to all orders in leading and next-to-leading logarithms in perturbation theory and so provides a suitable Ansatz for the full three-point vertex.

Free field representation of minimal models on a Riemann surface

We introduce a BRST invariant screened three-point vertex defined on a product of Fock spaces. The computation of correlation functions of minimal conformal models on higher genus Riemann surface is thereby reduced to a free field computation.

Perturbative aspects of the Chern-Simons field theory

The quantization of the non-abelian Chern-Simons theory in three dimensions is performed in the framework of the BRS formalism. General covariance is preserved on the physical subspace. The perturbative analysis at two loops confirms that the model is finite. To this order, the radiative corrections to the two- and three-point proper vertices vanish.

Numerical study of self-couplings in the broken phase of the lattice Ising model

A Monte Carlo study of a one-component scalar Φ4 model was made on a 104 hypercubic lattice in its Ising limit. We measured the renormalized mass and coupling of the three-point vertex in the spontaneously broken phase. By measuring them at non-zero momenta, we successfully settled problems caused by the finite vacuum expectation value of the scalar field. To suppress artificial fluctuation of observables, a uniform source was introduced. Our results are in good agreement with the one-loop relation between the vacuum expectation value, mass and the three-point coupling.

Interacting disordered electrons and the metal-insulator transition: The field theoretic approach reanalyzed

We consider the nonlinear σ-model for the interplay between interaction and disorder in electronic systems as derived by Finkelshtein. We use a parametrization of the model which facilitates setting up a loop expansion in exact analogy to more common models. We discuss symmetry properties, and address the questions of renormalizability and the number of renormalization constants. For the model with unitary symmetry we calculate the one-point and various two-point vertex functions to one-loop order. The introduction of five renormalization constants is necessary to make these finite. The three-point vertex functions are then shown to be rendered finite by the same five renormalization constants. This is a very strong indication that the theory is renormalizable, and that the resulting RG flow equations are physically meaningful.