Resulting Perturbation Series Research Articles

Diagrammatic approach to orbital quantum impurities interacting with a many-particle environment

Recently it was shown that an impurity exchanging orbital angular momentum with a surrounding bath can be described in terms of the angulon quasiparticle [Phys. Rev. Lett. 118, 095301 (2017)]. The angulon consists of a quantum rotor dressed by a many-particle field of boson excitations, and can be formed out of, for example, a molecule or a nonspherical atom in superfluid helium, or out of an electron coupled to lattice phonons or a Bose condensate. Here we develop an approach to the angulon based on the path-integral formalism, which sets the ground for a systematic, perturbative treatment of the angulon problem. The resulting perturbation series can be interpreted in terms of Feynman diagrams, from which, in turn, one can derive a set of diagrammatic rules. These rules extend the machinery of the graphical theory of angular momentum - well known from theoretical atomic spectroscopy - to the case where an environment with an infinite number of degrees of freedom is present. In particular, we show that each diagram can be interpreted as a 'skeleton', which enforces angular momentum conservation, dressed by an additional many-body contribution. This connection between the angulon theory and the graphical theory of angular momentum is particularly important as it allows to systematically and substantially simplify the analytical representation of each diagram. In order to exemplify the technique, we calculate the 1- and 2-loop contributions to the angulon self-energy, the spectral function, and the quasiparticle weight. The diagrammatic theory we develop paves the way to investigate next-to-leading order quantities in a more compact way compared to the variational approaches.

Perturbative QCD in acceptable schemes with holomorphic coupling

Perturbative QCD in mass independent schemes leads in general to running coupling a(Q2) which is nonanalytic (nonholomorphic) in the regime of low spacelike momenta |Q2|≲1 GeV 2. Such (Landau) singularities are inconvenient in the following sense: evaluations of spacelike physical quantities 𝒟(Q2) with such a running coupling a(κQ2)(κ~1) give us expressions with the same kind of singularities, while the general principles of local quantum field theory require that the mentioned physical quantities have no such singularities. In a previous work, certain classes of perturbative mass independent beta functions were found such that the resulting coupling was holomorphic. However, the resulting perturbation series showed explosive increase of coefficients already at N 4 LO order, as a consequence of the requirement that the theory reproduce the correct value of the τ lepton semihadronic strangeless decay ratio rτ. In this paper, we successfully extend the construction to specific classes of perturbative beta functions such that the perturbation series do not show explosive increase of coefficients, the perturbative coupling is holomorphic, and the correct value of rτ is reproduced. In addition, we extract, with Borel sum rule analysis of the V+A channel of the semihadronic strangeless decays of τ lepton, reasonable values of the corresponding D = 4 and D = 6 condensates.

A complete basis for a perturbation expansion of the general N-body problem

We discuss a basis set developed to calculate perturbation coefficients in an expansion of the general N-body problem. This basis has two advantages. First, the basis is complete order-by-order for the perturbation series. Second, the number of independent basis tensors spanning the space for a given order does not scale with N, the number of particles, despite the generality of the problem. At first order, the number of basis tensors is 25 for all N, i.e. the problem scales as N0, although one would initially expect an N6 scaling at first order. The perturbation series is expanded in inverse powers of the spatial dimension. This results in a maximally symmetric configuration at lowest order which has a point group isomorphic with the symmetric group, SN. The resulting perturbation series is order-by-order invariant under the N! operations of the SN point group which is responsible for the slower than exponential growth of the basis. In this paper, we demonstrate the completeness of the basis and perform the first test of this formalism through first order by comparing to an exactly solvable fully interacting problem of N particles with a two-body harmonic interaction potential.

Rovibrational Energies of Triatomic Molecules by Means of the Rayleigh–Schrödinger Perturbation Theory

A Rayleigh–Schrödinger perturbation theory approach based on the adiabatic (Born–Oppenheimer) separation of vibrational motions was previously developed and used to evaluate for a system of coupled oscillators the adiabatic energy levels and their nonadiabatic corrections. This method is applied here to calculate rotation–vibration energies of the triatomic molecular ions HeH+2 and ArNO+ consisting of a strongly bound diatomic fragment and a relatively loosely bound rare gas atom. In these systems the high-frequency stretching motion of the diatomic fragment can be separated from the other two low-frequency motions without substantial loss of accuracy. Treating the diatomic fragment as a rigid rotor, the low-frequency stretching motion is decoupled from the bending motion in analogy to the concept of the adiabatic (Born–Oppenheimer) separation of motions and the strong nonadiabatic couplings between these two motions are accounted for perturbationally. Although the resulting perturbation series may show poor convergence, they turn out to be accurately summable by applying standard techniques for the summation of divergent series. Comparison with the results obtained from full-dimensional calculations for the two ions shows that the approach is capable of providing accurate energies for quite a few of the bound rotation–vibration states and that in the case of the HeH+2 ion it is even able to predict the positions and widths of some low-lying resonance states with good accuracy. The perturbation approach yields zeroth-order energies and corrections in terms of the relevant quantum numbers. It thus allows a direct assignment of the energy levels without any reference to the corresponding eigenfunctions. The weak couplings between the high- and low-frequency motions can easily be treated by the same perturbative approach and numerically exact energies can finally be obtained.

Analysis of the ε-Expansion in Turbulence Theory: Approximate Renormalization Group for Diffusion of a Passive Scalar in a Random Velocity Field

The validity of the e-expansion in the turbulence problem is discussed using the example of diffusion of a passive scalar in a random velocity field. A generalization of Wilson's rule for calculating a diagram of arbitrary order is introduced. The resulting perturbation series, while having zero radius of convergence, is summed exactly yielding amplitudes which differ from those obtained to lowest order in the e-expansion by only a few percent. The properties of the expansion are analyzed in detail and it is shown that various subsets of diagrams, while differing by an infinite number of terms, give close results in the vicinity of the fixed point. This indicates nontrivial compensation of high-order interactions in turbulence. The irrelevance of high-order couplings is demonstrated for arbitrary values of e. The quality the approximation is tested by comparison with numerical experiments on diffusion of a passive scalar in a band-limited random velocity field in the limit of infinite Peclet number.

Dimensional perturbation theory for Regge poles

We apply dimensional perturbation theory to the calculation of Regge pole positions, providing a systematic improvement to earlier analytic first-order results. We consider the orbital angular momentum l as a function of spatial dimension D for a given energy E, and expand l in inverse powers of κ≡(D−1)/2. It is demonstrated for both bound and resonance states that the resulting perturbation series often converges quite rapidly, so that accurate quantum results can be obtained via simple analytic expressions given here through third order. For the quartic oscillator potential, the rapid convergence of the present l(D;E) series is in marked contrast with the divergence of the more traditional E(D;l) dimensional perturbation series, thus offering an attractive alternative for bound state problems.

Nonperturbative perturbation theory

In this talk we describe a recently proposed graphical perturbative calculational scheme for quantum field theory. The basic idea is to expand in the power of the interaction term. For example, to solve a λφ 4 theory in d-dimensional space-time, we introduce a small parameter δ and consider a λ( φ 2) 1+ δ field theory. We show how to expand such a theory as a series in powers of δ. The resulting perturbation series appears to have a finite radius of convergence and numerical results for low-dimensional models are good. We have computed the two-point and four-point Green's functions to second order in powers of δ and the 2n-point Green's functions (n > 2) to order δ. We explain how to renormalize the theory and show that, to first order in powers of δ, when δ > 0 and d ⪰ 4 the theory is free. This conclusion remains valid to second order in powers of δ, and we believe that it remains valid to all orders in powers of δ. The new perturbative scheme is consistent with global supersymmetry invariance. We examine a two-dimensional supersymmetric quantum field theory in which we do not know of any other means for doing analytical calculations. We illustrate the power of this new technique by computing the ground-state energy density E to second order in this new perturbation theory. We show that there is a beautiful and delicate cancellation between infinite classes of graphs which leads to the result that E = 0.

Logarithmic approximations to polynomial Lagrangians.

A new perturbative computational scheme for solving quantum field theory is proposed. The interaction term in the Lagrangean is expanded about a free-theory form, the expansion involving powers of logarithms of the fields. The resulting perturbation series appears to have a finite radius of convergence and numerical results for simple models are good.

Symmetry forcing and convergence properties of perturbation expansions for molecular interaction energies

AbstractThe convergence properties of perturbation theories for molecular interaction energies are tested by performing high‐accuracy high‐order numerical calculations for a ground‐state hydrogen atom interacting with a proton. It is shown that a strong symmetry forcing used in the Eisenschitz‐London‐Hirschfelder‐van der Avoird (EL‐HAV) theory leads to rapidly convergent perturbation expansion whereas a weak symmetry forcing, peculiar to the Murrell‐Shaw‐Musher‐Amos (MSMA) theory, is not able to guarantee the convergence of the resulting perturbation series. The perturbation expansion introduced recently by Jeziorski and Kolos and corresponding to an intermediate symmetry forcing is shown to converge rapidly ensuring the correct asymptotic behavior of the interaction energy calculated through second order. Despite the divergence of the resulting perturbation series the MSMA theory is shown to give very useful results at the distances corresponding to the van der Waals minimum. In this region, however, virtually the same results can be obtained by using a simpler theory employing a properly symmetrized wave function of the usual Rayleigh‐Schrödinger (RS) polarization theory.

Algebraic approximation in many-body perturbation theory

Many-body perturbation theory is developed within the algebraic approximation, i.e., parametrization of state functions by expansion in a finite basis set. This considerably extends the applicability of the theory to molecules, allowing all two-, three-, and four-body contributions to the energy to be evaluated through third order. Within this context, a comparison is presented between perturbation calculations and previously reported configuration-interaction calculations which employed the same basis sets. The [2/1] Pad\'e approximants to the energy are constructed and upper bounds to the energy expectation values are determined. Two zeroth-order Hamiltonians are used, and the convergence of the resulting perturbation series is compared. Both of these perturbation expansions yield Pad\'e approximants to the energy which are within 0.9% of the corresponding configuration-interaction results. The variation-perturbation upper bounds for the energy are all within 3.2% of the corresponding configuration-interaction bounds. Considering this agreement and the tractability of the diagrammatic perturbative scheme, it appears that the perturbation expansion is highly competitive.