Dilute Instanton Gas Approximation Research Articles

CP Conservation in the Strong Interactions

We discuss matters related to the point that topological quantization in the strong interaction is a consequence of an infinite spacetime volume. Because of the ensuing order of limits, i.e., infinite volume prior to summing over topological sectors, CP is conserved. Here, we show that this reasoning is consistent with the construction of the path integral from steepest-descent contours. We reply to some objections that aim to support the case for CP violation in strong interactions that are based on the role of the CP-odd theta-parameter in three-form effective theories, the correct sampling of all configurations in the dilute instanton gas approximation and the volume dependence of the partition function. We also show that the chiral effective field theory derived from taking the volume to infinity first is in no contradiction with analyses based on partially conserved axial currents.

θdependence in the small-Nlimit of2dCPN−1models

We present a systematic numerical study of $\theta$-dependence around $\theta=0$ in the small-$N$ limit of $2d$ $CP^{N-1}$ models, aimed at clarifying the possible presence of a divergent topological susceptibility in the continuum limit. We follow a twofold strategy, based on one side on direct simulations for $N = 2$ and $N = 3$ on lattices with correlation lengths up to $O(10^2)$, and on the other side on the small-$N$ extrapolation of results obtained for $N$ up to $9$. Based on that, we provide conclusive evidence for a finite topological susceptibility at $N = 3$, with a continuum estimate $\xi^2 \chi = 0.110(5)$. On the other hand, results obtained for $N = 2$ are still inconclusive: they are consistent with a logarithmically divergent continuum extrapolation, but do not yet exclude a finite continuum value, $\xi^2 \chi \sim 0.4$, with the divergence taking place for $N$ slightly below 2 in this case. Finally, results obtained for the non-quadratic part of $\theta$-dependence, in particular for the so-called $b_2$ coefficient, are consistent with a $\theta$-dependence matching that of the Dilute Instanton Gas Approximation at the point where $\xi^2 \chi$ diverges.

Nonperturbative flavor breaking in topological susceptibility at chiral crossover

We demonstrate that the QCD topological susceptibility nonperturbatively gets a significant contribution signaled by flavor-nonuniversal quark condensates at around the pseudo-critical temperature of the chiral crossover. It implies a remarkable flavor breaking in the axial anomaly as well as the QCD theta vacuum in high temperature QCD, which are almost flavor universal in the vacuum. A nontrivial flavor breaking is triggered by nonperturbative thermal loop corrections at around the chiral crossover, which is different from the trivial flavor violation just scaled by the quark mass ratio, observed at asymptotically high temperatures. This critical flavor violation cannot be dictated by the chiral perturbation theory with that lattice QCD usually compares, or the dilute instanton gas approximation based on that its astrophysical implications have conventionally been made. This would give an impact on the thermal history and the cosmological evolution of QCD axion including the estimate of the relic abundance as a dark matter candidate.

Phase structure of the twisted SU(3)/U(1)2 flag sigma model on \u211d \xd7 S1

We investigate the phase structure of the compactified 2-dimensional nonlinear SU(3)/U(1)2 flag sigma model with respect to two θ-terms. Based on the circle compactification with the ℤ3-twisted boundary condition, which preserves an ’t Hooft anomaly of the original uncompactified theory, we perform the semiclassical analysis based on the dilute instanton gas approximation (DIGA). We clarify classical vacua of the theory and derive fractional instanton solutions connecting these vacua. The resulting phase structure based on DIGA exhibits the quantum phase transitions and triple degeneracy at special points in the (θ1, θ2)-plane, which is consistent with the phase diagram obtained from the anomaly matching and global inconsistency conditions. This result indicates the adiabatic continuity between the flag sigma models on ℝ2 and ℝ × S1 with small compactification radius. We further estimate contributions from instanton-anti-instanton configuration (bion) and show the existence of the imaginary ambiguity, which is expected to be cancelled by that of the perturbative Borel resummation.

Instanton interaction in de Sitter space–time

Because of the presence of a cosmological horizon, the dilute instanton gas approximation used for the derivation of the Coleman–De Luccia tunneling rate in de Sitter space–time receives additional contributions due to the finite instanton separation. Here, I calculate the first corrections to the vacuum decay rate that arise from this effect and depend on the parameters of the theory and the cosmological constant of the background space–time.

Chiral observables and topology in hot QCD with two families of quarks

We present results on QCD with four dynamical flavors in the temperature range $150$ MeV $\lesssim T \lesssim 500$ MeV. We have performed lattice simulations with Wilson fermions at maximal twist and measured Polyakov loop, chiral condensate and disconnected susceptibility, on lattices with spacings as fine as 0.065 fm. For most observables spacing effects are below statistical errors, which enables us to identify lattice results with continuum estimates. Our estimate of the pseudocritical temperature compares favorably with continuum results from staggered and domain wall fermions, confirming that a dynamical charm does not contribute in the transition region. From the high temperature behaviour of the disconnected chiral susceptibility we infer the topological susceptibility, which encodes relevant properties of the QCD axion, a plausible Dark Matter candidate. The topological susceptibility thus measured exhibits a power-law decay for $T/T_c \gtrsim 2$, with an exponent close to the one predicted by the Dilute Instanton Gas Approximation (DIGA). Close to $T_c$ the temperature dependent effective exponent seems to approach the DIGA result from above, a behaviour which would support recent analytic calculations based on an Instantons-dyons model. These results constrain the mass of a hypothetic QCD post-inflationary axion, once an assumption concerning the relative contribution of axions to Dark Matter is made.

Topology in full QCD at high temperature: a multicanonical approach

We investigate the topological properties of Nf = 2 + 1 QCD with physical quark masses, at temperatures around 500 MeV. With the aim of obtaining a reliable sampling of topological modes in a regime where the fluctuations of the topological charge Q are very rare, we adopt a multicanonical approach, adding a bias potential to the action which enhances the probability of suppressed topological sectors. This method permits to gain up to three orders magnitude in computational power in the explored temperature regime. Results at different lattice spacings and physical spatial volumes reveal no significant finite size effects and the presence, instead, of large finite cut-off effects, with the topological susceptibility which decreases by 3-4 orders of magnitude while moving from a ≃ 0.06 fm towards the continuum limit. The continuum extrapolation is in agreeement with previous lattice determinations with smaller uncertainties but obtained based on ansatzes justified by several theoretical assumptions. The parameter b2, related to the fourth order coefficient in the Taylor expansion of the free energy density f (θ), has instead a smooth continuum extrapolation which is in agreement with the dilute instanton gas approximation (DIGA); moreover, a direct measurement of the relative weights of the different topological sectors gives an even stronger support to the validity of DIGA.

Topology (and axion's properties) from lattice QCD with a dynamical charm

We present results on QCD with four dynamical flavors in the temperature range 0.9≲T/Tc≲2. We have performed lattice simulations with Wilson fermions at maximal twist and measured the topological charge with gluonic and fermionic methods. The topological charge distribution is studied by means of its cumulants, which encode relevant properties of the QCD axion, a plausible Dark Matter candidate. The topological susceptibility measured with the fermionic method exhibits a power-law decay for T/Tc≳2, with an exponent close to the one predicted by the Dilute Instanton Gas Approximation (DIGA). Close to Tc the temperature dependent effective exponent approaches the DIGA result from above, in agreement with recent analytic calculations. These results constrain the axion window, once an assumption on the fraction of axions contributing to Dark Matter is made.

Topological susceptibility in finite temperature ( 2+1 )-flavor QCD using gradient flow

We compute the topological charge and its susceptibility in finite temperature (2+1)-flavor QCD on the lattice applying a gradient flow method. With the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson quarks, we perform simulations on a fine lattice with~$a\simeq0.07\,\mathrm{fm}$ at a heavy $u$, $d$ quark mass with $m_\pi/m_\rho\simeq0.63$ but approximately physical $s$ quark mass with $m_{\eta_{ss}}/m_\phi\simeq0.74$. In a temperature range from~$T\simeq174\,\mathrm{MeV}$ ($N_t=16$) to $697\,\mathrm{MeV}$ ($N_t=4$), we study two topics on the topological susceptibility. One is a comparison of gluonic and fermionic definitions of the topological susceptibility. Because the two definitions are related by chiral Ward-Takahashi identities, their equivalence is not trivial for lattice quarks which violate the chiral symmetry explicitly at finite lattice spacings. The gradient flow method enables us to compute them without being bothered by the chiral violation. We find a good agreement between the two definitions with Wilson quarks. The other is a comparison with a prediction of the dilute instanton gas approximation, which is relevant in a study of axions as a candidate of the dark matter in the evolution of the Universe. We find that the topological susceptibility shows a decrease in $T$ which is consistent with the predicted $\chi_\mathrm{t}(T) \propto (T/T_{\rm pc})^{-8}$ for three-flavor QCD even at low temperature $T_{\rm pc} < T\le1.5 T_{\rm pc}$.

The topological susceptibility in finite temperature QCD and axion cosmology

We study the topological susceptibility in 2+1 flavor QCD above the chiral crossover transition temperature using Highly Improved Staggered Quark action and several lattice spacings, corresponding to temporal extent of the lattice, $N_\tau=6,8,10$ and $12$. We observe very distinct temperature dependences of the topological susceptibility in the ranges above and below $250$ MeV. While for temperatures above $250$ MeV, the dependence is found to be consistent with dilute instanton gas approximation, at lower temperatures the fall-off of topological susceptibility is milder. We discuss the consequence of our results for cosmology wherein we estimate the bounds on the axion decay constant and the oscillation temperature if indeed the QCD axion is a possible dark matter candidate.

Topological susceptibility at high temperature on the lattice

QCD topological susceptibility at high temperature, $\chi_t(T)$, provides an important input for the estimate of the axion abundance in the present Universe. While the model independent determination of $\chi_t(T)$ should be possible from the first principles using lattice QCD, existing methods fail at high temperature, since not only the probability that non-trivial topological sectors appear in the configuration generation process but also the local topological fluctuations get strongly suppressed. We propose a novel method to calculate the temperature dependence of topological susceptibility at high temperature. A feasibility test is performed on a small lattice in the quenched approximation, and the results are compared with the prediction of the dilute instanton gas approximation. It is found that the method works well especially at very high temperature and the result is consistent with the instanton calculus down to $T\sim 2\, T_c$ within the statistical uncertainty.

ADHM construction of (anti-)self-dual instantons in eight dimensions

We study the ADHM construction of (anti-)self-dual instantons in eight dimensions. We propose a general scheme to construct the (anti-)self-dual gauge field configurations F∧F=±⁎8F∧F whose finite topological charges are given by the fourth Chern number. We show that our construction reproduces the known SO(8) one-instanton solution. We also construct multi-instanton solutions of the 't Hooft and the Jackiw–Nohl–Rebbi (JNR) types in the dilute instanton gas approximation. The well-separated configurations of multi-instantons reproduce the correct topological charges with high accuracy. We also show that our construction is generalized to (anti-)self-dual instantons in 4n(n=3,4,…) dimensions.

Axion cosmology, lattice QCD and the dilute instanton gas

Axions are one of the most attractive dark matter candidates. The evolution of their number density in the early universe can be determined by calculating the topological susceptibility χ(T) of QCD as a function of the temperature. Lattice QCD provides an ab initio technique to carry out such a calculation. A full result needs two ingredients: physical quark masses and a controlled continuum extrapolation from non-vanishing to zero lattice spacings. We determine χ(T) in the quenched framework (infinitely large quark masses) and extrapolate its values to the continuum limit. The results are compared with the prediction of the dilute instanton gas approximation (DIGA). A nice agreement is found for the temperature dependence, whereas the overall normalization of the DIGA result still differs from the non-perturbative continuum extrapolated lattice results by a factor of order ten. We discuss the consequences of our findings for the prediction of the amount of axion dark matter.

Modeling and simulations of quantum phase slips in ultrathin superconducting wires

We study quantum phase slips (QPS) in ultrathin superconducting wires. Starting from an effective one-dimensional microscopic model, which includes electromagnetic fluctuations, we map the problem to a (1+1)-dimensional gas of interacting instantons. We introduce a method to calculate the tunneling amplitude of quantum phase slips directly from Monte Carlo simulations. This allows us to go beyond the dilute instanton gas approximation and study the problem without any limitations of the density of QPS. We find that the tunneling amplitude shows a characteristic scaling behavior near the superconductor-insulator transition. We also calculate the voltage-charge relation of the insulating state, which is the dual of the Josephson current-phase relation in ordinary superconducting weak links. This evolves from a sinusoidal form in the regime of dilute QPS to more exotic shapes for higher QPS densities, where interactions are important.

QCD chiral transition,U(1)Asymmetry and the dirac spectrum using domain wall fermions

We report on a study of the finite-temperature QCD transition region for temperatures between 139 and 196 MeV, with a pion mass of 200 MeV and two space-time volumes: $24^3\times8$ and $32^3\times8$, where the larger volume varies in linear size between 5.6 fm (at T=139 MeV) and 4.0 fm (at T=195 MeV). These results are compared with the results of an earlier calculation using the same action and quark masses but a smaller, $16^3\times8$ volume. The chiral domain wall fermion formulation with a combined Iwasaki and dislocation suppressing determinant ratio gauge action are used. This lattice action accurately reproduces the $\sua$ and $\ua$ symmetries of the continuum. Results are reported for the chiral condensates, connected and disconnected susceptibilities and the Dirac eigenvalue spectrum. We find a pseudo-critical temperature, $T_c$, of approximately 165 MeV consistent with previous results and strong finite volume dependence below $T_c$. Clear evidence is seen for $\ua$ symmetry breaking above $T_c$ which is quantitatively explained by the measured density of near-zero modes in accordance with the dilute instanton gas approximation.

The vacuum tunnelling and the crossover of deconfinement in Friedberg–Lee model

We have discussed the vacuum tunnelling in Friedberg-Lee model. The tunnelling coefficient is derived in the field configuration space by calculating the transition amplitude using the path integral under the stationary phase approximation and the dilute instanton gas approximation. By studying the tunnelling effect between the two degenerating vacuums at the critical temperature and chemical potential, we find that the system could be deconfined by tunnelling, which could possibly change the first order deconfinement phase transition to crossover.

CHIRAL AND UA(1) PHASE TRANSITIONS IN π0, η AND η′ MESONS

We have generalized the Pisarsky–Wilczek model of studying the UA(1) phase transition in π0, η and η′ system by incorporating the chiral phase transition. The mass-squared matrix of the neutral pseudoscalar mesons is revisited explicitly from the chiral effective action which necessitates the mixing among the unmixed states to get the physical states. The derivation of the mass spectra of the neutral pseudoscalar mesons shows that apart from their masses, the mixing angles also become temperature dependent. Based on the dilute instanton gas approximation we show that, the UA(1) phase transition along with the chiral phase transition plays an important role in the process of hadronization of the quark–gluon plasma with the quarks of three flavors.

Non-Abelian Stokes Theorem and Quark Confinement in SU(N) Yang-Mills Gauge Theory

We derive a new version of the non-Abelian Stokes theorem for the Wilson loop in the SU(N) case by making use of the coherent state representation on the coset space SU(N)/U(1)N−1 = FN−1, the flag space. We consider the SU(N) Yang-Mills theory in the maximal Abelian gauge in which SU(N) is broken down to U(1)N−1. First, we show that the Abelian dominance in the string tension follows from this theorem and the Abelian-projected effective gauge theory that was derived by one of the authors. Next (but independently), combining the non-Abelian Stokes theorem with a novel reformulation of the Yang-Mills theory recently proposed by one of the authors, we proceed to derive the area law of the Wilson loop in four-dimensional SU(N) Yang-Mills theory in the maximal Abelian gauge. Owing to dimensional reduction, the planar Wilson loop at least for the fundamental representation in four-dimensional SU(N) Yang-Mills theory can be estimated by the diagonal (Abelian) Wilson loop defined in the two-dimensional CPN−1 model. This derivation shows that the fundamental quarks are confined by a single species of magnetic monopole. The origin of the area law is related to the geometric phase of the Wilczek-Zee holonomy for U(N−1). The calculations are performed using the instanton calculus (in the dilute instanton-gas approximation) and using the large N expansion (in the leading order).

The k -folded sine-Gordon model in finite volume

We consider the k-folded sine-Gordon model, obtained from the usual version by identifying the scalar field after k periods of the cosine potential. We examine (1) the ground state energy split, (2) the lowest lying multi-particle state spectrum and (3) vacuum expectation values of local fields in finite spatial volume, combining the Truncated Conformal Space Approach, the method of the Destri-de Vega nonlinear integral equation (NLIE) and semiclassical instanton calculations. We show that the predictions of all these different methods are consistent with each other and in particular provide further support for the NLIE method in the presence of a twist parameter. It turns out that the model provides an optimal laboratory for examining instanton contributions beyond the dilute instanton gas approximation. We also provide evidence for the exact formula for the vacuum expectation values conjectured by Lukyanov and Zamolodchikov.

Topological phase interference induced by a magnetic field along hard anisotropy axis in nanospin systems with different crystal symmetries

Based on the instanton technique in the spin-coherent-state path-integral representation, the spin-parity effects induced by the topological Wess-Zumino or Berry phase are studied theoretically in nanometer-scale single-domain ferromagnets in the presence of an external magnetic field along the hard anisotropy axis. We consider the magnetocrystalline anisotropy with the biaxial, trigonal, tetragonal, and hexagonal crystal symmetry, respectively. Both the Wentzel-Kramers-Brillouin (WKB) exponent and the preexponential factors are evaluated in the instanton's contribution to the ground-state tunnel splittings. The Euclidean transition amplitudes between energetically degenerate easy directions are obtained with the help of the dilute instanton gas approximation. The effective Hamiltonian approach is applied to present the final results of ground-state tunnel splittings for each kind of crystal symmetry. The Euclidean transition amplitudes and the ground-state tunnel splittings are found to depend on the parity of total spins of ferromagnets and oscillate with the external magnetic field for both the integer and half-integer total spins. We show that the topological phase interference or spin-parity effects can reflect in thermodynamic quantities of magnetic tunneling states. Possible relevance to experiments is also discussed.