Non-perturbative Nature Research Articles

Low-energy effective theory of Fermi surface coupled with U(1) gauge field in2+1dimensions

We study the low-energy effective theory for a non-Fermi-liquid state in $2+1$ dimensions, where a transverse U(1) gauge field is coupled with a patch of Fermi surface with $N$ flavors of fermion in the large $N$ limit. In the low-energy limit, quantum corrections are classified according to the genus of the two-dimensional surface on which Feynman diagrams can be drawn without a crossing in a double line representation and all planar diagrams are important in the leading order. The emerging theory has the similar structure to the four-dimensional SU(N) gauge theory in the large $N$ limit. Because of strong quantum fluctuations caused by the abundant low-energy excitations near the Fermi surface, low-energy fermions remain strongly coupled even in the large $N$ limit. As a result, there are infinitely many quantum corrections that contribute to the leading frequency dependence of the Green's function of fermion on the Fermi surface. On the contrary, the boson self-energy is not modified beyond the one-loop level and the theory is stable in the large $N$ limit. The nonperturbative nature of the theory also shows up in correlation functions of gauge-invariant operators.

New intermediate models for rotating shallow water and an investigation of the preference for anticyclones

New intermediate models for the rotating shallow water (RSW) equations are derived by considering the nonlinear interactions between subsets of the eigenmodes for the linearized equations. It is well-known that the two-dimensional quasi-geostrophic (QG) equation results when the nonlinear interactions are restricted to include only the vortical eigenmodes. Continuing past QG in a non-perturbative manner, the new models result by including subsets of interactions which include inertial-gravity wave (IG) modes. The such simplest model adds nonlinear interactions between one IG mode and two vortical modes. In sharp contrast to QG, the latter model behaves similar to the full RSW equations for decay from balanced initial conditions as well as unbalanced random initial conditions with divergence-free velocity. Quantitative agreement is observed for statistics that measure structure size, intermittency and cyclone/anticyclone asymmetry. In particular, dominance of anticyclones is observed for Rossby numbersRoin the range 0.1 <Ro< 1 (away from the QG parameter regimeRo→ 0). A hierarchy of models is explored to determine the effects of wave-vortical and wave–wave interactions on statistics such as the skewness of vorticity in decaying turbulence. Possible advantages over previously derived intermediate models include (i) the non-perturbative nature of the new models (not restricting thema priorito any particular parameter regime) and (ii) insight into the physical and mathematical consequences of vortical–wave interactions.

Heavy-quark free energies, internal-energy and entropy contributions

We present lattice QCD results on heavy-quark free energies, extract from its temperature dependence the entropy and internal-energy contributions, and discuss the onset of medium effects that lead to screening of static quark–antiquark sources in a thermal medium. The detailed analysis of the temperature and distance dependence of the different contributions indicate the complex non-perturbative nature of strongly interacting matter. We shall discuss the necessity to include those effects in studies on the behavior of heavy quarks, heavy-quark bound states and their dissociation in the quark–gluon plasma phase.

Towards a controlled study of the QCD critical point

The phase diagram of QCD, as a function of temperature T and quark chemical potential μ, may contain a critical point (μE, TE) whose non-perturbative nature makes it a natural object of lattice studies. However, the sign problem prevents the application of standard Monte Carlo techniques at nonzero baryon density. We have been pursuing an approach free of the sign problem, where the chemical potential is taken as imaginary and the results are Taylor expanded in μ/T about μ = 0, then analytically continued to real μ. Within this approach we have determined the sensitivity of the critical chemical potential μE to the quark mass, . Our study indicates that the critical point moves to a larger chemical potential as the quark mass decreases. This finding, contrary to common wisdom, implies that the deconfinement crossover, which takes place in QCD at μ = 0 when the temperature is raised, will remain a crossover in the μ-region where our Taylor expansion can be trusted. If this result, obtained on a coarse lattice, is confirmed by simulations on finer lattices now in progress, then we predict that no chiral critical point will be found for μB ≲ 500 MeV, unless the phase diagram contains additional transitions.

Hadron properties and Dyson–Schwinger equations

An overview of the theory and phenomenology of hadrons and QCD is provided from a Dyson–Schwinger equation viewpoint. Following a discussion of the definition and realization of light-quark confinement, the nonperturbative nature of the running mass in QCD and inferences from the gap equation relating to the radius of convergence for expansions of observables in the current-quark mass are described. Some exact results for pseudoscalar mesons are also highlighted, with details relating to the U A ( 1 ) problem, and calculated masses of the lightest J = 0 , 1 states are discussed. Studies of nucleon properties are recapitulated upon and illustrated: through a comparison of the ln -weighted ratios of Pauli and Dirac form factors for the neutron and proton; and a perspective on the contribution of quark orbital angular momentum to the spin of a nucleon at rest. Comments on prospects for the future of the study of quarks in hadrons and nuclei round out the contribution.

Lattice QCD at finite temperature and density

QCD at finite temperature and density is becoming increasingly important for various experimental programmes, ranging from heavy ion physics to astro-particle physics. The non-perturbative nature of non-abelian quantum field theories at finite temperature leaves lattice QCD as the only tool by which we may hope to come to reliable predictions from first principles. This requires careful extrapolations to the thermodynamic, chiral and continuum limits in order to eliminate systematic effects introduced by the discretization procedure. After an introduction to lattice QCD at finite temperature and density, its possibilities and current systematic limitations, a review of present numerical results is given. In particular, plasma properties such as the equation of state, screening masses, static quark free energies and spectral functions are discussed, as well as the critical temperature and the QCD phase structure at zero and finite density.

Probing electroweak physics for allB→XMdecays in the endpoint region

Using soft-collinear effective theory we describe at leading order in $1/{m}_{b}$ all the semi-inclusive hadronic $B\ensuremath{\rightarrow}XM$ decays near the endpoint, where an energetic light meson $M$ recoils against an inclusive jet $X$. Here we extend to the decays in which spectator quarks go into the jet $X$, and also include the decays involving $\ensuremath{\eta}$, ${\ensuremath{\eta}}^{\ensuremath{'}}$ mesons that receive additional contributions from gluonic operators. The predicted branching ratios and $CP$ asymmetries depend on fewer hadronic parameters than the corresponding two-body $B$ decays. This makes semi-inclusive hadronic $B\ensuremath{\rightarrow}XM$ decays a powerful probe of the potential nonperturbative nature of charming penguins as well as a useful probe of new physics effects in electroweak flavor changing transitions. A comparison with $B\ensuremath{\rightarrow}KX$ data from BABAR points to an enhanced charming penguin, albeit with large experimental errors.

Linear growth of the trace anomaly in Yang-Mills thermodynamics

In the lattice work by Miller and in the work by Zwanziger a linear growth of the trace anomaly for high temperatures was found in pure SU(2) and SU(3) Yang-Mills theories. While future numerical work is needed to confirm or to rule out the linear growth, the aforementioned results point to the remarkable property that the corresponding systems are strongly interacting even at high temperatures. We show that within an analytical approach to Yang-Mills thermodynamics this linear rise is obtained and is directly connected to the presence of a temperature-dependent ground state, which describes (part of) the nonperturbative nature of the Yang-Mills system. Our predictions are in approximate agreement with Miller and Zwanziger.

Weinberg eigenvalues and pairing with low-momentum potentials

The non-perturbative nature of nucleon–nucleon interactions evolved to low momentum has recently been investigated in free space and at finite density using Weinberg eigenvalues as a diagnostic. This analysis is extended here to the in-medium eigenvalues near the Fermi surface to study pairing. For a fixed value of density and cutoff, the eigenvalues increase arbitrarily in magnitude close to the Fermi surface, signaling the pairing instability. When using normal-phase propagators, the Weinberg analysis with complex energies becomes a form of stability analysis and the pairing gap can be estimated from the largest attractive eigenvalue. With Nambu–Gorkov Green's functions, the largest attractive eigenvalue goes to unity close to the Fermi surface, indicating the presence of bound states (Cooper pairs), and the corresponding eigenvector leads to the self-consistent gap function.

Ampere's law and energy loss in AdS/CFT duality

We note that the energy loss in N = 4 SYM measures directly the spatial string tension σ S = π λ T 2 / 2 which is at the origin of the area law for large spatial Wilson loops. We show that the latter reflects on the nonperturbative nature of Ampere's law in N = 4 SYM both in vacuum and at finite temperature.

Colour, copies and confinement

In this paper we construct a wide class of Gribov copies in Coulomb gauge SU(2) gauge theory. Infinitesimal copies are studied in some detail and their non-perturbative nature is made manifest. As an application it is shown that the copies prevent a non-perturbative definition of colour charge.

The physics of hadronic tau decays

Hadronic $\ensuremath{\tau}$ decays provide a clean laboratory for the precise study of quantum chromodynamics (QCD). Observables based on the spectral functions of hadronic $\ensuremath{\tau}$ decays can be related to QCD quark-level calculations to determine fundamental quantities like the strong-coupling constant, parameters of the chiral Lagrangian $\ensuremath{\mid}{V}_{us}\ensuremath{\mid}$, the mass of the strange quark, and to simultaneously test the concept of quark-hadron duality. Using the best available measurements and a revisited analysis of the theoretical framework, the value ${\ensuremath{\alpha}}_{s}({m}_{\ensuremath{\tau}}^{2})=0.345\ifmmode\pm\else\textpm\fi{}{0.004}_{\mathrm{exp}}\ifmmode\pm\else\textpm\fi{}{0.009}_{\mathrm{th}}$ is obtained. Taken together with the determination of ${\ensuremath{\alpha}}_{s}({M}_{Z}^{2})$ from the global electroweak fit, this result leads to the most accurate test of asymptotic freedom: the value of the logarithmic slope of ${\ensuremath{\alpha}}_{s}^{\ensuremath{-}1}(s)$ is found to agree with QCD at a precision of 4%. The $\ensuremath{\tau}$ spectral functions can also be used to determine hadronic quantities that, due to the nonperturbative nature of long-distance QCD, cannot be computed from first principles. An example for this is the contribution from hadronic vacuum polarization to loop-dominated processes like the anomalous magnetic moment of the muon. This article reviews the measurements of nonstrange and strange $\ensuremath{\tau}$ spectral functions and their phenomenological applications.

Intrinsic Versus Extrinsic Anomalous Hall Effect in Ferromagnets

A unified theory of the anomalous Hall effect (AHE) is presented for multiband ferromagnetic metals with dilute impurities. In the clean limit, the AHE is mostly due to extrinsic skew scattering. When the Fermi level is located around anticrossing of band dispersions split by spin-orbit interaction, the intrinsic AHE to be calculated ab initio is resonantly enhanced by its nonperturbative nature, revealing the extrinsic-to-intrinsic crossover which occurs when the relaxation rate is comparable to the spin-orbit coupling.

Convergence of the Born series with low-momentum interactions

The non-perturbative nature of nucleon–nucleon interactions as a function of a momentum cutoff is studied using Weinberg eigenvalues as a diagnostic. This investigation extends an earlier study of the perturbative convergence of the Born series to partial waves beyond the 3S 1– 3D 1 channel and to positive energies. As the cutoff is lowered using renormalization-group or model-space techniques, the evolution of non-perturbative features at large cutoffs from strong short-range repulsion and the iterated tensor interaction are monitored via the complex Weinberg eigenvalues. When all eigenvalues lie within the unit circle, the expansion of the scattering amplitude in terms of the interaction is perturbative, with the magnitude of the largest eigenvalue setting the rate of convergence. Major decreases in the magnitudes of repulsive eigenvalues are observed as the Argonne v 18 , CD-Bonn or Nijmegen potentials are evolved to low momentum, even though two-body observables are unchanged. For chiral EFT potentials, running the cutoff lower tames the impact of the tensor force and of new non-perturbative features entering at N 3LO. The efficacy of separable approximations to nuclear interactions derived from the Weinberg analysis is studied as a function of cutoff, and the connection to inverse scattering is demonstrated.

Incorporation of a wave-packet propagation method into theS-matrix framework: Investigation of the effects of excited state dynamics on intense-field ionization

We propose a theoretical method for investigation of ionization of atoms and molecules in intense laser fields that copes with the effects of excited state dynamics (or intramolecular electronic dynamics). The time-evolving wave packet composed of only bound electronic states, $\ensuremath{\mid}{\ensuremath{\Phi}}_{i}(t)⟩$, is introduced into a framework of the intense-field $S$-matrix theory. Then, the effects of both Coulomb field and radiation field on the bound electron(s) are well described by $\ensuremath{\mid}{\ensuremath{\Phi}}_{i}(t)⟩$, while the effects of a radiation field on a freed electron are also treated in a nonperturbative way. We have applied the theory to ionization of H and $\mathrm{H}_{2}{}^{+}$ in ultrashort intense laser pulses. Although only a small number of Gaussian functions are used in the expansion of $\ensuremath{\mid}{\ensuremath{\Phi}}_{i}(t)⟩$, the present method can quantitatively reproduce the features of enhanced ionization of $\mathrm{H}_{2}{}^{+}$ obtained by an accurate grid propagation method. This agreement supports the view that field-induced population transfer between the lowest two electronic states triggers the enhancement of ionization at large internuclear distances. We also applied the method to calculate the photoelectron momentum distribution of H in an intense near-infrared field. A broad low intensity component due to ``rescattering'' appears in the distribution of the momentum perpendicular to the polarization direction of an applied laser field, as observed in the experiments of single ionization of noble gas atoms. The present method provides a practical way of properly describing the nonperturbative nature of field-induced dynamics of an electron (or electrons) in the presence of both Coulomb and radiation fields.

Is nuclear matter perturbative with low-momentum interactions?

The nonperturbative nature of inter-nucleon interactions is explored by varying the momentum cutoff of a two-nucleon potential. Conventional force models, which have large cutoffs, are nonperturbative because of strong short-range repulsion, the iterated tensor interaction, and the presence of bound or nearly-bound states. But for low-momentum interactions with cutoffs around 2 fm −1 , the softened potential combined with Pauli blocking leads to corrections in nuclear matter in the particle-particle channel that are well converged at second order in the potential, suggesting that perturbation theory can be used in place of Brueckner resummations. Calculations of nuclear matter using the low-momentum two-nucleon force V low k with a corresponding leading-order three-nucleon (3N) force from chiral effective field theory (EFT) exhibit nuclear binding in the Hartree–Fock approximation, and become less cutoff dependent with the inclusion of the dominant second-order contributions. The role of the 3N force is essential to obtain saturation, and the contribution to the total potential energy is compatible with EFT power-counting estimates.

Reconnection of colliding cosmic strings

For vortex strings in the Abelian Higgs model and D-strings in superstring theory, both of which can be regarded as cosmic strings, we give analytical study of reconnection (recombination, inter-commutation) when they collide, by using effective field theories on the strings. First, for the vortex strings, via a string sigma model, we verify analytically that the reconnection is classically inevitable for small collision velocity and small relative angle. Evolution of the shape of the reconnected strings provides an upper bound on the collision velocity in order for the reconnection to occur. These analytical results are in agreement with previous numerical results. On the other hand, reconnection of the D-strings is not classical but probabilistic. We show that a quantum calculation of the reconnection probability using a D-string action reproduces the nonperturbative nature of the worldsheet results by Jackson, Jones and Polchinski. The difference on the reconnection -- classically inevitable for the vortex strings while quantum mechanical for the D-strings -- is suggested to originate from the difference between the effective field theories on the strings.

Erratum: General effective action for high-density quark matter [Phys. Rev. D70, 114029 (2004)

We derive a general effective action for quark matter at nonzero temperature and/or nonzero density. For this purpose, we distinguish irrelevant from relevant quark modes, as well as hard from soft gluon modes by introducing two separate cut-offs in momentum space, one for quarks, $\Lambda_q$, and one for gluons, $\Lambda_g$. We exactly integrate out irrelevant quark modes and hard gluon modes in the functional integral representation of the QCD partition function. Depending on the specific choice for $\Lambda_q$ and $\Lambda_g$, the resulting effective action contains well-known effective actions for hot and/or dense quark matter, for instance the ``Hard Thermal Loop'' or the ``Hard Dense Loop'' action, as well as the high-density effective theory proposed by Hong and others. We then apply our effective action to review the calculation of the color-superconducting gap parameter to subleading order in weak coupling, where the strong coupling constant $g \ll 1$. In this situation, relevant quark modes are those within a layer of thickness $2 \Lambda_q$ around the Fermi surface. The non-perturbative nature of the gap equation invalidates naive attempts to estimate the importance of the various contributions via power counting on the level of the effective action. Nevertheless, once the gap equation has been derived within a particular many-body approximation scheme, the cut-offs $\Lambda_q, \Lambda_g$ provide the means to rigorously power count different contributions to the gap equation. We recover the previous result for the QCD gap parameter for the choice $\Lambda_q \alt g \mu \ll \Lambda_g \alt \mu$, where $\mu$ is the quark chemical potential. We also point out how to improve this result beyond subleading order in weak coupling.

General effective action for high-density quark matter

We derive a general effective action for quark matter at nonzero temperature and/or nonzero density. For this purpose, we distinguish irrelevant from relevant quark modes, as well as hard from soft gluon modes by introducing two separate cutoffs in momentum space, one for quarks, ${\ensuremath{\Lambda}}_{q}$, and one for gluons, ${\ensuremath{\Lambda}}_{g}$. We exactly integrate out irrelevant quark modes and hard gluon modes in the functional integral representation of the QCD partition function. Depending on the specific choice for ${\ensuremath{\Lambda}}_{q}$ and ${\ensuremath{\Lambda}}_{g}$, the resulting effective action contains well-known effective actions for hot and/or dense quark matter, for instance the ``Hard Thermal Loop'' or the ``Hard Dense Loop'' action, as well as the high-density effective theory proposed by Hong and others. We then apply our effective action to review the calculation of the color-superconducting gap parameter to subleading order in weak coupling, where the strong coupling constant $g\ensuremath{\ll}1$. In this situation, relevant quark modes are those within a layer of thickness $2{\ensuremath{\Lambda}}_{q}$ around the Fermi surface. The nonperturbative nature of the gap equation invalidates naive attempts to estimate the importance of the various contributions via power counting on the level of the effective action. Nevertheless, once the gap equation has been derived within a particular many-body approximation scheme, the cutoffs ${\ensuremath{\Lambda}}_{q},{\ensuremath{\Lambda}}_{g}$ provide the means to rigorously power count different contributions to the gap equation. We recover the previous result for the QCD gap parameter for the choice ${\ensuremath{\Lambda}}_{q}\ensuremath{\lesssim}g\ensuremath{\mu}\ensuremath{\ll}{\ensuremath{\Lambda}}_{g}\ensuremath{\lesssim}\ensuremath{\mu}$, where $\ensuremath{\mu}$ is the quark chemical potential. We also point out how to improve this result beyond subleading order in weak coupling.

Nonperturbative aspects of the QCD analytic invariant charge

In the framework of the analytic approach to Quantum Chromodynamics a new model for the strong running coupling has recently been developed. Its underlying idea is to impose the analyticity requirement on the perturbative expansion of the renormalization group $\beta$ function for restoring the correct analytic properties of the latter. The proposed model possesses a number of appealing features. Namely, the analytic invariant charge has no unphysical singularities at any loop level; it contains no free parameters; it has universal behavior both in ultraviolet and infrared regions at any loop level; and it possesses a fair higher loop and scheme stability. The extension of this model to the timelike region revealed the asymmetrical behavior of the running coupling in the intermediate- and low-energy domains of spacelike and timelike regions, that is essential when one handles the experimental data. The developed approach enables one to describe, in a consistent way, various strong interaction processes both of perturbative and intrinsically nonperturbative nature.