Exponential Corrections Research Articles

One-point functions in finite volume/temperature: a case study

We consider finite volume (or equivalently, finite temperature) expectation values of local operators in integrable quantum field theories using a combination of numerical and analytical approaches. It is shown that the truncated conformal space approach, when supplemented with a recently proposed renormalization group, can be sufficiently extended to the low-energy regime that it can be matched with high precision by the low-temperature expansion proposed by Leclair and Mussardo. Besides verifying the consistency of the two descriptions, their combination leads to an evaluation of expectation values which is valid to a very high precision for all volume/temperature scales. As a side result of the investigation, we also discuss some unexpected singularities in the framework recently proposed by Pozsgay and Tak\'acs for the description of matrix elements of local operators in finite volume, and show that while some of these singularities are resolved by the inclusion of the class of exponential finite size corrections known as \mu-terms, these latter corrections themselves lead to the appearance of new singularities. We point out that a fully consistent description of finite volume matrix elements is expected to be free of singularities, and therefore a more complete and systematic understanding of exponential finite size corrections is necessary.

Quantum Loop Expansion to High Orders, Extended Borel Summation, and Comparison with Exact Results

We compare predictions of the quantum loop expansion to (essentially) infinite orders with (essentially) exact results in a simple quantum mechanical model. We find that there are exponentially small corrections to the loop expansion, which cannot be explained by any obvious "instanton"-type corrections. It is not the mathematical occurrence of exponential corrections but their seeming lack of any physical origin which we find surprising and puzzling.

Three-particle scattering amplitudes from a finite volume formalism

We present a quantization condition for the spectrum of a system composed of three identical bosons in a finite volume with periodic boundary conditions. This condition gives a relation between the finite volume spectrum and infinite volume scattering amplitudes. The quantization condition presented is an integral equation that in general must be solved numerically. However, for systems with an attractive two-body force that supports a two-body bound state, a diboson, and for energies below the diboson breakup, the quantization condition reduces to the well-known L\"uscher formula with exponential corrections in volume that scale with the diboson binding momentum. To accurately determine infinite volume phase shifts, it is necessary to extrapolate the phase shifts obtained from the L\"uscher formula for the boson-diboson system to the infinite volume limit. For energies above the breakup threshold, or for systems with no two-body bound state (with only scattering states and resonances) the L\"uscher formula gets power-law volume corrections and consequently fails to describe the three-particle system. These corrections are nonperturbatively included in the quantization condition presented.

Large-scale Molecular Simulations of Hypervelocity Impact of Materials

We describe the application of the ReaxFF reactive force field with short-range distance-dependent exponential inner wall corrections and the non-adiabatic electron Force Field (eFF) for studying the hypervelocity impact (HVI) effects on material properties. In particular, to understanding nonequilibrium energy/mass transfer, high strain/heat rate material decomposition, defects formation, plastic flow, phase transitions, and electronic excitation effects that arise from HVI impact of soft and hard materials on different material surfaces. Novel results are presented on the single shock Hugoniot and shock chemistry of Nylon6-6, on the hypervelocity shock sensitivity of energetic materials with planar interfacial defects and on HVI chemistry of silicon carbide surfaces with diamondoid nanoparticles. Both methods provide a means to elucidate the chemical, atomic and molecular processes that occur within the bulk and at the surfaces of materials subjected to HVI conditions and constitute a critical tool to enabling technologies required for the next generation of energy, spatial, transportation, medical, and military systems and devices, among many others. This has proven to be extremely challenging, if not impossible, for experimental observations, mainly because the material states that occur are hard to isolate and their time scales for changes are too rapid (<1 ps). First principles quantum mechanics (QM) simulation methods have also been bounded by the prohibitive scaling cost of propagating the total Schrödinger equation for more than 100 atoms at finite temperatures and pressures.

Eguchi-Hanson-like space-times inF(R)gravity

We consider a model of $F(R)$ gravity in which exponential and power corrections to Einstein-$\ensuremath{\Lambda}$ gravity are included. We show that this model has four-dimensional Eguchi-Hanson-type instanton solutions in Euclidean space. We then seek solutions to the five-dimensional equations for which space-time contains a hypersurface corresponding to the Eguchi-Hanson space. We obtain analytic solutions of the $F(R)$ gravitational field equations and, by assuming certain relationships between the model parameters and integration constants, find several classes of exact solutions. Finally, we investigate the asymptotic behavior of the solutions and compute the second derivative of the $F(R)$ function with respect to the Ricci scalar to confirm Dolgov-Kawasaki stability.

Sine–Gordon form factors in finite volume

We compare form factors in sine–Gordon theory, obtained via the bootstrap, to finite volume matrix elements computed using the truncated conformal space approach. For breather form factors, this is essentially a straightforward application of a previously developed formalism that describes the volume dependence of operator matrix elements up to corrections exponentially decaying with the volume. In the case of solitons, it is necessary to generalize the formalism to include effects of non-diagonal scattering. In some cases it is also necessary to take into account some of the exponential corrections (so-called μ-terms) to get agreement with the numerical data. For almost all matrix elements the comparison is a success, with the puzzling exception of some breather matrix elements that contain disconnected pieces. We also give a short discussion of the implications of the observed behavior of μ-terms on the determination of operator matrix elements from finite volume data, as occurs e.g. in the context of lattice field theory.

Finite-size corrections for quantum strings on $ {\text{Ad}}{{\text{S}}_4} \times \mathbb{C}{P^3} $

We revisit the calculation of curvature corrections to the pp-wave energy of type IIA string states on AdS_4 x CP^3 initiated in arXiv:0807.1527. Using the near pp-wave Hamiltonian found in arXiv:0912.2257, we compute the first non-vanishing correction to the energy of a set of bosonic string states at order 1/R^2, where R is the curvature radius of the background. The leading curvature corrections give rise to cubic, order 1/R, and quartic, order 1/R^2, terms in the Hamiltonian, for which we implement the appropriate normal ordering prescription. Including the contributions from all possible fermionic and bosonic string states, we find that there exist logarithmic divergences in the sums over mode numbers which cancel between the cubic and quartic Hamiltonian. We show that the cubic Hamiltonian naturally requires that the cutoff for summing over heavy modes must be twice the one for light modes. With this prescription the strong-weak coupling interpolating function h(\lambda), entering the magnon dispersion relation, does not receive a one-loop correction, in agreement with the algebraic curve spectrum. However, the single magnon dispersion relation exhibits finite-size exponential corrections.

Y-system and quasi-classical strings

Recently Kazakov, Vieira and the author conjectured the Y-system set of equations describing the planar spectrum of AdS/CFT. In this paper we solve the Y-system equations in the strong coupling scaling limit. We show that the quasi-classical spectrum of string moving inside AdS3 x S1 matches precisely with the prediction of the Y-system. Thus the Y-system, unlike the asymptotic Bethe ansatz, describes correctly the spectrum of one-loop string energies including all exponential finite size corrections. This gives a very non-trivial further support in favor of the conjecture.

The automorphic NS5-brane

Understanding the implications of SL(2, Z) S-duality for the hyper-multiplet moduli space of type II string theories has led to much progress recently in uncovering D-instanton contributions. In this work, we suggest that the duality group SL(3, Z), which includes both S-duality and Ehlers symmetry, may determine the contributions of D5 and NS5-branes. We support this claim by automorphizing the perturbative corrections to the extended universal hypermultiplet, a five-dimensional universal SO(3)\SL(3, R) subspace which includes the string coupling, overall volume, Ramond zero-form and six-form and NS axion. Using the non-Abelian Fourier expansion of the Eisenstein series attached to the principal series of SL(3, R), worked out many years ago by Vinogradov, Takhtajan and Bump, we extract the contributions of D(-1)-D5 and NS5-brane instantons, corresponding to the Abelian and non-Abelian coefficients, respectively. In particular, the contributions of k NS5-branes can be summarized into a vector of wave functions Psi(k,l), l = 0,..., k - 1, as expected on general grounds. We also point out that for more general models with a symmetric moduli space K\G, the minimal theta series of G generates an infinite series of exponential corrections of the form required for small D(-1)-D1-D3-D5-NS5 instanton bound states. As a mathematical spin-off, we make contact with earlier results in the literature about the spherical vectors for the principal series of SL(3, R) and for minimal representations.

Finite size giant magnons in the string dual of 𝒩 = 6 superconformal Chern-Simons theory

We find the exact solution for a finite size Giant Magnon in the $SU(2)\times SU(2)$ sector of the string dual of the $\mathcal{N}=6$ superconformal Chern-Simons theory recently constructed by Aharony, Bergman, Jafferis and Maldacena. The finite size Giant Magnon solution consists of two magnons, one in each $SU(2)$. In the infinite size limit this solution corresponds to the Giant Magnon solution of arXiv:0806.4959. The magnon dispersion relation exhibits finite-size exponential corrections with respect to the infinite size limit solution.

On string solutions of the Bethe equations in the % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgaiuaacqWF % neVtryqr1ngBPrgaiyaacqGF9aqpcqGF0aanaaa!4928! $$ \mathcal{N} = 4 $$ supersymmetric Yang-Mills theory

The Bethe equations which arise in description of the spectrum of the dilatation operator for the su(2) sector of the $$ \mathcal{N} = 4 $$ supersymmetric Yang-Mills theory are considered in the anti-ferromagnetic regime. These equations are a deformation of those for the Heisenberg XXX magnet. We prove that in the thermodynamic limit, roots of the deformed equations group into strings. We prove that the corresponding Yang's action is convex, which implies the uniqueness of a solution for centers of the strings. The state formed by strings of length (2n+1) is considered, and the density of their distribution is found. It is shown that the energy of such a state decreases as n grows. It is observed that the nonanalyticity of the left-hand sides of the Bethe equations leads to an additional contribution to the density and energy of strings of even length. We conclude that the structure of the anti-ferromagnetic vacuum is determined by the behavior of exponential corrections to string solutions in the thermodynamic limit and, possibly, involves strings of length 2. Bibliography: 14 titles.

Finite-size effects from giant magnons

In order to analyze finite-size effects for the gauge-fixed string sigma model on AdS 5 × S 5 , we construct one-soliton solutions carrying finite angular momentum J. In the infinite J limit the solutions reduce to the recently constructed one-magnon configuration of Hofman and Maldacena. The solutions do not satisfy the level-matching condition and hence exhibit a dependence on the gauge choice, which however disappears as the size J is taken to infinity. Interestingly, the solutions do not conserve all the global charges of the psu ( 2 , 2 | 4 ) algebra of the sigma model, implying that the symmetry algebra of the gauge-fixed string sigma model is different from psu ( 2 , 2 | 4 ) for finite J, once one gives up the level-matching condition. The magnon dispersion relation exhibits exponential corrections with respect to the infinite J solution. We also find a generalisation of our one-magnon configuration to a solution carrying two charges on the sphere. We comment on the possible implications of our findings for the existence of the Bethe ansatz describing the spectrum of strings carrying finite charges.

Nonperturbative part of the plaquette in pure gauge theory

We define the nonperturbative part of a quantity as the difference between its numerical value and the perturbative series truncated by dropping the order of minimal contribution and the higher orders. For the anharmonic oscillator, the double-well potential, and the single plaquette gauge theory, the nonperturbative part can be parametrized as $A{\ensuremath{\lambda}}^{B}{\mathrm{e}}^{\ensuremath{-}C/\ensuremath{\lambda}}$ and the coefficients can be calculated analytically. For lattice QCD in the quenched approximation, the perturbative series for the average plaquette is dominated at low order by a singularity in the complex coupling plane and the asymptotic behavior can only be reached by using extrapolations of the existing series. We discuss two extrapolations that provide a consistent description of the series up to order 20--25. These extrapolations favor the idea that the nonperturbative part scales like $(a/{r}_{0}{)}^{4}$ with $a/{r}_{0}$ defined with the force method. We discuss the large uncertainties associated with this statement. We propose a parametrization of $\mathrm{ln}(a/{r}_{0})$ as the two-loop universal terms plus a constant and exponential corrections. These corrections are consistent with ${a}_{1\mathrm{\text{\ensuremath{-}}}\mathrm{loop}}^{2}$ and play an important role when $\ensuremath{\beta}&lt;6$. We briefly discuss the possibility of calculating them semiclassically at large $\ensuremath{\beta}$.

Finite volume corrections toππscattering

Lattice QCD studies of hadron-hadron interactions are performed by computing the energy levels of the system in a finite box. The shifts in energy levels proportional to inverse powers of the volume are related to scattering parameters in a model independent way. In addition, there are nonuniversal exponentially suppressed corrections that distort this relation. These terms are proportional to ${e}^{\ensuremath{-}{m}_{\ensuremath{\pi}}L}$ and become relevant as the chiral limit is approached. In this paper we report on a one-loop chiral perturbation theory calculation of the leading exponential corrections in the case of $I=2$ $\ensuremath{\pi}\ensuremath{\pi}$ scattering near threshold.

Exact expressions for quantum corrections to spinning strings

Abstract The one-loop worldsheet quantum corrections to the energy of spinning strings on R × S 3 within AdS 5 × S 5 are reexamined. The explicit expansion in the effective 't Hooft coupling λ ′ = λ / J 2 is rigorously derived. The expansion contains both analytic and non-analytic terms in λ ′ , as well as exponential corrections. Furthermore, we pin down the origin of the terms that are not captured by the quantum string Bethe ansatz, which only produces analytic terms in λ ′ . It is shown that the analytic terms arise from string fluctuations within the S 3 , whereas the non-analytic and exponential terms, which are not captured by the Bethe ansatz, originate from the fluctuations in all directions within the supersymmetric sigma model on AdS 5 × S 5 . We also comment on the case of spinning string in AdS 3 × S 1 .

Finite-volume effects for two-hadron states in moving frames

We determine the finite-volume corrections to the spectrum and matrix elements of two-hadron states in a moving frame, i.e., one in which the total momentum of the two hadrons is non-zero. The analysis is performed entirely within field theory and the results are accurate up to exponential corrections in the volume. Our results for the spectrum are equivalent to those of Rummukainen and Gottlieb which had been obtained using a relativistic quantum mechanical approach. A technical step in our analysis is a simple derivation of the summation formulae relating the loop summations over the momenta of the two hadrons in finite volume to the corresponding integrals in infinite volume.

On exponential corrections to the 1/Nexpansion in two-dimensional Yang Mills

We compute $e^{-AN}$ corrections to the Gross-Taylor 1/N expansion of the paritition function of two-dimensional SU(N) and U(N) Yang Mills theory. We find a very similar structure of mixing between holomorphic and anti-holomorphic sectors as that described by Vafa for the 1/N expansion. Some of the non-perturbative terms are suggestive of D-strings wrapping the $T^2$ of the 2dYM but blowing up into a fuzzy geometry by the Myers effect in the directions transverse to the $T^2$.

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

The correlator of vector and nonsinglet axial-vector currents in the external electromagnetic field is calculated within the instanton liquid model of QCD vacuum. In general the correlator has two Lorentz structures: longitudinal w_L and transversal w_T with respect to 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 momentum transfer, q, and show the absence of power corrections to this structure at large q^2. Instead there arise the exponential corrections to w_T at large q^2 reflecting nonlocal properties of 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.

Effects of kinks on DNA elasticity

We study the elastic response of a wormlike polymer chain with reversible kinklike structural defects. This is a generic model for (a) the double-stranded DNA with sharp bends induced by binding of certain proteins, and (b) effects of trans-gauche rotations in the backbone of the single-stranded DNA. The problem is solved both analytically and numerically by generalizing the well-known analogy to the quantum rotator. In the small stretching force regime, we find that the persistence length is renormalized due to the presence of the kinks. In the opposite regime, the response to the strong stretching is determined solely by the bare persistence length with exponential corrections due to the "ideal gas of kinks." This high-force behavior changes significantly in the limit of high bending rigidity of the chain. In that case, the leading corrections to the mechanical response are likely to be due to the formation of multikink structures, such as kink pairs.

The evolution of travelling waves in generalized Fisher equations via matched asymptotic expansions: Exponential corrections

In this paper we address an initial boundary value problem for a generalized Fisher equation. In particular we develop the matched asymptotic theory presented in Leach and Needham (2000) to consider the correction terms to the asymptotic approach to the wave- front of speed v = v*(≥ 2) as t (time) → ∞. We establish the precise form of these corrections, and demonstrate that the rate of approach to the wave-front is algebraic in t when v* = 2 (there being two cases), but exponential in t when v* > 2.