Instanton Calculations Research Articles

Quantum corrections to instanton calculations in the Abelian Higgs model

In the (1+1)-dimensional Abelian Higgs model we calculate the final state quantum corrections to the fermion number violating cross-section in the background of the instanton. The arising determinants are treated in the Schwinger proper time formalism using some refined approximation techniques which are controlled in several ways and are also applied to the well known case of the quantum double well for comparison.

Breaking cosmic strings without monopoles.

It is shown that topologically stable cosmic strings can, in fact, appear to end or to break, even in theories without monopoles. This can occur whenever the spatial topology of the universe is nontrivial. For the case of Abelian-Higgs strings, we describe the gauge and scalar field configurations necessary for a string to end on a black hole. We give a lower bound for the rate at which a cosmic string will break via black hole pair production, using an instanton calculation based on the Euclidean C-metric.

Instanton calculations versus exact results in four dimensional SUSY gauge theories

We relate the non-perturbative exact results in supersymmetry to perturbation theory using several different methods: instanton calculations at weak or strong coupling, a method using gaugino condensation and another method relating strong and weak coupling. This allows many precise numerical checks of the consistency of these methods, especially the amplitude of instanton effects, and of the network of exact solutions in supersymmetry. However, there remain difficulties with the instanton computations at strong coupling.

Instanton-anti-instanton pair-induced asymptotics of perturbation theory in QCD.

The instanton--antiinstanton pair induced asymptotics of perturbation theory expansion for QCD correlators is considered. It is argued that though the true asymptotics is dominated by renormalon, the instanton-induced contribution may dominate in the intermediate asymptotics $n= 5\div 15$. Obtained asymptotic formulae are valid for $N_f \le N_c$. For $N_f =N_c$ the finite nonperturbative expression for instanton--antiinstanton contribution was also found. At $q^2<0$ the imaginary part of correlators in the case $N_f =N_c$ is suppressed like $1/\log (q/\Lambda)$, but the present accuracy of instanton calculations allows to fix it unambiguously. The series of corrections to the instanton induced asymptotics of the order of $\sim (\log(n)/n)^k$ is found.

A new approach to instanton calculations in the O(3) nonlinear σ-model

We construct all instantons in the 0(3) σ -model on a cylindrical space-time, known not to exist on a finite time interval. The scale parameter, ρ , is related to the boundary condition in time. This may cure the ρ → 0 divergent instanton gas, through a proper inclusion of in and out states in the path integral.

Electroweak instantons at non-zero Weinberg angle

Previous calculations of instanton effects in electroweak theory have concentrated on the case of zero Weinberg angle θ w , where the U(1) hypercharge field decouples. In this paper we extend the instanton calculation to non-zero θ w , by constructing a perturbation expansion. This allows for the first time the study of photon production at B and L number violating vertices. We find that the orientation of the instanton solution in isospin space has to be carefully considered to avoid unphysical results.

Comments on information loss and remnants.

The information loss and remnant proposals for resolving the black hole information paradox are reconsidered. It is argued that in typical cases information loss implies energy loss, and thus can be thought of in terms of coupling to a spectrum of ``fictitious'' remnants. This suggests proposals for information loss that do not imply Planckian energy fluctuations in the low energy world. However, if consistency of gravity prevents energy nonconservation, these remnants must then be consiered to be real. In either case, the catastrophe corresponding to infinite pair production remains a potential problem. Using Reissner-Nordstr\"om black holes as a paradigm for a theory of remnants, it is argued that couplings in such a theory may give finite production despite an infinite spectrum. Evidence for this is found in analyzing the instanton for Schwinger production; fluctuations from the infinite number of states lead to a divergent stress tensor, spoiling the instanton calculation. Therefore naive arguments for infinite production fail.

Instantons and surface tension at a first-order transition

We study the dynamics of the first-order phase transition in the two-dimensional 15-state Potts model, both at and off equilibrium. We find that phase changes take place through nucleation in both cases, and finite volume effects are described well through an instanton computation. Thus a dynamical measurement of the surface tension is possible. We find that the order-disorder surface tension is compatible with perfect wetting. An accurate treatment of fluctuations about the instanton solution is seen to be of great importance.

Measure of strong CP violation.

We investigate a controversial issue on the measure of CP violation in strong in teractions. In the presence of nontrivial topological gauge configurations, the $\theta$-term in QCD has a profound effect: it breaks the CP symmetry. The CP-violating amplitude is shown to be determined by the vacuum tunneling process, where the semiclassical method makes most sense. We discuss a long-standing dispute on whether the instanton dynamics satisfies or not the anomalous Ward identity (AWI). The strong CP violation measure, when complying with the vacuum alignment, is proportional to the topological susceptibility. We obtain an effective CP-violating lagrangian different from that provided by Baluni. To solve the IR divergence problem of the instanton computation, We present a ``classically gauged'' Georgi-Manohar model and derive an effective potential which uniquely determines an explicit $U(1)_A$ symmetry breaking sector. The CP violation effects are analyzed in this model. It is shown that the strong CP problem and the $U(1)$ problem are closely related. Some possible solutions to both problems are also discussed with new insights.

Study of the reliability of the semiclassical instanton contribution to the inclusive cross section

The reliability of the semiclassical instanton calculation of the inclusive cross section is studied. This is done in the two-dimensional linear sigma model, where the contribution of constrained instantons to the semiclassical cross section is compared with an exact unitarity bound. The conclusion is that, at energies in which the semiclassically calculated cross section becomes unsuppressed (at least in some range of parameters), the instanton calculation cannot be trusted.

A proof of exponential suppression of high-energy transitions in the anharmonic oscillator

We derive rigorous bounds on the matrix elements of the position operator between a low-lying and an arbitrary excited state of the quartic anharmonic oscillator. For both the single-and the double-well potentials, these bounds decrease monotonically and exponentially fast with the energy difference between these states, measured in units of the perturbative frequency. The order of the exponential decay changes smoothly from 1 to 3 4 at some non-perturbative large energy scale. Our bounds prove that transitions, induced by an external infinitesimal but rapidly-oscillating force, never become strong contrary to the predictions of the tree-level and leading-order instanton calculations. We explain why these induced transitions are the quantum-mechanical analog of two-particle collision processes with large multiplicity in the final state.

Translational Invariance Breaking in QCD by Instantons

It is noted that existing instanton calculations in QCD are incompatible with translational invariance

Translational Invariance Breaking in QCD by Instantons

It is noted that existing instanton calculations in QCD are incompatible with translational invariance.

An illustrative model for non-perturbative scattering in theories with weak coupling

A simple two-dimensional model with one scalar field and a slightly metastable vacuum is used as a test ground to analyze non-perturbative multiparticle scattering at high energy induced by a non-trivial saddle point configuration in the euclidean path integral analogous to the instanton-induced multiparticle processes in the electroweak theory. Existence of an additional small parameter in the theory makes it possible to find the behavior of the scattering amplitudes up to the analog of the sphaleron energy. In particular it is illustrated that large quantum corrections arising in the existing instanton calculations of such amplitudes develop because the quantization is performed near the stationary configuration which is not distorted by the sources of external particles. It is explicitly shown that once the proper saddle point configuration in the presence of external sources is found the quantization around it yields only small corrections. Also the explicit solution shows what happens to some known first quantum effects when they are incorporated in the non-perturbative result.

Constraint dependence of the instanton calculations and exponentiation of hard-soft corrections at high energies

We study the dependence of the final state exponential factor for the total instanton induced cross section on the constraints used to separate the instanton zero modes. We find the non-trivial dependence of the semiclassical exponent on the translational zero mode constraint, starting in the order (E E 0) 8 3 . We argue that, to compensate this dependence in the physical cross section, the hard-soft corrections should enter the exponent at this order. We identify the terms in the hard-soft gauge propagator which may give rise to such behaviour.

QCD sum rules at low temperature.

We discuss finite-temperature QCD sum rules at low temperatures, including perturbative as well as nonperturbative thermal corrections. The electric and magnetic condensates of the thermalized QCD state are extracted from present lattice calculations of the energy density and pressure. Alternative parametrizations based on instanton calculations and string behavior at finite temperature are also discussed. Throughout, a particular emphasis is put on the nature and size of the temperature corrections. In the ${1}^{\ensuremath{-}}$ channel the temperature behavior of the $\ensuremath{\rho}$ parameters are investigated. In the range of applicability of the sum-rule procedure, the $\ensuremath{\rho}$ parameters are found to vary slowly with temperature.

Large order behavior of non-perturbative gravity

We study the large order behavior and Borel summability of the topological expansion of models of 2D gravity coupled to matter. We treat in particular the cases of pure gravity, and gravity coupled to Yang-Lee, Ising, and tricritical Ising matter. We then discuss arbitrary one-matrix models and gravity coupled to general ( p, q) conformal matter, in particular showing that the genus- k contribution to their partition functions goes asymptotically for large k as Γ (2k− 1 2 ) . Finally, we show that the large order behavior analysis for the one-matrix models coincides with an instanton calculation of barrier penetration effects.

Baryon-number violation at the Superconducting Super Collider? Recent claims reexamined.

Recent instanton calculations suggest that baryon-number violation is unsuppressed at energies near 10 TeV. We show how to diagrammatically analyze corrections to the basic approximations. We find additional multiplicative contributions of $\mathrm{exp}(\ifmmode\pm\else\textpm\fi{}\frac{\mathrm{const}}{{\ensuremath{\alpha}}_{W}})$ to the cross section at such energies which may keep the rate from becoming large. In addition, the ${p}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{p}_{j}$ structure of these corrections is problematic for recent techniques suggested for improving the instanton calculation and controlling corrections at high energy. Despite these corrections, the initial exponential growth of the cross section first obtained by Ringwald survives.

Big wormholes and little interactions

In an earlier paper we suggested a mechanism by which a large density of wormholes at a small distance scale could prevent the formation of any wormholes at any larger scale. There is an apparent paradox here, in that the total effect of small wormholes should be equivalent to that of a high-dimension interaction, which although large at the small scale should be very small at a very large scale; it is difficult to see how such a small interaction could have a large effect. In this paper, we work out what happens in some detail, including a comparison (in a model theory) of an instanton computation and hamiltonian one. We find that not only is there no paradox, small wormholes are even more efficient at eliminating large wormholes than our earlier estimates had indicated; even a small density at small distance scale is enough to do the job.

Instantons and the spectrum of Bloch electrons in a magnetic field.

A semiclassical path-integral treatment of the spectrum of a charged particle moving in a two-dimensional periodic potential in the presence of a transverse magnetic field is presented. Complex instanton solutions to the Euclidean equations of motion are identified, and it is shown that the dilute-instanton-gas sum reproduces features of the spectrum studied by previous authors. Comparison of the path-integral results to a perturbative calculation of the splitting of the lowest Landau level is used to relate the self-similar nature of the spectrum to a type of duality in the parameter \ensuremath{\alpha}, the number of flux quanta per unit cell. As a byproduct of the instanton calculation the generating functional for a sum over lattice paths of given length and algebraic area modulo a fundamental area determined by the magnetic field is evaluated.