Linear Delta Expansion Research Articles

Generalized Binomial Transform Applied to the Divergent Series

The divergent series for a function defined through Lapalce integral and the ground state energy of the quartic anharmonic oscillator to large orders are studied to test the generalized binomial transform which is the renamed version of $\delta$-expansion proposed recently. We show that, by the use of the generalized binomial transform, the values of functions in the limit of zero of an argument is approximately computable from the series expansion around the infinity of the same argument. In the Laplace integral, we investigate the subject in detail with the aid of Mellin transform. In the anharmonic oscillator, we compute the strong coupling limit of the ground state energy and also the expansion coefficients at strong coupling from the weak coupling perturbation series. The obtained result is compared with that of the linear delta expansion.

Supergraph approach in a higher-order linear delta expansion calculation of the effective potential forF-type broken supersymmetry

In this work, we adopt the simplest model that spontaneously breaks supersymmetry, namely, the minimal O'Raifeartaigh model. The effective potential is computed in the framework of the linear delta expansion approach up to the order ${\ensuremath{\delta}}^{2}$, conjugated with superspace and supergraph techniques. The latter can be duly mastered even if supersymmetry is no longer exact, and the efficacy of the superfield approach in connection with the linear delta expansion procedure is confirmed according to our investigation. That opens up a way for a semi-nonperturbative superspace computation that allows us to deal with spontaneously broken supersymmetric models and encourages us to go further and apply this treatment to the minimal supersymmetric Standard Model precision tests.

Superspace approach to the renormalization of the O’Raifeartaigh model up to the second order in the linear delta expansion parameter

We adopt a superspace/supergraph formalism to pursue the investigation of the structure of one- and two-loop divergences in the frame of the minimal O'Raifeartaigh model that realizes the F-term spontaneous supersymmetry breaking. The linear delta expansion(LDE) procedure is introduced and renormalization is carried out up to the second order in the LDE expansion parameter. In agreement with the nonrenormalization theorem for the (chiral/antichiral) matter potential of ${\mathcal N}=1, \ D=4$ supersymmetry, our explicit supergraph calculations confirm that only the K\"{a}hler potential is actually renomalized.

Iterative approach for the eigenvalue problems

An approximation method based on the iterative technique is developed within the framework of linear delta expansion (LDE) technique for the eigenvalues and eigenfunctions of the one-dimensional and three-dimensional realistic physical problems. This technique allows us to obtain the coefficient in the perturbation series for the eigenfunctions and the eigenvalues directly by knowing the eigenfunctions and the eigenvalues of the unperturbed problems in quantum mechanics. Examples are presented to support this. Hence, the LDE technique can be used for nonperturbative as well as perturbative systems to find approximate solutions of eigenvalue problems.

Linear delta expansion applied to the O’Raifeartaigh model

We reassess the method of the linear delta expansion for the calculation of effective potentials in superspace, by adopting the improved version of the super-Feynman rules in the framework of the O'Raifeartaigh model for spontaneous supersymmetry breaking. The effective potential is calculated using both the fastest apparent convergence and the principle of minimal sensitivity criteria and the consistency and efficacy of the method are checked in deriving the Coleman-Weinberg potential.

An extension of the linear delta expansion to superspace

We introduce and discuss the method of linear delta expansion for the calculation of effective potentials in superspace, by adopting the improved version of the super-Feynman rules. Calculations are carried out up to two loops and an expression for the optimized Kaehler potential in the Wess-Zumino model is worked out.

Linear delta expansion technique for the solution of anharmonic oscillations

The linear delta expansion technique has been developed for solving the differential equation of motion for symmetric and asymmetric anharmonic oscillators. We have also demonstrated the sophistication and simplicity of this new perturbation technique.

Dispersion relation of the nonlinear Klein-Gordon equation through a variational method

We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equation in the case of strong nonlinearities using a method based on the linear delta expansion. All the results obtained in this article are fully analytical, never involve the use of special functions, and can be used to obtain systematic approximations to the exact results to any desired degree of accuracy. We compare our findings with similar results in the literature and show that our approach leads to better and simpler results.

Improved Lindstedt–Poincaré method for the solution of nonlinear problems

We apply the Linear Delta Expansion (LDE) to the Lindstedt–Poincaré (“distorted time”) method to find improved approximate solutions to nonlinear problems. We find that our method works very well for a wide range of parameters in the case of the anharmonic oscillator (Duffing equation), of the non linear pendulum and of more general anharmonic potentials. The approximate solutions found with this method converge more rapidly to the exact ones than in the simple Lindstedt–Poincaré method.

Presenting a new method for the solution of nonlinear problems

We present a method for the resolution of (oscillatory) nonlinear problems. It is based on the application of the linear delta expansion to the Lindstedt–Poincaré method. By applying it to the Duffing equation, we show that our method substantially improves the approximation given by the simple Lindstedt–Poincaré method.

Quantum field dynamics of the slow rollover in the linear delta expansion

We show how the linear delta expansion, as applied to the slow-roll transition in quantum mechanics, can be recast in the closed time-path formalism. This results in simpler, explicit expressions than were obtained in the Schr\"odinger formulation and allows for a straightforward generalization to higher dimensions. Motivated by the success of the method in the quantum-mechanical problem, where it has been shown to give more accurate results for longer than existing alternatives, we apply the linear delta expansion to four-dimensional field theory. At small times all methods agree. At later times, the first-order linear delta expansion is consistently higher that Hartree-Fock, but does not show any sign of a turnover. A turnover emerges in second-order of the method, but the value of $<\hat{\Phi}^2(t)>$ at the turnover is larger that that given by the Hartree-Fock approximation. Based on this calculation, and our experience in the corresponding quantum-mechanical problem, we believe that the Hartree-Fock approximation does indeed underestimate the value of $<\hat{\Phi}^2(t)>$ at the turnover. In subsequent applications of the method we hope to implement the calculation in the context of an expanding universe, following the line of earlier calculations by Boyanovsky {\sl et al.}, who used the Hartree-Fock and large-N methods. It seems clear, however, that the method will become unreliable as the system enters the reheating stage.

Convergence of the linear delta expansion in the critical O(N) field theory.

The linear delta expansion is applied to the three-dimensional O(N) scalar field theory at its critical point in a way that is compatible with the large-N limit. For a range of the arbitrary mass parameter, the linear delta expansion for <phi-->(2)> converges, with errors decreasing as a power of the order n in delta. If the principal of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with n.

Convergence of the linearδexpansion for the shift inTcfor Bose-Einstein condensation

The leading correction from interactions to the transition temperature T_c for Bose-Einstein condensation can be obtained from a nonperturbative calculation in the critical O(N) scalar field theory in 3 dimensions with N=2. We show that the linear delta expansion can be applied to this problem in such a way that in the large-N limit it converges to the exact analytic result. If the principal of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the delta expansion. For N=2, we calculate the shift in T_c to fourth order in delta. The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations.

Convergent resummed linear delta expansion in the critical O(N) (phi 2 i) 2 3d Model.

The linear delta expansion (LDE) is applied to the critical O(N) phi4 three-dimensional field theory which has been widely used to study the temperature of condensation of dilute weakly interacting homogeneous Bose gases. We study the higher order convergence of the LDE as it is usually applied to this problem. We show how to improve both the large N and finite N=2 LDE results with an efficient resummation technique which accelerates convergence. In the large N limit, it reproduces the known exact result within numerical integration accuracy. In the finite N=2 case, our improved results support the recent numerical Monte Carlo estimates for the transition temperature.

Effect of weak interactions on the ultrarelativistic Bose-Einstein condensation temperature

We calculate the ultrarelativistic Bose-Einstein condensation temperature of a complex scalar field with weak $\ensuremath{\lambda}({\ensuremath{\Phi}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\Phi}{)}^{2}$ interaction. We show that at high temperature and finite density we can use dimensional reduction to produce an effective three-dimensional theory which then requires nonperturbative analysis. For simplicity and ease of implementation we illustrate this process with the linear delta expansion.

Quantum dynamics of the slow rollover transition in the linear delta expansion

We apply the linear delta expansion to the quantum mechanical version of the slow rollover transition which is an important feature of inflationary models of the early universe. The method, which goes beyond the Gaussian approximation, gives results which stay close to the exact solution for longer than previous methods. It provides a promising basis for extension to a full field theoretic treatment.

Non-perturbative calculations of a global theory at finite density and temperature

We use an optimised hopping parameter expansion for the free energy (linear delta expansion) to study the phase transitions at finite temperature and finite charge density in a global U(1) scalar Higgs sector on the lattice at large lattice couplings. We are able to plot out phase diagrams in lattice parameter space and find that the standard second-order phase transition with temperature at zero chemical potential becomes first order as the chemical potential increases.

Convergence of Scaled Delta Expansion: Anharmonic Oscillator

We prove that the linear delta expansion for energy eigenvalues of the quantum mechanical anharmonic oscillator converges to the exact answer if the order dependent trial frequency Ω is chosen to scale with the order as Ω = CN γ; 1/3 < γ < 1/2, C > 0 as N → ∞. It converges also for γ = 1/3, if C ≥ α c g 1/3, α c ≃ 0.570875, where g is the coupling constant in front of the operator q 4/4. The extreme case with γ = 1/3, C = γ c g 1/3 corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones.

Variational approach to the SU(2) mass gap on the lattice

A brief introduction is given to the basic idea of the linear delta expansion, which is illustrated in the case of a simple integral. The method is then applied to the calculation of the scalar glueball mass in lattice SU(2).

Optimization in the extended variational approach: TheSU(2)-SO(3) phase structure on the lattice

We use the linear delta expansion with a trial action based on single links to explore the phase structure of the mixedSU(2)-SO(3) lattice gauge theory. The method can be regarded as a systematic expansion going beyond mean field theory or the variational method, to which it is closely related in first order. AtO(δ2) the line of first-order phase transitions is shown to terminate, and atO(δ3) the boundaries between the different phases are well reproduced.