Irreducible Effective Action Research Articles

Conformal anomaly actions for dilaton interactions

We discuss, in conformally invariant field theories such as QCD with massless fermions, a possible link between the perturbative signature of the conformal anomaly, in the form of anomaly poles of the 1-particle irreducible effective action, and its descrip- tion in terms of Wess-Zumino actions with a dilaton. The two descriptions are expected to capture the UV and IR behaviour of the conformal anomaly, in terms of fundamental and effective degrees of freedom respectively, with the dilaton effective state appearing in a nonlinear realization. As in the chiral case, conformal anomalies seem to be related to the appearance of these effective interactions in the 1PI action in all the gauge-invariant sectors of the Standard Model. We show that, as a consequence of the underlying anomalous symmetry, the infinite hierarchy of recurrence relations involving self-interactions of the dilaton is entirely determined only by the first four of them. This relation can be generalized to any even space-time dimension.

Effective potential and Goldstone bosons in de Sitter space

We investigate nonperturbative infrared effects for the $O(N)$ linear sigma model in de Sitter space using the two-particle irreducible effective action at the Hartree truncation level. This approximation resums the infinite series of so-called superdaisy diagrams. For the proper treatment of ultraviolet divergences, we first study the renormalization of this approximation on a general curved background. Then, we calculate radiatively corrected masses and the effective potential. As a result, spontaneous symmetry breaking is possible; on the other hand, the Goldstone modes acquire a positive definite mass term due to the screening effects of interaction. Possible infrared divergence is self-regulated by the mass term. Furthermore, there is a symmetry-restoring phase transition as a function of the Hubble parameter. In our approximation, the phase transition is of first order.

Gauge theories in anti-selfdual variables

Some years ago the Nicolai map, viewed as a change of variables from the gauge connection in a fixed gauge to the anti-selfdual part of the curvature, has been extended by the first named author to pure Yang-Mills from its original definition in $ \mathcal{N} $ = 1 supersymmetric Yang-Mills. We study here the perturbative one-particle irreducible effective action in the anti-selfdual variables of any gauge theory, in particular pure Yang-Mills, QCD and $ \mathcal{N} $ = 1 supersymmetric Yang-Mills. We prove that the one-loop one-particle irreducible effective action of a gauge theory mapped to the anti-selfdual variables in any gauge is identical to the one of the original theory. This is due to the conspiracy between the Jacobian of the change to the anti-selfdual variables and an extra functional determinant that arises from the non-linearity of the coupling of the anti-selfdual curvature to an external source in the Legendre transform that defines the one-particle irreducible effective action. Hence we establish the one-loop perturbative equivalence of the mapped and original theories on the basis of the identity of the one-loop one-particle irreducible effective actions. Besides, we argue that the identity of the perturbative one-particle irreducible effective actions extends order by order in perturbation theory.

Shear viscosity in the Nambu–Jona-Lasinio model withΦ-derivable approximations

Within the Nambu--Jona-Lasinio model the temperature dependence of the shear viscosity is calculated to the first nontrivial order in the $1/{N}_{c}$ expansion with $\ensuremath{\Phi}$-derivable approximations. The two-particle irreducible effective action is computed to next-to-leading order, from which the integral equations for the 3- and 4-point vertices are obtained. These sum infinite sets of diagrams contributing to the shear viscosity at the same order in the $1/{N}_{c}$ expansion. We find that the shear viscosity decreases rapidly when the chiral crossover is approached. Comparing with the hadron phase, the quark-gluon plasma phase has a low shear viscosity, which is consistent with the measurements. The ratio of the shear viscosity to entropy density is also calculated.

The nonlinear glasma

We study the evolution of quantum fluctuations in the glasma, created immediately after the collision of heavy nuclei. We show how the presence of instabilities leads to an enhancement of nonlinear interactions among initially small fluctuations. The nonlinear dynamics leads to an enhanced growth of fluctuations in a large momentum region, exceeding by far the originally unstable band. We investigate the dependence on the coupling constant at weak coupling using classical statistical lattice simulations for $SU(2)$ gauge theory and show how these nonlinearities can be analytically understood within the framework of two-particle irreducible effective action techniques. The dependence on the coupling constant is only logarithmic, in accordance with analytic expectations. Concerning the isotropization of bulk quantities, our results indicate that the system exhibits an order-one anisotropy on parametrically large time scales. Despite this fact, we find that gauge invariant pressure correlation functions seem to exhibit a power law behavior characteristic for wave turbulence.

Nonperturbative infrared effects for light scalar fields in de Sitter space

We study the ϕ4 scalar field theory in de Sitter space using the two-particle irreducible effective action formalism. This formalism enables us to investigate some nonperturbative quantum aspects of the theory. We use the mean field and gap equations to calculate the physical mass and the effective potential. We find that nonperturbative infrared effects in de Sitter space produce a curvature-induced mass and work to restore the broken Z2 symmetry.

New method to calculate then-particle irreducible effective action

In this paper, we present a new method to calculate the n-Loop n-particle irreducible effective action. The key is an organizational trick that involves the introduction of a set of fictitious bare vertices that are set to zero at the end of the calculation. Using these fictitious vertices, we prove that the Schwinger-Dyson equations are the same as the equations of motion obtained from the n-particle irreducible effective action, up to the level at which they respect the symmetries of the original theory. This result allows us to obtain the effective action directly from the Schwinger-Dyson equations, which are comparatively easy to calculate. As a check of our method, we reproduce the known results for the n-Loop n-particle irreducible effective action with n = 4 and n = 5. We also use the technique to calculate the 6-Loop 6-particle irreducible effective action.

Functional renormalization group flow equation: Regularization and coarse-graining in phase space

Starting from the basic path integral in phase space we reconsider the functional approach to the renormalization group flow of the one particle irreducible effective average action. On employing a balanced coarse-graining procedure for the canonical variables we obtain a functional integral with a non trivial measure which leads to a modified flow equation. We first address quantum mechanics for boson and fermion degrees of freedom, and we then extend the construction to quantum field theories. For this modified flow equation we discuss the reconstruction of the bare action and the implications on the computation of the vacuum energy density.

Functional renormalization group for quantized anharmonic oscillator

Functional renormalization group methods formulated in the real-time formalism are applied to the O( N) symmetric quantum anharmonic oscillator, considered as a 0 + 1 dimensional quantum field-theoric model, in the next-to-leading order of the gradient expansion of the one- and two-particle irreducible effective action. The infrared scaling laws and the sensitivity-matrix analysis show the existence of only a single, symmetric phase. The Taylor expansion for the local potential converges fast while it is found not to work for the field-dependent wavefunction renormalization, in particular for the double-well bare potential. Results for the gap energy for the bare anharmonic oscillator potential hint on improving scheme-independence in the next-to-leading order of the gradient expansion, although the truncated perturbation expansion in the bare quartic coupling provides strongly scheme-dependent results for the infrared limits of the running couplings.

Far-from-equilibrium dynamics of an ultracold Fermi gas

Nonequilibrium dynamics of an N-fold spin-degenerate ultracold Fermi gas is described in terms of beyond-mean-field Kadanoff-Baym equations for correlation functions. Using a nonperturbative expansion in powers of 1/N, the equations are derived from the two-particle irreducible effective action in Schwinger-Keldysh formulation. The definition of the nonperturbative approximation on the level of the effective action ensures vital conservation laws as, e.g., for the total energy and particle number. As an example, the long-time evolution of a homogeneous, twofold spin-degenerate Fermi gas is studied in one spatial dimension after an initial preparation far from thermal equilibrium. Analysis of the fluctuation-dissipation relation shows that, at low energies, the gas does not thermalise.

Matter-wave turbulence: Beyond kinetic scaling

Turbulent scaling phenomena are studied in an ultracold Bose gas away from thermal equilibrium. Fixed points of the dynamical evolution are characterized in terms of universal scaling exponents of correlation functions. The scaling behavior is determined analytically in the framework of quantum field theory, using a nonperturbative approximation of the two-particle irreducible effective action. While perturbative Kolmogorov scaling is recovered at higher energies, scaling solutions with anomalously large exponents arise in the infrared regime of the turbulence spectrum. The extraordinary enhancement in the momentum dependence of long-range correlations could be experimentally accessible in dilute ultracold atomic gases. Such experiments have the potential to provide insight into dynamical phenomena directly relevant also in other present-day focus areas like heavy-ion collisions and early-universe cosmology.

Towards next-to-leading order transport coefficients from the four-particle irreducible effective action

Transport coefficients can be obtained from 2-point correlators using the Kubo formulae. It has been shown that the full leading order result for electrical conductivity and (QCD) shear viscosity is contained in the re-summed 2-point function that is obtained from the 3-loop 3PI re-summed effective action. The theory produces all leading order contributions without the necessity for power counting, and in this sense it provides a natural framework for the calculation. In this article we study the 4-loop 4PI effective action for a scalar theory with cubic and quartic interactions in the presence of spontaneous symmetry breaking. We obtain a set of integral equations that determine the re-summed 2-point vertex function. A next-to-leading order contribution to the viscosity could be obtained from this set of coupled equations.

Renormalized nonequilibrium quantum field theory: Scalar fields

We discuss the renormalisation of the initial value problem in quantum field theory using the two-particle irreducible (2PI) effective action formalism. The nonequilibrium dynamics is renormalised by counterterms determined in equilibrium. We emphasize the importance of the appropriate choice of initial conditions and go beyond the Gaussian initial density operator by defining self-consistent initial conditions. We study the corresponding time evolution and present a numerical example which supports the existence of a continuum limit for this type of initial conditions.

Leading order QCD shear viscosity from the three-particle irreducible effective action

In this article we calculate the leading order shear viscosity in QCD using the resummed 3PI effective action. We work to 3-loop order in the effective action. We show that the integral equations that resum the pinching and collinear contributions are produced naturally by the formalism. All leading order terms are included, without the need for any kind of power counting arguments.

Kaon condensation in the color–flavor-locked phase of quark matter, the Goldstone theorem, and the 2PI Hartree approximation

At very high densities, QCD is in the color–flavor-locked phase, which is a color-superconducting phase. The diquark condensates break chiral symmetry in the same way as it is broken in vacuum QCD and gives rise to an octet of pseudo-Goldstone bosons and a superfluid mode. The lightest of these are the charged and neutral kaons. For energies below the superconducting gap, the kaons are described by an O ( 2 ) × O ( 2 ) -symmetric effective scalar field theory with chemical potentials. We use this effective theory to study Bose-condensation of kaons and their properties as functions of the temperature and the chemical potentials. We use the 2-particle irreducible effective action formalism in the Hartree approximation. The renormalization of the gap equations and the effective potential is studied in detail and we show that the counterterms are independent of temperature and chemical potentials. We determine the phase diagram and the medium-dependent quasiparticle masses. It is shown that the Goldstone theorem is satisfied to a very good approximation. The effects of imposing electric charge neutrality is examined as well.

Fourth-order full quantum correlations from a Langevin–Schwinger–Dyson equation

It is well known that some quantum and statistical fluctuations of a quantum field may be recovered by adding suitable stochastic sources to the mean field equations derived from the Schwinger–Keldysh (closed-time path) effective action. In this paper we show that this method can be extended to higher correlations and higher (n-particle irreducible) effective actions. As an example, we investigate third- and fourth-order correlations by adding stochastic sources to the Schwinger–Dyson equations derived from the 2-particle irreducible effective action. This method is a simple way to investigate the nonlinear dynamics of quantum fluctuations.

Two-particle irreducible effective action approach to nonlinear current-conservingapproximations in driven systems

Using closed-time-path two-particle irreducible coarse-grained effective action (CTP 2PICGEA) techniques, we study the response of an open interacting electronic system totime-dependent external electromagnetic fields. We show that the CTP 2PI CGEA isinvariant under a simultaneous gauge transformation of the external field and the fullSchwinger–Keldysh propagator, and that this property holds even when the loopexpansion of the CTP 2PI CGEA is truncated at arbitrary order. The effective actionapproach provides a systematic way of calculating the propagator and responsefunctions of the system, via the Schwinger–Dyson equation and the Bethe–Salpeterequations, respectively. We show that, due to the invariance of the CTP 2PI CGEAunder external gauge transformations, the response functions calculated fromit satisfy the Ward–Takahashi hierarchy, thus warranting the conservation ofthe electronic current beyond the expectation value level. We also clarify theconnection between nonlinear response theory and the WT hierarchy, and discuss anexample of an ad hoc approximation that violates it. These findings may be usefulin the study of current fluctuations in correlated electronic pumping devices.

Ultracold gases far from equilibrium

Ultracold atomic quantum gases belong to the most exciting challenges of modern physics. Their theoretical description has drawn much from (semi- ) classical field equations. These mean-field approximations are in general reliable for dilute gases in which the atoms collide only rarely with each other, and for situations where the gas is not too far from thermal equilibrium. With present-day technology it is, however, possible to drive and observe a system far away from equilibrium. Functional quantum field theory provides powerful tools to achieve both, analytical understanding and numerical computability, also in higher dimensions, of far-from-equilibrium quantum many-body dynamics. In the article, an outline of these approaches is given, including methods based on the two-particle irreducible effective action as well as on renormalisation-group theory. Their relation to near-equilibrium kinetic theory is discussed, and the distinction between quantum and classical statistical fluctuations is shown to naturally emerge from the functional-integral description. Example applications to the evolution of an ultracold atomic Bose gas in one spatial dimension underline the power of the methods.

Effective convergence of the two-particle irreducible1/Nexpansion for nonequilibrium quantum fields

The $1/N$ expansion of the two-particle irreducible effective action offers a powerful approach to study quantum field dynamics far from equilibrium. We investigate the effective convergence of the $1/N$ expansion in the $O(N)$ model by comparing results obtained numerically in $1+1$ dimensions at leading, next-to-leading and next-to-next-to-leading order in $1/N$ as well as in the weak coupling limit. A comparison in classical statistical field theory, where exact numerical results are available, is made as well. We focus on early-time dynamics and quasiparticle properties far from equilibrium and observe rapid effective convergence already for moderate values of $1/N$ or the coupling.

Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach

Dynamic equations for quantum fields far from equilibrium are derived by use of functional renormalisation group techniques. The obtained equations are non-perturbative and lead substantially beyond mean-field and quantum Boltzmann type approximations. The approach is based on a regularised version of the generating functional for correlation functions where times greater than a chosen cutoff time are suppressed. As a central result, a time evolution equation for the non-equilibrium effective action is derived, and the time evolution of the Green functions is computed within a vertex expansion. It is shown that this agrees with the dynamics derived from the 1 / N -expansion of the two-particle irreducible effective action.