Seiberg-Witten/Whitham Equations and Instanton Corrections in N=2 Supersymmetric Yang-Mills Theory

We obtain the instanton correction recursion relations for the low energy effective prepotential in pure N=2 SU(n) supersymmetric Yang-Mills gauge theory from Whitham hierarchy and Seiberg-Witten/Whitham equations. These formulae provide us a powerful tool to calculate arbitrary order instanton corrections coefficients from the perturbative contributions of the effective prepotential in Seiberg-Witten gauge theory. We apply this idea to evaluate one- and two- order instanton corrections coefficients explicitly in SU(n) case in detail through the dynamical scale parameter expressed in terms of Riemann's theta-function.     

Nonperturbative corrections to 4D string theory effective actions from SL(2,Z) duality and supersymmetry

We find the D(-1)- and D1-brane instanton contributions to the hypermultiplet moduli space of type IIB string compactifications on Calabi-Yau threefolds. These combine with known perturbative and world sheet instanton corrections into a single modular invariant function that determines the hypermultiplet low-energy effective action.     

Nonperturbative contribution to the quark form factor within the instanton model

The nonperturbative effects in the quark form factor are considered in the Wilson loop formalism, within the framework of the instanton liquid model. For the integration path corresponding to this form factor, the explicit expression for the vacuum expectation value of the Wilson operator is found to the leading order. It is shown that the instantons produce the power-like corrections to the perturbative result, which are comparable in magnitude with the perturbative part at the scale of order of the inverse average instanton size. It is demonstrated that the instanton contributions to the quark form factor are exponentiated to high orders in the small instanton density parameter.Received: 30 September 2002, Published online: 22 October 2003PACS: 12.38.-t Quantum chromodynamics - 11.15.Tk Other nonperturbative techniques     

On the short distance behavior of the critical Ising model perturbed by a magnetic field

We apply here a recently developed approach to compute the short distance corrections to scaling for the correlators of all primary operators of the critical two-dimensional Ising model in a magnetic field. The essence of the method is the fact that if one deals with OPE Wilson coefficients instead of correlators, all-order IR safe formulas can be obtained for the perturbative expansion with respect to the magnetic field. This approach yields in a natural way the expected fractional powers of the magnetic field, which are clearly absent in the naive perturbative expression for correlators. The technique of the Mellin transform has been used to compute the IR behavior of the regularized integrals. As a corollary of our results, by comparing the existing numerical data for the lattice model we give an estimate of the vacuum expectation value of the energy operator of the continuous theory.     

Multi-instantons and exact results IV: Path integral formalism

This is the fourth paper in a series devoted to the large-order properties of anharmonic oscillators. We attempt to draw a connection of anharmonic oscillators to field theory, by investigating the partition function in the path integral representation around both the Gaussian saddle point, which determines the perturbative expansion of the eigenvalues, as well as the nontrivial instanton saddle point. The value of the classical action at the saddle point is the instanton action which determines the large-order properties of perturbation theory by a dispersion relation. In order to treat the perturbations about the instanton, one has to take into account the continuous symmetries broken by the instanton solution because they lead to zero-modes of the fluctuation operator of the instanton configuration. The problem is solved by changing variables in the path integral, taking the instanton parameters as integration variables (collective coordinates). The functional determinant (Faddeev–Popov determinant) of the change of variables implies nontrivial modifications of the one-loop and higher-loop corrections about the instanton configuration. These are evaluated and compared to exact WKB calculations. A specific cancellation mechanism for the first perturbation about the instanton, which has been conjectured for the sextic oscillator based on a nonperturbative generalized Bohr–Sommerfeld quantization condition, is verified by an analytic Feynman diagram calculation.     

Wilson's operator expansion: Can it fail?

Within the framework of some simple models we discuss the status of the operator product expansion (OPE) in the presence of nonperturbative effects. We consider, in particular, the 4d Higgs model, 2d sigma model and the Schwinger model. The general formulation of OPE is presented and it is demonstrated that there exists a consistent procedure allowing one to define unambiguously both coefficient functions and matrix elements of composite operators. One of the key elements of the procedure is the introduction of an auxiliary parameter, the normalization point μ. For the simplest T-products discussed in the literature earlier we construct the corresponding OPE explicitly. Then we check its validity by comparing the results for the two-point functions with independent direct calculations of the same correlators. Although the general procedure is standard and does not vary from one theory to another, numerically the relative role of perturbative and nonperturbative contributions in vacuum condensates is different in different theories. The two extremes considered are the λ?⁴ theory with no spontaneous breaking of the symmetry and the O(N) sigma model in the limit N → ∞. In the former case there is only perturbative contribution to (?²), while in the latter case the perturbative pieces are suppressed by 1/N factors. Numerically QCD is much closer to the O(N) sigma model in the large-N limit. Comments on specific features of QCD are presented.     

Non-perturbative non-locality: V±A correlators in euclidean position space

Using an OPE modification, which takes non-perturbative non-locality into account, the difference and sum of vector and axial-vector correlators are evaluated in euclidean position space. For distances up to 0.8 fm the calculated behavior is close to the instanton liquid result and the ALEPH data, in contrast to the implications of the conventional OPE.     

Notes on Certain (0,2) Correlation Functions

In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of elements of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.     

Instanton interpolating current for σ-tetraquark

We perform a QCD sum rule analysis for the light scalar meson σ   (f₀(600)$f_0(600)$) with a tetraquark current related to the instanton picture for QCD vacuum. We demonstrate that instanton current, including equal weights of scalar and pseudoscalar diquark–antidiquarks, leads to a strong cancelation between the contributions of high dimension operators in the operator product expansion (OPE). Furthermore, in the case of this current direct instanton contributions do not spoil the sum rules. Our calculation, obtained from the OPE up to dimension 10 operators, gives the mass of σ-meson around 780 MeV.     

$VA\widetilde{V}$ correlator within the instanton vacuum model

The correlator of vector and non-singlet axial-vector currents in the external electromagnetic field is calculated within the instanton liquid model of the QCD vacuum. In general the correlator has two Lorentz structures: longitudinal, w_L, and transversal, w_T, with respect to the axial-vector index. Within the instanton model the saturation of the anomalous w_L structure is demonstrated. It is known that in the chiral limit the transversal structure w_T is free from perturbative corrections. In this limit within the instanton model we calculate the transversal invariant function w_T at arbitrary space-like momentum transfer q and show the absence of power corrections to this structure at large q². Instead there arise exponential corrections to w_T at large q² reflecting non-local properties of the QCD vacuum. The slope of w_T at zero virtuality, the QCD vacuum magnetic susceptibility of the quark condensate and its momentum dependence are estimated.Received: 7 May 2005, Published online: 6 July 2005     

Perturbative–nonperturbative interference in the static QCD interaction at small distances

Short distance static quark–antiquark interaction is studied systematically using the background perturbation theory with nonperturbative background described by field correlators. A universal linear term 6N_csσr/2π is observed at small distance r due to the interference between perturbative and nonperturbative contributions. Possible modifications of this term due to additional subleading terms are discussed and implications for systematic corrections to OPE are formulated.     

Gluonic Distribution in the Constituent Quark and Nucleon Induced by the Instantons

Abstract Instanton effects can give large contribution to strong interacting processes, especially at the energy scale where perturbative QCD is no longer valid. However instanton contribution to the gluon contribution in constituent quark and nucleon has never been calculated before. Based on both the constituent quark picture and the instanton model for QCD vacuum, we calculate the unpolarized and polarized gluon distributions in the constituent quark and in the nucleon for the first time. We find that the pion field plays an important role in producing both the unpolarized and the polarized gluon distributions.     

Estimating ground-state energies in supersymmetric quantum mechanics: (2) Unbroken case

In this second paper studying supersymmetric quantum mechanics, we concentrate on cases where supersymmetry is not spontaneously broken. This is done to further study the importance of various contributions to the ground-state energy. We perform an instanton calculation in detail and show that estimates of the ground-state energy based on the instanton contribution are poor-usually a naive zero-point energy calculation gives as good an estimate. We test the method developed in the previous paper and show that it gives good results for these cases, as before. We question the common assumptions that instanton effects are exponentially small compared to the perturbative scale, and that instanton contributions are necessarily suppressed by powers ofh.     

Magnetic-field dependence of the specific heat in planar spin chains

The specific heat of theXY model develops a characteristic double peak as a function of an applied magnetic field at relatively low temperatures. We argue that such a behavior is generic for planar spin chains, including chains with single-site anisotropy, and should be amenable to experimental observation. We also show that anharmonic corrections in the standard spin-wave theory exhibit certain infrared anomalies which limit the validity of perturbative calculations.     

Improved calculations of quark distributions in hadrons: the case of the pion

The earlier introduced method of calculation of quark distributions in hadrons, based on QCD sum rules, is improved. The imaginary part of the virtual photon forward scattering amplitude on some hadronic current is considered in the case, when initial and final virtualities of the current , and are different, . The operator product expansion (OPE) in , is performed. The sum rule for quark distribution is obtained using double dispersion representation of the amplitude on one side in terms of calculated in QCD OPE and on the other side in terms of physical states contributions. Double Borel transformation in , is applied to the sum rule, killing background non-diagonal transition terms which deteriorated the accuracy in previous calculations. The case of the valence quark distribution in the pion is considered, which was impossible to treat by the previous method. OPE up to dimension 6 operators is performed and leading order perturbative corrections are accounted. Valence u-quark distribution in was found at intermediate x, , and normalization point . These results may be used as input for evolution equations. Received: 14 July 1999 / Published online: 6 March 2000     

Duality between QCD perturbative series and power corrections

We elaborate on the relation between perturbative and power-like corrections to short-distance sensitive QCD observables. We confront theoretical expectations with explicit perturbative calculations existing in literature. As is expected, the quadratic correction is dual to a long perturbative series and one should use one of them but not both. However, this might be true only for very long perturbative series, with number of terms needed in most cases exceeding the number of terms available. What has not been foreseen, the quartic corrections might also be dual to the perturbative series. If confirmed, this would imply a crucial modification of the dogma. We confront this quadratic correction against existing phenomenology (QCD (spectral) sum rules scales, determinations of light quark masses and of α_s from τ-decay). We find no contradiction and (to some extent) better agreement with the data and with recent lattice calculations.     

On de Sitter vacua in type IIA orientifold compactifications

This Letter discusses the orientifold projection of the quantum corrections to type IIA strings compactified on rigid Calabi–Yau threefolds. It is shown that N=2$N=2$ membrane instanton effects give a holomorphic contribution to the superpotential, while the perturbative corrections enter into the Kähler potential. At the level of the scalar potential the corrections to the Kähler potential give rise to a positive energy contribution similar to adding anti-D3-branes in the KKLT scenario. This provides a natural mechanism to lift an AdS vacuum to a meta-stable dS vacuum.     

Aspects of UA (1) breaking in the Nambu and Jona-Lasinio model

The six-quark instanton induced ’t Hooft interaction, which breaks the unwanted U_A (1) symmetry of QCD, is a source of perturbative corrections to the leading order result formed by the four-quark forces with the U_L (3) × U_R (3) chiral symmetry. A detailed quantitative calculation is carried out to bosonize the model by the functional integral method. We concentrate our efforts on finding ways to integrate out the auxiliary bosonic variables. The functional integral over these variables cannot be evaluated exactly. We show that the modified stationary phase approach leads to a resummation within the perturbative series and calculate the integral in the “two-loop” approximation. The result is a correction to the effective mesonic Lagrangian which may be important for the low-energy spectrum and dynamics of the scalar and pseudoscalar nonets.     

A general scheme for computation of instanton effects

A general approach to the computation of instanton effects without invoking an ad hoc cut-off on their size is illustrated here by the computation of instanton contributions to the static $\text{Q}$Q potential and vacuum polarization. Our results suggest that the instanton effects are much smaller than perturbative effects in the region where both may be reliably computed.