Resummation Techniques Research Articles

Solving the anharmonic oscillator: tuning the boundary condition

We outline a remarkably efficient method for generating solutions to quantum anharmonic oscillators with an x2M potential. We solve the Schrödinger equation in terms of a free parameter which is then tuned to give the correct boundary condition by generating a power series expansion of the wavefunction in x and applying a modified Borel resummation technique to obtain the large x behaviour. The process allows us to calculate energy eigenvalues to an arbitrary level of accuracy. High degrees of precision are achieved even with modest computing power. Our technique extends to all levels of excitation and produces the correct solution to the double well oscillators even though they are dominated by non-perturbative effects.

Momentum distributions and spectroscopic factors of doubly closed shell nuclei in correlated basis function theory

The momentum distributions, natural orbits, spectroscopic factors, and quasihole wave functions of the $^{12}\mathrm{C}$, $^{16}\mathrm{O}$, $^{40}\mathrm{Ca}$, $^{48}\mathrm{Ca}$, and $^{208}\mathrm{Pb}$ doubly closed shell nuclei have been calculated in the framework of the correlated basis function theory, by using the Fermi hypernetted chain resummation techniques. The calculations have been done by using the realistic Argonne ${v}_{8}^{'}$ nucleon-nucleon potential, together with the Urbana IX three-body interaction. Operator dependent correlations, which consider channels up to the tensor ones, have been used. We found noticeable effects produced by the correlations. For high momentum values, the momentum distributions show large enhancements with respect to the independent particle model results. Natural orbit occupation numbers are depleted by about 10% with respect to the independent particle model values. The effects of the correlations on the spectroscopic factors are larger on the most deeply bound states.

A NONPERTURBATIVE CALCULATION FOR THE EFFECTIVE POTENTIAL OF THE $\left(\frac{g}{4}\phi^{4}-J\phi\right)_{1+1}$ SCALAR FIELD THEORY USING THE OSCILLATOR REPRESENTATION METHOD WITH BOREL RESUMMATION

We established a resummed formula for the effective potential of [Formula: see text] scalar field theory that can mimic the true effective potential not only at the critical region but also at any point in the coupling space. We first extend the effective potential from the oscillator representation method, perturbatively, up to g3 order. We supplement perturbations by the use of a resummation algorithm, originally due to Kleinert, Thoms and Janke, which has the privilege of using the strong coupling as well as the large coupling behaviors rather than the conventional resummation techniques which use only the large order behavior. Accordingly, although the perturbation series available is up to g3 order, we found a good agreement between our resummed effective potential and the well-known features from constructive field theory. The resummed effective potential agrees well with the constructive field theory results concerning existing and order of phase transition in the absence of an external magnetic field. In the presence of the external magnetic field, as in magnetic systems, the effective potential shows nonexistence of phase transition and gives the behavior of the vacuum condensate as a monotonic increasing function of J, in complete agreement with constructive field theory methods.

Finite-temperature φ 4 theory from the 2PI effective action: Two-loop truncation

Using resummation techniques based on the 2PI effective action we study the scalar φ 4 theory at finite temperature. We present an analytical as well as numerical study for a renormalized two-loop truncation of the action. Both the spectral properties and critical behaviour of the theory are investigated. Within the truncation, we explicitly check that the physical observables are UV-finite.

Critical Exponents from the Resummed Effective Potential of the $$({\frac{g}{4}}\phi^{4}-J\phi)_{1+1}$$ Scalar Field Theory

We establish a unified way for the calculation of the critical exponents, without the use of epsilon expansion, through the improvement of the perturbative effective potential of the 1+1 dimensional \(({\frac{g}{4}}\phi^{4}-J\phi)\) scalar field theory. First, we obtain the perturbation series for the effective potential up to g3. We improved the perturbative effective potential by establishing a parameter-free resummation algorithm, originally due to Kleinert, Thoms and Janke, which has the privilege of using the strong coupling as well as the large coupling behaviors rather than the conventional resummation techniques which use only the large order behavior. Accordingly, although the perturbation series available is up to g3 order, we found a complete agreement between our resummed effective potential and the well known features from constructive field theory. We prove that the 1-PI correlation functions and the effective potential ought to have the same large order as well as strong coupling behaviors. We computed the critical exponents and our results show a good agreement with the exact Ising model values.

The fractal self-similar-Borel algorithm for the effective potential of the scalar field theory in one time plus one space dimensions

Abstract We develop a new resummation algorithm that can give accurate results for the resummation of a divergent series. First, we use a Borel resummation algorithm, originally due to Kleinert, Thoms and Janke, which has the advantage of using the strong coupling as well as the large order behaviors rather than the conventional resummation techniques which use only the large order behavior. The obtained series is supplemented by the fractal self-similar method. We applied the new algorithm to the effective potential of the ( g 4 ) ( ϕ 4 ) 1 + 1 scalar field theory. We were able to predict the critical reduced temperature 1 G c ≈ φ = 5 - 1 2 , which reflects the fractal structure of the magnetization at the critical point. Also, our prediction is in complete agreement with lattice calculations in spite of the use of a relatively short input perturbation series which is up to g3 order.

Improved perturbation method and its application to the IIB matrix model

We present a new scheme for extracting approximate values in “the improved perturbation method”, which is a sort of resummation technique capable of evaluating a series outside the radius of convergence. We employ the distribution profile of the series that is weighted by nth-order derivatives with respect to the artificially introduced parameters. By those weightings the distribution becomes more sensitive to the “plateau” structure in which the consistency condition of the method is satisfied. The scheme works effectively even in such cases that the system involves many parameters. We also propose that this scheme has to be applied to each observable separately and be analyzed comprehensively. We apply this scheme to the analysis of the IIB matrix model by the improved perturbation method obtained up to eighth order of perturbation in the former works. We consider here the possibility of spontaneous breakdown of Lorentz symmetry, and evaluate the free energy and the anisotropy of space–time extent. In the present analysis, we find an SO ( 10 ) -symmetric vacuum besides the SO ( 4 ) - and SO ( 7 ) -symmetric vacua that have been observed. It is also found that there are two distinct SO ( 4 ) -symmetric vacua that have almost the same value of free energy but the extent of space–time is different. From the approximate values of free energy, we conclude that the SO ( 4 ) -symmetric vacua are most preferred among those three types of vacua.

Model C critical dynamics of disordered magnets

The critical dynamics of model C in the presence of disorder is considered. It is known that in the asymptotics a conserved secondary density decouples from the nonconserved order parameter for disordered systems. However couplings between order parameter and secondary density cause considerable effects on non-asymptotic critical properties. Here, a general procedure for a renormalization group treatment is proposed. Already the one-loop approximation gives a qualitatively correct picture of the diluted model C dynamical criticality. A more quantitative description is achieved using two-loop approximation. In order to get reliable results resummation technique has to be applied.

IMPROVEMENT OF RENORMALIZATION-SCALE UNCERTAINTIES WITHIN EMPIRICAL DETERMINATIONS OF THE b-QUARK MASS

Accurate determinations of the [Formula: see text]b-quark mass mb(mb) from σ(e+e-→ hadrons ) experimental data currently contain three comparable sources of uncertainty; the experimental uncertainty from moments of this cross-section, the uncertainty associated with αs(Mz), and the theoretical uncertainty associated with the renormalization scale. Through resummation of all logarithmic terms explicitly determined in the perturbative series by the renormalization-group (RG) equation, it is shown that the renormalization-scale dependence is virtually eliminated as a source of theoretical uncertainty in mb(mb). This resummation also reduces the estimated effect of higher-loop perturbative contributions, further reducing the theoretical uncertainties in mb(mb). Furthermore, such resummation techniques improve the agreement between the values of the [Formula: see text]b-quark mass extracted from the various moments of R(s) = σ(e+e-→ hadrons )/σpt [σpt = 4πα2/(3s)], obviating the need to choose an optimum moment for determining mb(mb). Based on this analysis, the resulting value of the b-mass is mb(mb) = 4.207 GeV ± 40 MeV , where the dominant uncertainty now arises from the experimental moments. Resummation techniques are also shown to reduce renormalization-scale dependence in the relation between b-quark [Formula: see text] and pole mass and in the relation between the pole and 1S mass.

QCD Effects in High Energy Processes

In this talk, some important QCD effects in Higgs physics, supersymmetry and top physics, as well as the factorization and resummation techniques in QCD are reviewed.

Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities

We prove the existence of small amplitude periodic solutions, for a large Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak quadratic and velocity dependent nonlinearity and with Dirichelet boundary conditions. Such nonlinear PDE can be regarded as a simple model describing oscillations of flexible structures like suspension bridges in presence of an uniform wind flow. The periodic solutions are explicitly constructed by a convergent perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mechanics. The periodic solutions are defined only in a Cantor set, and resummation techniques of divergent powers series are used in order to control the small divisors problem.

Critical dynamics and effective exponents of magnets with extended impurities

We investigate the asymptotic and effective static and dynamic critical behavior of (d=3)-dimensional magnets with quenched extended defects, correlated in $\epsilon_d$ dimensions (which can be considered as the dimensionality of the defects) and randomly distributed in the remaining $d-\epsilon_d$ dimensions. The field-theoretical renormalization group perturbative expansions being evaluated naively do not allow for the reliable numerical data. We apply the Chisholm-Borel resummation technique to restore convergence of the two-loop expansions and report the numerical values of the asymptotic critical exponents for the model A dynamics. We discuss different scenarios for static and dynamic effective critical behavior and give values for corresponding non-universal exponents.

Extrapolation of lattice QCD results beyond the power-counting regime

Resummation of the chiral expansion is necessary to make accurate contact with current lattice simulation results of full QCD. Resummation techniques including relativistic formulations of chiral effective field theory and finite-range regularization (FRR) techniques are reviewed, with an emphasis on using lattice simulation results to constrain the parameters of the chiral expansion. We illustrate how the chiral extrapolation problem has been solved and use FRR techniques to identify the power-counting regime (PCR) of chiral perturbation theory. To fourth-order in the expansion at the 1% tolerance level, we find 0 \le m_pi \le 0.18 GeV for the PCR, extending only a small distance beyond the physical pion mass.

Global analysis of data on the proton structure functiong1and the extraction of its moments

Inspired by recent measurements with the CLAS detector at Jefferson Lab, we perform a self-consistent analysis of world data on the proton structure function ${g}_{1}$ in the range $0.17<{Q}^{2}<30\text{ }(\mathrm{G}\mathrm{e}\mathrm{V}/c{)}^{2}$. We compute for the first time low-order moments of ${g}_{1}$ and study their evolution from small to large values of ${Q}^{2}$. The analysis includes the latest data on both the unpolarized inclusive cross sections and the ratio $R={\ensuremath{\sigma}}_{L}/{\ensuremath{\sigma}}_{T}$ from Jefferson Lab, as well as a new model for the transverse asymmetry ${A}_{2}$ in the resonance region. The contributions of both leading and higher twists are extracted, taking into account effects from radiative corrections beyond the next-to-leading order by means of soft-gluon resummation techniques. The leading twist is determined with remarkably good accuracy and is compared with the predictions obtained using various polarized parton distribution sets available in the literature. The contribution of higher twists to the ${g}_{1}$ moments is found to be significantly larger than in the case of the unpolarized structure function ${F}_{2}$.

Periodic Solutions for Completely Resonant Nonlinear Wave Equations with Dirichlet Boundary Conditions

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=\phi(u)$, with $\phi(u)=\Phi u^{3} + O(u^{5})$ odd and analytic, $\Phi\neq0$, and we construct small amplitude periodic solutions with frequency $\o$ for a large Lebesgue measure set of $\o$ close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and nonresonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect the nonlinear wave equations $u_{xx}-u_{tt}+ M u = \phi(u)$, $M\neq0$, is that not only the P equation but also the Q equation is infinite-dimensional

Supercooled vortex liquid and quantitative theory of melting of the flux-line lattice in type-II superconductors

A metastable homogeneous state exists down to zero temperature in systems of repelling objects. Zero ''fluctuation temperature'' liquid state therefore serves as a (pseudo) ''fixed point'' controlling the properties of vortex liquid below and even around melting point. There exists Madelung constant for the liquid in the limit of zero temperature which is higher than that of the solid by an amount approximately equal to the latent heat of melting. This picture is supported by an exactly solvable large $N$ Ginzburg - Landau model in magnetic field. Based on this understanding we apply Borel - Pade resummation technique to develop a theory of the vortex liquid in type II superconductors. Applicability of the effective lowest Landau level model is discussed and corrections due to higher levels is calculated. Combined with previous quantitative description of the vortex solid the melting line is located. Magnetization, entropy and specific heat jumps along it are calculated. The magnetization of liquid is larger than that of solid by $% 1.8%$ irrespective of the melting temperature. We compare the result with experiments on high $T_{c}$ cuprates $YBa_{2}Cu_{3}O_{7}$, $DyBCO$, low $% T_{c}$ material $(K,Ba)BiO_{3}$ and with Monte Carlo simulations.

Two-pion bound state in the sigma channel at finite temperature

We study how we can understand the change of the spectral function and the pole location of the correlation function for sigma at finite temperature, which were previously obtained in the linear sigma model with a resummation technique called optimized perturbation theory. There are two relevant poles in the sigma channel. One pole is the original sigma pole which shows up as a broad peak at zero temperature and becomes lighter as the temperature increases. The behavior is understood from the decreasing of the sigma condensate, which is consistent with the Brown-Rho scaling. The other pole changes from a virtual state to a bound state of pion-pion as the temperature increases which causes the enhancement at the pion-pion threshold. The behavior is understood as the emergence of the pion-pion bound state due to the enhancement of the pion-pion attraction by the induced emission in medium. The latter pole, not the former, eventually degenerates with pion above the critical temperature of the chiral transition. This means that the observable "sigma" changes from the former to the latter pole, which can be interpreted as the level crossing of "sigma" and pion-pion at finite temperature.

The 1+1 SU(2) Yang–Mills path integral

The path integral for SU(2) invariant two-dimensional Yang–Mills theory is recast in terms of the chromoelectric field strength by integrating the gauge fields from the theory. Implementing Gauss's law as a constraint in this process induces a topological term in the action that is no longer invariant under large gauge transformations. For the case that the partition function is considered over a circular spatial degree of freedom, it is shown that the effective action of the path integral is quantum mechanically WKB exact and localizes onto a set of chromoelectric zero modes satisfying antiperiodic boundary conditions. Summing over the zero modes yields a partition function that can be reexpressed using the Poisson resummation technique, allowing an easy determination of the energy spectrum, which is found to be identical to that given by other approaches.

Surface critical behaviour of semi-infinite systems with cubic anisotropy at the ordinary transition

The critical behavior at the ordinary transition in semi-infinite n-component anisotropic cubic models is investigated by applying the field theoretic approach in d=3 dimensions up to the two-loop approximation. Numerical estimates of the resulting two-loop series expansions for the critical exponents of the ordinary transition are computed by means of Pade resummation techniques. For $n<n_{c}$ the system belongs to the universality class of the isotropic n-component model, while for $n>n_{c}$ the cubic fixed point becomes stable, where $n_{c}<3$ is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behavior of the semi-infinite systems with cubic anisotropy is characterized by a new set of surface critical exponents for $n>n_{c}$.

On the preparation of pure states in resonant microcavities

Abstract We consider the time evolution of the radiation field (R) and a two-level atom (A) in a resonant microcavity in terms of the Jaynes-Cummings model with an initial general pure quantum state for the radiation field. It is then shown, using the Cauchy-Schwarz inequality and also a Poisson resummation technique, that perfect coherence of the atom can in general never be achieved. The atom and the radiation field are, however, to a good approximation in a pure state ∥ψ〉A⊗R=∥ψ〉A×∥ψ〉R in the middle of what has been traditionally called the ‘collapse region’, independent of the initial state of the atoms, provided that the initial pure state of the radiation field has a photon number probability distribution which is sufficiently peaked and phase differences that do not vary significantly around this peak. An approximate analytic expression for the quantity Tr[ρ2 A(t)], where ρA (t) is the reduced density matrix for the atom, is derived. We also show that under quite general circumstances an initial entangled pure state will be disentangled to the pure state ∥ψ〉PSA⊗R.