Large-order Perturbation Theory Research Articles

The breakdown of resummed perturbation theory at high energies

Calculations of high-energy processes involving the production of a large number of particles in weakly-coupled quantum field theories have previously signaled the need for novel non-perturbative behavior or even new physical phenomena. In some scenarios, already tree-level computations may enter the regime of large-order perturbation theory and therefore require a careful investigation. We demonstrate that in scalar quantum field theories with a unique global minimum, where suitably resummed perturbative expansions are expected to capture all relevant physical effects, perturbation theory may still suffer from severe shortcomings in the high-energy regime. As an example, we consider the computation of multiparticle threshold amplitudes of the form 1 → n in φ6 theory with a positive mass term, and show that they may violate unitarity of the quantum theory for large n, even after the resummation of all leading-n quantum corrections. We further argue that this is a generic feature of scalar field theories with higher-order self-interactions beyond φ4, thereby rendering the latter unique with respect to its high-energy behavior.

Two-dimensional hydrogen-like atom in a uniform magnetic field: Large-order perturbation theory

The well-known hypervirial perturbation method (HPM), based on hypervirial relations and the Hellmann-Feynman theorem, is suitable for the calculation of perturbation corrections of large order for the non-relativistic, spinless, two-dimensional hydrogen-like atom in a uniform magnetic field. We show analytical results in terms of the quantum numbers and large-order corrections for particular states. This approach appears to be simpler and more efficient than the recently proposed one based on the Green’s functions for this particular model.

Unified Treatment of Even and Odd Anharmonic Oscillators of Arbitrary Degree

We present a unified treatment, including higher-order corrections, of anharmonic oscillators of arbitrary even and odd degree. Our approach is based on a dispersion relation which takes advantage of the PT symmetry of odd potentials for imaginary coupling parameter, and of generalized quantization conditions which take into account instanton contributions. We find a number of explicit new results, including the general behavior of large-order perturbation theory for arbitrary levels of odd anharmonic oscillators, and subleading corrections to the decay width of excited states for odd potentials, which are numerically significant.

Two-loop Euler–Heisenberg QED pair-production rate

We study the divergence of large-order perturbation theory in the worldline expression for the two-loop Euler–Heisenberg QED effective Lagrangian in a constant magnetic field. The leading rate of divergence is identical, up to an overall factor, to that of the one-loop case. From this we deduce, using Borel summation techniques, that the leading behaviour of the imaginary part of the two-loop effective Lagrangian for a constant E field, giving the pair-production rate, is proportional to the one-loop result. This also serves as a test of the mass renormalization, and confirms the earlier analysis by Ritus.

Large-order perturbation theory for a non-Hermitian 𝓟𝓣-symmetric Hamiltonian

A precise calculation of the ground-state energy of the complex 𝒫𝒯-symmetric Hamiltonian H=p2+14x2+iλx3, is performed using high-order Rayleigh–Schrödinger perturbation theory. The energy spectrum of this Hamiltonian has recently been shown to be real using numerical methods. Here we present convincing numerical evidence that the Rayleigh–Schrödinger perturbation series is Borel summable, and show that Padé summation provides excellent agreement with the real energy spectrum. Padé analysis provides strong numerical evidence that the once-subtracted ground-state energy considered as a function of λ2 is a Stieltjes function. The analyticity properties of this Stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable Hamiltonian H=p2+14 x2−εx3.

On the use of algebraic approximants to sum divergent series for Fermi resonances in vibrational spectroscopy

Padé summation of large-order perturbation theory can often yield highly accurate energy eigenvalues for molecular vibrations. However, for eigenstates involved in Fermi resonances the convergence of the Padé approximants can be very slow. This is because the energy is a multivalued function of the perturbation parameter while Padé approximants are single valued, and Fermi resonances occur when a branch point lies close to the physical value of the parameter. Algebraic approximants are multivalued generalizations of Padé approximants. Using the (200) state of H2S and the (400) state of H2O as examples of Fermi resonances, it is demonstrated here that algebraic approximants greatly improve the summation convergence.

A New Approach on Wave Function and Energy Level Calculation Through Perturbation Theory

A new approach to non-relativistic large order perturbation theory for constructing energy and wave function correction up to K th order has been developed. Convergent argument of this series has been demonstrated for the ground state of second order and fourth order Schroedinger equation. Further a comparison with the Bender and Wu computational work reflects that the present series is not the same as Rayleigh-Schroedinger perturbation series.

Large-order dimensional perturbation theory for diatomic molecules within the Born-Oppenheimer approximation

A renormalization of the $D$-dimensional Hamiltonian is developed to ensure that the large-$D$ limit corresponds to a single well at any value of the internuclear distance $R.$ This avoids convergence problems caused by a symmetry-breaking transition that is otherwise expected to occur when $R$ is approximately equal to the equilibrium bond distance ${R}_{\mathrm{eq}},$ with larger $R$ giving a double well. This symmetry breaking has restricted the applicability of large-order perturbation theory in $1/D$ to cases where $R$ is significantly less than ${R}_{\mathrm{eq}}.$ The renormalization greatly extends the range of $R$ for which the large-order expansion can be summed. A numerical demonstration is presented for H${}_{2}^{+}.$ The $1/D$ expansions are summed using Pad\'e-Borel approximants with modifications that explicitly model known singularity structure.

Self-consistent field perturbation theory of molecular vibrations

Perturbation theory is used to perform non-iterative calculations of energy eigenvalues of the coupled ordinary differential equations that result from imposing separability assumptions in terms of normal coordinates on vibrational wavefunctions. Various model Hamiltonians with 2 or 3 coupled normal modes are studied and the increase of computational cost with the number of degrees of freedom is analysed. Quadratic Padé approximants of the perturbation expansions are rapidly convergent, and directly yield complex numbers for resonance eigenvalues. For a 3-mode system, results are obtained within partial separability assumptions, with a pair of modes left coupled. Large-order perturbation theory with partial separability is suggested as an alternative to low-order exact perturbation theory.

Dimensional perturbation theory for vibration–rotation spectra of linear triatomic molecules

A very efficient large-order perturbation theory is formulated for the nuclear motion of a linear triatomic molecule. All coupling between vibration and rotation is included. To demonstrate the method, all of the experimentally observed rotational energies, with values of J almost up to 100, for the ground and first excited vibrational states of CO2 and for the ground vibrational states of N2O and of OCS are calculated. The perturbation expansions reported here are rapidly convergent. The perturbation parameter is D−1/2, where D is the dimensionality of space. Increasing D is qualitatively similar to increasing the angular momentum quantum number J. Therefore, this approach is especially suited for states with high rotational excitation. The computational cost of the method scales only in proportion to JNv5/3, where Nv is the size of the vibrational basis set.

Self-consistent-field perturbation theory for the Schrödinger equation

A method is developed for using large-order perturbation theory to solve the systems of coupled differential equations that result from the variational solution of the Schr\"odinger equation with wave functions of product form. This is a noniterative, computationally efficient way to solve self-consistent-field (SCF) equations. Possible applications include electronic structure calculations using products of functions of collective coordinates that include electron correlation, vibrational SCF calculations for coupled anharmonic oscillators with selective coupling of normal modes, and ab initio calculations of molecular vibration spectra without the Born-Oppenheimer approximation.

Dimensional perturbation theory: an efficient method for computing vibration-rotation spectra

Large-order perturbation theory in D −1 2 , where D is the dimensionality of space, is shown to be an accurate and computationally efficient method for computing molecular vibration-rotation spectra. Results are presented for the ground vibrational state of CO 2 with J as high as 80. The rate of convergence of the perturbation expansion increases with J and the computational cost scales on JN v 5 3 , where N v is the number of vibrational basis functions. Therefore, this approach seems to be especially well suited for treating highly excited rotational states.

Perturbation theory for coupled anharmonic oscillators

Perturbation theory is applied to a pair of coupled oscillators with cubic anharmonicity. Large-order perturbation theory is shown to be more efficient computationally than numerical diagonalization of the Hamiltonian. Quadratic Padé summation of the energy expansions yields convergent results for the real and the imaginary parts of resonance eigenvalues.

Instanton-anti-instanton interaction and asymptotics of the perturbation theory expansion for the double well oscillator

An instanton-anti-instanton pair is considered as a source of singularity in the Borel transform for the ground state energy of the anharmonic oscillator. The problem of defining the short range instanton-anti-instanton interaction reduces to a calculation of the smooth part of the Borel function, which on the other hand is determined by the first few terms of ordinary perturbation theory. The preasymptotics (∼ 1 n ) of the large order perturbation theory contribution to the ground state energy of the anharmonic oscillator is found analytically. To this end the subleading correction to the long range classical instanton-anti-instanton interaction, the one-loop quantum contribution to instanton-anti-instanton interaction, and the second quantum correction to a single instanton density have been taken into account.

Dimensional perturbation theory for weakly bound systems

The standard dimensional continuation of the Schr\"odinger equation, in which the kinetic energy is generalized to arbitrary D while the potential energy remains unchanged, leads to dissociation in the limit D\ensuremath{\rightarrow}\ensuremath{\infty} for systems that are only weakly bound at D=3. Using the bound states of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ as an example, we show that accurate energies can be obtained from large-order perturbation theory in 1/D if the repulsive part of the potential is attenuated with a D-dependent factor that prevents the large-D dissociation.

Two-electron atoms: o(4,2) operator replacements and large-order perturbation theory with respect to the replaced kinetic-energy operator.

The Schr\odinger equation for two-electron atoms is transformed into an alternative equation depending on a free dimensionless parameter \ensuremath{\beta}, according to the method of o(4,2) operator replacements. These operator replacements are well defined except for the case where both the orbital and spin momenta are zero, i.e., except for singlet S states. When \ensuremath{\beta} goes to positive infinity, the solutions of this present equation should correspond to the solutions of the Schr\odinger equation. When \ensuremath{\beta} goes to negative infinity, the present equation becomes exactly solvable. These exact solutions are the zero-order solutions for a Rayleigh-Schr\odinger perturbative expansion where the perturbation is the nondiagonal part of the replaced kinetic-energy operator. An essential property of this method is that the perturbative operator is bounded, and therefore the convergence radius of the series is nonzero. The method is purely nonvariational. A numerical application for the triplet S even-parity ground-state helium atom is performed.

Large-order perturbation theory for a physical quantity

We discuss renormalization scheme and normalization scale dependence of the large-order asymptotics of perturbative expansions due to renormalons based on an exact calculation of the Borel transform of the imaginary part of the electromagnetic current correlation function within an expansion in the number of fermions in quantum electrodynamics. Arguments are given that the leading uncertainty of perturbative series in quantum chromodynamics due to ultraviolet renormalons can be suppressed by an appropriate choice of normalization scale. Combining momentum analyticity with the Borel transformed renormalization-group equation, we relate the large-order behaviour of total cross sections to that of their polarization operator.

Renormalization scheme-invariant large-order perturbation theory and infrared renormalons in QCD

We discuss the renormalization scheme dependence of large-order estimates for perturbative series of physical quantities and obtain the renormalization scheme-invariant asymptotics of its effective β-function. As an application we consider infrared renormalons in QCD. For the first infrared renormalon of the electromagnetic current-current correlation function we find the 1 n -corrections to the leading large- n asymptotics.

Determination of the D=2 “operator” from e+t> −data

We present an estimate of the dimension D=2 “operator” induced by large order perturbation theory using two different sum rules for the e + e − into I=1 hadrons total cross-section. We conclude that this “operator” can only affect the existing ϱ-meson spectral sum rules results and the τ-decay semi-inclusive process by an amount respectively less than 12% and 3.6%. This effect is consistent with the estimated perturbative error used previously in the determination of α s from τ-decay.

Large-order perturbation theory for the electromagnetic current-current correlation function.

The constraints imposed by asymptotic freedom and analyticity on the large-order behavior of perturbation theory for the electromagnetic current-current correlation function are examined. By suitably applying the renormalization group, the coefficients of the asymptotic expansion in the deep Euclidean region may be expressed explicitly in terms of the perturbative coefficients of the Minkowski space discontinuity (the $R$-ratio in $e^+ e^-$ scattering). This relation yields a ``generic'' prediction for the large-order behavior of the Euclidean perturbation series and suggests the presence of non-perturbative $1/q^2$ correction in the Euclidian correlation function. No such ``generic'' prediction can be made for the physically measurable $R$-ratio. A novel functional method is developed to obtain these results.