Nonperturbative Scheme Research Articles

Localized Growth Modes, Dynamic Textures, and Upper Critical Dimension for the Kardar-Parisi-Zhang Equation in the Weak-Noise Limit

A weak-noise scheme is applied to the Kardar-Parisi-Zhang equation for a growing interface in all dimensions. It is shown that the solutions can be interpreted in terms of a growth morphology of a dynamically evolving texture of localized growth modes with superimposed diffusive modes. By applying Derrick's theorem, it is conjectured that the upper critical dimension is four.

A NEW GENERAL APPROXIMATION SCHEME IN QUANTUM THEORY: APPLICATION TO THE ANHARMONIC AND THE DOUBLE WELL OSCILLATORS

A self-consistent, nonperturbative approximation scheme is proposed which is potentially applicable to arbitrary interacting quantum systems. For the case of self-interaction, the scheme consists in approximating the original interaction HI(ϕ) by a suitable "potential" V(ϕ) which satisfies the following two basic requirements, (i) exact solvability (ES): the "effective" Hamiltonian H0 generated by V(ϕ) is exactly solvable i.e., the spectrum of states |n〉 and the eigenvalues En are known and (ii) equality of quantum averages (EQA): 〈n|HI(ϕ)|n〉 = 〈n|V(ϕ)|n〉 for arbitrary n. The leading order (LO) results for |n〉 and En are thus readily obtained and are found to be accurate to within a few percent of the "exact" results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H0 as the unperturbed Hamiltonian and the modified interaction, λH′(ϕ)≡λ(HI(ϕ) - V(ϕ)), as the perturbation where λ is the coupling strength. The condition of convergence of the IPT for arbitrary λ is satisfied due to the EQA requirement which ensures that 〈n|λH′(ϕ)|n〉 = 0for arbitrary λ and n. This is in contrast to the divergence (which occurs even for infinitesimal λ!) in the naive perturbation theory where the original interaction λHI(ϕ) is chosen as the perturbation. We apply the method to the different cases of the anharmonic and the double well potentials, e.g. quartic-, sextic- and octic-anharmonic oscillators and quartic-, sextic-double well oscillators. Uniformly accurate results for the energy levels over the full allowed range of λ and n are obtained. The results compare well with the exact results predicted by supersymmetry for the case of the sextic anharmonic potential and the double well partner potential. Further improvement in the accuracy of the results by the use of IPT, is demonstrated. We also discuss the vacuum structure and stability of the resulting theory in the above approximation scheme.

Quantum time uncertainty in a gravity's rainbow formalism

The existence of a minimum time uncertainty is usually argued to be a consequence of the combination of quantum mechanics and general relativity. Most of the studies that point to this result are nonetheless based on perturbative quantization approaches, in which the effect of matter on the geometry is regarded as a correction to a classical background. In this paper, we consider rainbow spacetimes constructed from doubly special relativity by using a modification of the proposals of Magueijo and Smolin. In these models, gravitational effects are incorporated (at least to a certain extent) in the definition of the energy-momentum of particles without adhering to a perturbative treatment of the backreaction. In this context, we derive and compare the expressions of the time uncertainty in quantizations that use as evolution parameter either the background or the rainbow time coordinates. These two possibilities can be regarded as corresponding to perturbative and nonperturbative quantization schemes, respectively. We show that, while a nonvanishing time uncertainty is generically unavoidable in a perturbative framework, an infinite time resolution can in fact be achieved in a nonperturbative quantization for the whole family of doubly special relativity theories with unbounded physical energy.

Unquenching the Quark-Antiquark Green?s Function

We propose a nonperturbative resummation scheme for the four-point connected quark-antiquark Green's function $G^4$ that shows how the Bethe-Salpeter equation may be `unquenched' with respect to quark-antiquark loops. This mechanism allows to dynamically account for hadronic meson decays and multiquark structures whilst respecting the underlying symmetries. An initial approximation to the four-point Schwinger-Dyson equation -- suitable for phenomenological application -- is examined numerically in a couple of aspects. It is demonstrated that this approximation explicitly maintains the correct asymptotic limits and contains the physical resonance structures in the near timelike region in the quark-antiquark channel whereas no resonances are found in the diquark channel, respectively.

Renormalization ofNNscattering with one pion exchange and energy dependent boundary conditions

A nonperturbative renormalization scheme for nucleon-nucleon interaction based on boundary conditions at short distances is presented and applied to the one pion exchange potential. It is free of off-shell ambiguities and ultraviolet divergences, provides finite results at any step of the calculation, and allows us to remove the short distance cutoff in a suitable way. Low energy constants and their nonperturbative evolution can be directly obtained from experimental threshold parameters in a completely unique and model independent way when the long range explicit pion effects are eliminated. This allows us to compute scattering phase shifts which are, by construction, consistent with the effective range expansion to a given order in the center of mass (c.m.) momentum $p$. In the singlet $^{1}S_{0}$ and triplet $^{3}S_{1}\text{\ensuremath{-}}^{3}D_{1}$ channels ultraviolet fixed points and limit cycles are obtained, respectively, for the threshold parameters. Data are described satisfactorily up to c.m. momenta of about $p\ensuremath{\sim}{m}_{\ensuremath{\pi}}$.

Nonperturbative renormalization for domain wall fermions and the chiral condensate

We study the chiral condensate, $<\bar\psi\psi>$, and various quark bilinear vertex functions for domain wall fermions at different lattice scales, with both the Wilson and DBW2 gauge actions, in both quenched and dynamical fermion simulations. We use the vertex functions to calculate renormalization factors within a non-perturbative scheme.

Renormalized nonperturbative fermion model in covariant light-front dynamics

Within the framework of the covariant formulation of light-front dynamics, we develop a nonperturbative renormalization scheme in the fermion model supposing that the composite fermion is a superposition of the ``bare'' fermion and a fermion+boson state. We first assume the constituent boson to be spinless. Then we address the case of gauge bosons in the Feynman and in the light-cone gauges. For all these cases the fermion state vector and the necessary renormalization counterterms are calculated analytically. It turns out that, in light-front dynamics, to restore the rotational invariance, an extra counterterm is needed, having no analogue in the Feynman approach. For gauge bosons the results obtained in the two gauges are compared with each other. In general, the number of spin components of the two-body $(\mathrm{fermion}+\mathrm{boson})$ wave function depends on the gauge. But due to the two-body Fock space truncation, only one nonzero component survives for each gauge. And moreover, the whole solutions for the state vector, found for the Feynman and light-cone gauges, are the same (except for the normalization factor). The counterterms are, however, different.

Modelling the quark propagator

The quark propagator is at the core of lattice hadron spectrum calculations as well as studies in other nonperturbative schemes. We investigate the quark propagator with an improved staggered action (Asqtad) and an improved gluon action, which provides good quality data down to small quark masses. This is used to construct ansa¨tze suitable for model hadron calculations as well as adding to our intuitive understanding of QCD.

Finite-temperature regularization

We present a non-perturbative regularization scheme for Quantum Field Theories which amounts to an embedding of the original unregularized theory into a spacetime with an extra compactified dimension of length L∼Λ−1 (with Λ the ultraviolet cutoff), plus a doubling in the number of fields, which satisfy different periodicity conditions and have opposite Grassmann parity. The resulting regularized action may be interpreted, for the fermionic case, as corresponding to a finite-temperature theory with a supersymmetry, which is broken because of the boundary conditions. We test our proposal both in a perturbative calculation (the vacuum polarization graph for a D-dimensional fermionic theory) and in a non-perturbative one (the chiral anomaly).

Remarks on the gauge dependence of the RI/MOM renormalization procedure

The RI/MOM non-perturbative renormalization scheme is studied on the lattice in SU(3) quenched QCD with Wilson fermions. The gauge dependence of some fermion bilinear renormalization constants is discussed by comparing data which have been gauge-fixed in two different realizations of the Landau gauge and in a generic covariant gauge. The very good agreement between the various sets of results and the theory indicates that the numerical uncertainty induced by the lattice gauge-fixing procedure is below our statistical errors which are of the order of few percent.

Parameter-free calculation of hadronic masses from instantons

We propose a non-perturbative calculation scheme which is based on the semi-classical approximation of QCD and can be used to evaluate quantities of interest in hadronic physics. As a first application, we evaluate the mass of the pion and of the nucleon. Such masses are related to a particular combination of Green functions which, in some limit, is dominated by the contribution of \emph{very small-sized} instantons. The size distribution of these pseudo-particles is determined by the 't Hooft tunneling amplitude formula and therefore our calculation is free from any model parameters. We prove that instanton forces generate a light pion and a nucleon with realistic mass ($M_n \sim 970 MeV$). In connection with sum-rules approaches, we discuss the overlap of instantons with pion and nucleon resonances.

Moments expansion study of the Rabi Hamiltonian

The interaction of a single bosonic mode with a two-level fermion system is investigated. For the condensed-matter physicist the boson in question is typically a phonon whereas the fermion system is represented by a two-level electron structure. In this work we shall use both the connected moments expansion (CMX) and the alternate moments expansion (AMX) as well as the non-perturbative Lanczos tridiagonal scheme to study the ground-state spectrum of this system. Comparisons will be made with other approximation schemes as well as “exact” methods.

The Functional Callan–Symanzik Equation for the Coulomb Gas

A non-perturbative scheme, based on the functional generalization of the Callan–Symanzik equation is developed to treat the Coulomb interaction in an electron gas. The one-particle irreducible vertex functions are shown to satisfy an evolution equation whose initial condition is given by means of the classical action and the final point corresponds to the physical system. This equation is truncated by expanding it in momenta and excitation energies, leaving the electric charge as an arbitrary, not necessarily small parameter. Exact coupled partial differential equations up to first order in the frequencies and excitation energies are derived. The numerical integration of these equations is left to a later stage. Nevertheless, in order to demonstrate the relation with the perturbation expansion the one-loop Lindhard function and screening are reproduced in the independent mode approximation of the evolution equation.

Is the axial anomaly really determined in a continuous non-perturbative regularization?

In the framework of a gauge invariant continuous and non-perturbative regularization scheme based on the smearing of point like interactions by means of cutoff functions, we show that the axial anomaly, though cutoff independent, depends on the shape of the cutoff functions. The standard value for the strength of the axial anomaly is recovered if we assume that the regularized gauge invariant axial current is in addition local.

The topological susceptibility and fπ from lattice QCD

Abstract We study the topological susceptibility, χ , in QCD with two quark flavours using lattice field configurations that have been produced with an O ( a ) improved quark action. We find clear evidence for the expected suppression at small quark mass, and examine the variation of χ with this mass. The resulting estimate of the pion decay constant, f π =105±6 +18 −10 MeV, is consistent with the experimental value of ≃93 MeV. We compare χ to the large- N c prediction and find consistency over a large range of quark masses. We discuss the benefits of the non-perturbative action improvement scheme and of the stategy of keeping the lattice spacing (nearly) fixed as the quark mass is varied. We compare our results with other studies and suggest why such a quark mass dependence has not always been seen.

φ→π+π−decay within a chiral unitary approach

Starting from the chiral perturbation theory Lagrangian, but keeping different masses for the charged and neutral mesons ${(m}_{u}\ensuremath{\ne}{m}_{d}),$ and using a previously developed nonperturbative unitary scheme that generates the lightest meson-meson resonances, we construct $K\overline{K}\ensuremath{\rightarrow}K\overline{K}$ and $K\overline{K}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ in the vector channel. This allows us to obtain the kaon-loop contribution to $\ensuremath{\varphi}\ensuremath{-}\ensuremath{\rho}$ mixing and study the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varphi}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay. The dominant contribution to this decay comes from the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varphi}}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ process. However, there can be large interferences with the subdominant contributions coming from $\ensuremath{\varphi}\ensuremath{-}\ensuremath{\rho}$ and $\ensuremath{\varphi}\ensuremath{-}\ensuremath{\omega}$ mixing, or of these two contributions among themselves. As a consequence, a reliable measurement of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varphi}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay could be used to differentiate between some $\ensuremath{\varphi}\ensuremath{-}\ensuremath{\omega}$ mixing scenarios proposed in the literature.

Meson-loop effects in the NJL model at zero and nonzero temperature

We compare two different possibilities of including meson-loop corrections in the Nambu-Jona-Lasinio model: a strict 1/N c expansion in the next-to-leading order and a nonperturbative scheme corresponding to a one-meson-loop approximation to the effective action. Both schemes are consistent with chiral symmetry, in particular, with the Goldstone theorem and the Gell-Mann-Oakes-Renner relation. The numerical part at zero temperature focuses on the pion and the ρ-meson sector. For the latter, meson-loop corrections are crucial in order to include the dominant ρ → ππ-decay channel, while the standard Hartree + RPA approximation only contains unphysical $$m_\pi ,f_\pi ,\left\langle {\bar \psi \psi } \right\rangle $$ -decay channels. We find that $$q\bar q$$ , and quantities related to the ρ-meson self-energy can be described reasonably with one parameter set in the 1/N c-expansion scheme, whereas we did not succeed in obtaining such a fit in the nonperturbative scheme. We also investigate the temperature dependence of the quark condensate. Here, we find consistency with the chiral perturbation theory to the lowest order. Similarities and differences of both schemes are discussed.

A consistent nonperturbative approach to thermal damping-rates

We propose a nonperturbative scheme for the calculation of thermal damping-rates using exact renormalization group (RG)-equations. Special emphasis is put on the thermal RG where first results for the rate were given by Pietroni [Phys. Rev. Lett. 81 (1998) 2424]. We point out that in order to obtain a complete result that also reproduces the known perturbative behaviour one has to take into account effects that were neglected by Pietroni. We propose a well-defined way of doing the calculations that reproduces perturbation theory in lowest order but goes considerably beyond perturbative results and should be applicable also at second order phase-transitions.

Nonperturbative renormalization of a fermion mass term in the mass-independent renormalization scheme

We develop a new nonperturbative renormalization scheme for the fermion mass term in the Dyson-Schwinger equation (DSE) approach for the study of the fermion propagator. The renormalized DSE for the fermion propagator is derived in the mass-independent renormalization (MIR) scheme. Our method is suitable for studying the spontaneous breaking of chiral symmetry in gauge field theories, since the MIR scheme connects the system with a finite renormalized mass and the chiral limit of the theory in a smooth way. We demonstrate that our method correctly reproduces the known asymptotic behavior of the renormalized quark propagator when applied to asymptotic-free quark models for QCD. We apply the MIR scheme to the dual Ginzburg-Landau theory and demonstrate its applicability.

Meson-meson interactions in a nonperturbative chiral approach

A nonperturbative method which combines constraints from chiral symmetry breaking and coupled channel unitarity is used to describe the meson-meson interaction up to about 1.2 GeV. The approach uses the $\mathcal{O}{(p}^{2})$ and $\mathcal{O}{(p}^{4})$ chiral Lagrangians. The seven free parameters of the $\mathcal{O}{(p}^{4})$ Lagrangian are fitted to the data. The results are in good agreement with a vast amount of experimental analyses. The amplitudes develop poles in the complex plane corresponding to the ${f}_{0}, {a}_{0}, \ensuremath{\rho}, {K}^{*}, \ensuremath{\varphi}, \ensuremath{\sigma},$ and $\ensuremath{\kappa}$ resonances, the latter two being very broad. The total and partial decay widths of the resonances are also well reproduced. Further extensions and applications of this chiral nonperturbative scheme are also discussed.