FLAVIO IASSUO TAKAKURA - Academia.edu (original) (raw)
Papers by FLAVIO IASSUO TAKAKURA
We introduce and study a pure gauge abelian theory in 2+1D in which massive quantum vortex states... more We introduce and study a pure gauge abelian theory in 2+1D in which massive quantum vortex states do exist in the spectrum of excitations. This theory can be mapped in a three dimensional gas of point particles with a logarithmic interaction, in the grand-canonical ensemble. We claim that this theory is the 2+1D analog of the Sine-Gordon, the massive vortices being the counterparts of Sine-Gordon solitons. We show that a symmetry breaking, order parameter, similar to the vacuum expectation value of a Higgs field does exist. Work supported in part by CNPq-Brazilian National Research Council.
Proceedings of International Workshop on Astroparticle and High Energy Physics — PoS(AHEP2003)
The anomaly-induced inflation (modified Starobinsky model) is based on the application of the eff... more The anomaly-induced inflation (modified Starobinsky model) is based on the application of the effective quantum field theory approach to the Early Universe. We present a brief general review of the model and show that it does not require a fine-tuning for the parameters of the theory or initial data, gives a real chance to meet a graceful exit to the FRW phase and also has positive features with respect to the metric perturbations.
The Tenth Marcel Grossmann Meeting, 2006
ABSTRACT We obtain a new self-similar solution to the Einstein's equations in four-dimens... more ABSTRACT We obtain a new self-similar solution to the Einstein's equations in four-dimensions, representing the collapse of a spherically symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of naked singularities and black holes. The black holes are identified as the Schwarzschild ones.
We obtain a new self-similar solution to the Einstein's equations in four-dimensions, represe... more We obtain a new self-similar solution to the Einstein's equations in four-dimensions, representing the collapse of a spherically symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of naked singularities and black holes. Since the black holes are identified as the Schwarzschild ones, one may naturally see how these black holes are produced as remnants of the scalar field collapse.
Hydrodynamic turbulence is studied as a constrained system from the point of view of metafluid dy... more Hydrodynamic turbulence is studied as a constrained system from the point of view of metafluid dynamics. We present a Lagrangian description for this new theory of turbulence inspired from the analogy with electromagnetism. Consequently it is a gauge theory. This new approach to the study of turbulence tends to renew the optimism to solve the difficult problem of turbulence. As a constrained system, turbulence is studied in the Dirac and Faddeev-Jackiw formalisms giving the Dirac brackets. An important result is that we show these brackets are the same in and out of the inertial range, giving the way to quantize turbulence.
Physical review D: Particles and fields, Jan 15, 1995
We solve the equations of motion for a scalar field with domain wall boundary conditions in a Fri... more We solve the equations of motion for a scalar field with domain wall boundary conditions in a Friedmann-Robertson-Walker (FRW) spacetime. We find (in agreement with Basu and Vilenkin) that no domain wall solutions exist in de Sitter spacetime for h ≡ H/m ≥ 1/2, where H is the Hubble parameter and m is the scalar mass. In the general FRW case we develop a systematic perturbative expansion in h to arrive at an approximate solution to the field equations. We calculate the energy momentum tensor of the domain wall configuration, and show that the energy density can become negative at the core of the defect for some values of the non-minimal coupling parameter ξ. We develop a translationally invariant theory for fluctuations of the wall, obtain the effective Lagrangian for these fluctuations, and quantize them using the Bunch-Davies vacuum in the de Sitter case. Unlike previous analyses, we find that the fluctuations act as zero-mass (as opposed to tachyonic) modes. This allows us to calculate the distortion and the normal-normal correlators for the surface. The normal-normal correlator decreases logarithmically with the distance between points for large times and distances, indicating that the interface becomes rougher than in Minkowski spacetime.
Physical Review E, 1995
The response and nonconserved dynamics of a two-phase interface in the presence of a temperature ... more The response and nonconserved dynamics of a two-phase interface in the presence of a temperature gradient oriented normally to the interface are considered. Two types of boundary conditions on the order parameter are considered, and the structure of the effective free energy and the Langevin equation for the collective coordinate specifying the interface position are analyzed.
Physical Review B, 1998
We study the non-equilibrium dynamics of solitons in model Hamiltonians for Peierls dimerized qua... more We study the non-equilibrium dynamics of solitons in model Hamiltonians for Peierls dimerized quasi-one dimensional conducting polymers and commensurate charge density wave systems. The real time equation of motion for the collective coordinate of the soliton and the associated Langevin equation is found in a consistent adiabatic expansion in terms of the ratio of the optical phonon or phason frequency to the soliton mass. The equation of motion for the soliton collective coordinate allows to obtain the frequency dependent soliton conductivity. In lowest order we find that although the coefficient of static friction vanishes, there is dynamical dissipation represented by a non-Markovian dissipative kernel associated with two-phonon processes. The correlation function of the noise in the quantum Langevin equation and the dissipative kernel are related by a generalized quantum fluctuation dissipation relation. To lowest adiabatic order we find that the noise is gaussian, additive and colored. We numerically solve the equations of motion in lowest adiabatic order and compare to the Markovian approximation which is shown to fail both in the φ 4 and the Sine Gordon models even at large temperatures.
Physical Review E, 1998
We study the large λ limit of the loop-dependent characteristic functional Z(λ) =< exp(iλ c v • d... more We study the large λ limit of the loop-dependent characteristic functional Z(λ) =< exp(iλ c v • d x) >, related to the probability density function (PDF) of the circulation around a closed contour c. The analysis is carried out in the framework of the Martin-Siggia-Rose field theory formulation of the turbulence problem, by means of the saddle-point technique. Axisymmetric instantons, labelled by the component σzz of the strain field-a partially annealed variable in our formalism-are obtained for a circular loop in the xy plane, with radius defined in the inertial range. Fluctuations of the velocity field around the saddle-point solutions are relevant, leading to the lorentzian asymptotic behavior Z(λ) ∼ 1/λ 2. The O(1/λ 4) subleading correction and the asymmetry between right and left PDF tails due to parity breaking mechanisms are also investigated.
Physical Review E, 2001
We study the nonequilibrium dynamics of a charge interacting with its own radiation, which origin... more We study the nonequilibrium dynamics of a charge interacting with its own radiation, which originates the radiation damping. The real-time equation of motion for the charge and the associated Langevin equation is found in classical limit. The equation of motion for the charge allows one to obtain the frequency-dependent coefficient of friction. In the lowest order we find that although the coefficient of static friction vanishes, there is dynamical dissipation represented by a non-Markovian dissipative kernel.
Physics Letters A, 2010
In this work we study Lagrangian formulations for the noncommutative versions of the irrotational... more In this work we study Lagrangian formulations for the noncommutative versions of the irrotational and rotational fluid models. These formulations will be obtained by using the Faddeev-Jackiw symplectic formalism. In this context, interesting results will be revealed, for example, a chiral behavior into both fluid models comes up. In fact, distinct and non-dynamically equivalent Lagrangian descriptions of the noncommutative versions of the fluid models can be proposed.
Physical Review D, 1992
ABSTRACT Quantum corrections to the tachyon amplitude are computed perturbatively using Polyakov&... more ABSTRACT Quantum corrections to the tachyon amplitude are computed perturbatively using Polyakov's method of quantization for the bosonic string. The tachyonic pole is eliminated for dimension {ital D}{ge}13.
Physical Review D, 1998
We study the non-equilibrium dynamics of domain walls in real time for φ 4 and Sine Gordon models... more We study the non-equilibrium dynamics of domain walls in real time for φ 4 and Sine Gordon models in 1 + 1 dimensions in the dilute regime. The equation of motion for the collective coordinate is obtained by integrating out the meson excitations around the domain wall to oneloop order. The real-time non-equilibrium relaxation is studied analytically and numerically to this order. The constant friction coefficient vanishes but there is dynamical friction and relaxation caused by off-shell non-Markovian effects. The validity of a Markovian description is studied in detail. The proper Langevin equation is obtained to this order, the noise is Gaussian and additive but colored. We analyze the classical and hard thermal loop contributions to the self-energy and noise kernels and show that at temperatures larger than the meson mass the hard contributions are negligible and the finite temperature contribution to the dynamics is governed by the classical soft modes of the meson bath. The long time relaxational dynamics is completely dominated by classical Landau damping resulting in that the corresponding time scales are not set by the temperature but by the meson mass. The noise correlation function and the dissipative kernel obey a generalized form of the Fluctuation-Dissipation relation.
Optics Communications, 1999
We consider the coupling of the electromagnetic vacuum field with an oscillating perfectlyreflect... more We consider the coupling of the electromagnetic vacuum field with an oscillating perfectlyreflecting mirror in the nonrelativistic approximation. As a consequence of the frequency modulation associated to the motion of the mirror, low frequency photons are generated. We calculate the photon emission rate by following a nonperturbative approach, in which the coupling between the field sidebands is taken into account. We show that the usual perturbation theory fails to account correctly for the contribution of TM-polarized vacuum fluctuations that propagate along directions nearly parallel to the plane surface of the mirror. As a result of the modification of the field eigenfunctions, the resonance frequency for photon emission is shifted from its unperturbed value.
Nuclear Physics B - Proceedings Supplements, 2004
We discuss the stability of the anomaly-induced inflation (modified Starobinsky model) with respe... more We discuss the stability of the anomaly-induced inflation (modified Starobinsky model) with respect to the arbitrary choice of initial data and with respect to the small perturbations of the conformal factor and tensor modes of the metric in the later period of inflation and, partially, in the present Universe.
Nuclear Physics B - Proceedings Supplements, 2004
The behaviour of gravitational waves in the anomaly-induced inflationary phase is studied. The me... more The behaviour of gravitational waves in the anomaly-induced inflationary phase is studied. The metric perturbations exhibit a stable behaviour, with a very moderate growth in the amplitude of the waves. The spectral indice is computed, revealing an almost flat spectrum.
Nuclear Physics B, 2003
We analyze various phases of inflation based on the anomaly-induced effective action of gravity (... more We analyze various phases of inflation based on the anomaly-induced effective action of gravity (modified Starobinsky model), taking the cosmological constant Λ and k = 0, ±1 topologies into account. The total number of the inflationary e-folds may be enormous, but at the last 65 of them the inflation greatly slows down due to the contributions of the massive particles. For the supersymmetric particle content, the stability of inflation holds from the initial point at the sub-Planck scale until the supersymmetry breaks down. After that the universe enters into the unstable regime with the eventual transition into the stable FRW-like evolution with small positive cosmological constant. It is remarkable, that all this follows automatically, without fine-tuning of any sort, independent on the values of Λ and k. Finally, we consider the stability under the metric perturbations during the last 65 e-folds of inflation and find that the amplitude of the ones with the wavenumber below a certain cutoff have an acceptable range.
Modern Physics Letters A, 2008
Improving the beginning steps of a previous work, we settle the dual embedding method (DEM) as an... more Improving the beginning steps of a previous work, we settle the dual embedding method (DEM) as an alternative and efficient method for obtaining dual equivalent actions also in D = 3. We show that we can obtain dual equivalent actions which were previously obtained in the literature using the gauging iterative Noether dualization method (NDM). We believe that, with the arbitrariness property of the zero mode, the DEM is more profound since it can reveal a whole family of dual equivalent actions. The result confirms the one obtained previously which is important since it has the same structure that appears in the Abelian Higgs model with an anomalous magnetic interaction.
Journal of Physics A: Mathematical and General, 2005
We construct a supersymmetrized version of the model to the radiation damping [1] introduced by t... more We construct a supersymmetrized version of the model to the radiation damping [1] introduced by the present authors [2]. We dicuss its symmetries and the corresponding conserved Noether charges. It is shown this supersymmetric version provides a supersymmetric generalization of the Galilei algebra obtained in [2]. We have shown that the supersymmetric action can be splited into dynamically independent external and internal sectors.
Journal of Physics A: Mathematical and General, 2005
... 0 = ˜ν(0) ˜θ ∂ ˜V (0)(ai,pi, η) ∂ ˜ξ(0) ˜θ = µθ ∂V (0)(ai,pi) ∂ξ(0)θ + µθ ∂G(1)(ai,pi, η) ∂ξ(... more ... 0 = ˜ν(0) ˜θ ∂ ˜V (0)(ai,pi, η) ∂ ˜ξ(0) ˜θ = µθ ∂V (0)(ai,pi) ∂ξ(0)θ + µθ ∂G(1)(ai,pi, η) ∂ξ(0)θ + µθ ∂G(2)(ai,pi, η) ∂ξ(0)θ + ··· + G(1)(ai,pi, η) ∂η + ∂G(2)(ai,pi, η) ∂η + ··· , 21) ... For a quadratic correction term, we get µθ ∂G(1)(ai,pi, η) ∂ξ(0)θ + ∂G(2)(ai,pi, η) ∂η = 0. (23) ...
We introduce and study a pure gauge abelian theory in 2+1D in which massive quantum vortex states... more We introduce and study a pure gauge abelian theory in 2+1D in which massive quantum vortex states do exist in the spectrum of excitations. This theory can be mapped in a three dimensional gas of point particles with a logarithmic interaction, in the grand-canonical ensemble. We claim that this theory is the 2+1D analog of the Sine-Gordon, the massive vortices being the counterparts of Sine-Gordon solitons. We show that a symmetry breaking, order parameter, similar to the vacuum expectation value of a Higgs field does exist. Work supported in part by CNPq-Brazilian National Research Council.
Proceedings of International Workshop on Astroparticle and High Energy Physics — PoS(AHEP2003)
The anomaly-induced inflation (modified Starobinsky model) is based on the application of the eff... more The anomaly-induced inflation (modified Starobinsky model) is based on the application of the effective quantum field theory approach to the Early Universe. We present a brief general review of the model and show that it does not require a fine-tuning for the parameters of the theory or initial data, gives a real chance to meet a graceful exit to the FRW phase and also has positive features with respect to the metric perturbations.
The Tenth Marcel Grossmann Meeting, 2006
ABSTRACT We obtain a new self-similar solution to the Einstein's equations in four-dimens... more ABSTRACT We obtain a new self-similar solution to the Einstein's equations in four-dimensions, representing the collapse of a spherically symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of naked singularities and black holes. The black holes are identified as the Schwarzschild ones.
We obtain a new self-similar solution to the Einstein's equations in four-dimensions, represe... more We obtain a new self-similar solution to the Einstein's equations in four-dimensions, representing the collapse of a spherically symmetric, minimally coupled, massless, scalar field. Depending on the value of certain parameters, this solution represents the formation of naked singularities and black holes. Since the black holes are identified as the Schwarzschild ones, one may naturally see how these black holes are produced as remnants of the scalar field collapse.
Hydrodynamic turbulence is studied as a constrained system from the point of view of metafluid dy... more Hydrodynamic turbulence is studied as a constrained system from the point of view of metafluid dynamics. We present a Lagrangian description for this new theory of turbulence inspired from the analogy with electromagnetism. Consequently it is a gauge theory. This new approach to the study of turbulence tends to renew the optimism to solve the difficult problem of turbulence. As a constrained system, turbulence is studied in the Dirac and Faddeev-Jackiw formalisms giving the Dirac brackets. An important result is that we show these brackets are the same in and out of the inertial range, giving the way to quantize turbulence.
Physical review D: Particles and fields, Jan 15, 1995
We solve the equations of motion for a scalar field with domain wall boundary conditions in a Fri... more We solve the equations of motion for a scalar field with domain wall boundary conditions in a Friedmann-Robertson-Walker (FRW) spacetime. We find (in agreement with Basu and Vilenkin) that no domain wall solutions exist in de Sitter spacetime for h ≡ H/m ≥ 1/2, where H is the Hubble parameter and m is the scalar mass. In the general FRW case we develop a systematic perturbative expansion in h to arrive at an approximate solution to the field equations. We calculate the energy momentum tensor of the domain wall configuration, and show that the energy density can become negative at the core of the defect for some values of the non-minimal coupling parameter ξ. We develop a translationally invariant theory for fluctuations of the wall, obtain the effective Lagrangian for these fluctuations, and quantize them using the Bunch-Davies vacuum in the de Sitter case. Unlike previous analyses, we find that the fluctuations act as zero-mass (as opposed to tachyonic) modes. This allows us to calculate the distortion and the normal-normal correlators for the surface. The normal-normal correlator decreases logarithmically with the distance between points for large times and distances, indicating that the interface becomes rougher than in Minkowski spacetime.
Physical Review E, 1995
The response and nonconserved dynamics of a two-phase interface in the presence of a temperature ... more The response and nonconserved dynamics of a two-phase interface in the presence of a temperature gradient oriented normally to the interface are considered. Two types of boundary conditions on the order parameter are considered, and the structure of the effective free energy and the Langevin equation for the collective coordinate specifying the interface position are analyzed.
Physical Review B, 1998
We study the non-equilibrium dynamics of solitons in model Hamiltonians for Peierls dimerized qua... more We study the non-equilibrium dynamics of solitons in model Hamiltonians for Peierls dimerized quasi-one dimensional conducting polymers and commensurate charge density wave systems. The real time equation of motion for the collective coordinate of the soliton and the associated Langevin equation is found in a consistent adiabatic expansion in terms of the ratio of the optical phonon or phason frequency to the soliton mass. The equation of motion for the soliton collective coordinate allows to obtain the frequency dependent soliton conductivity. In lowest order we find that although the coefficient of static friction vanishes, there is dynamical dissipation represented by a non-Markovian dissipative kernel associated with two-phonon processes. The correlation function of the noise in the quantum Langevin equation and the dissipative kernel are related by a generalized quantum fluctuation dissipation relation. To lowest adiabatic order we find that the noise is gaussian, additive and colored. We numerically solve the equations of motion in lowest adiabatic order and compare to the Markovian approximation which is shown to fail both in the φ 4 and the Sine Gordon models even at large temperatures.
Physical Review E, 1998
We study the large λ limit of the loop-dependent characteristic functional Z(λ) =< exp(iλ c v • d... more We study the large λ limit of the loop-dependent characteristic functional Z(λ) =< exp(iλ c v • d x) >, related to the probability density function (PDF) of the circulation around a closed contour c. The analysis is carried out in the framework of the Martin-Siggia-Rose field theory formulation of the turbulence problem, by means of the saddle-point technique. Axisymmetric instantons, labelled by the component σzz of the strain field-a partially annealed variable in our formalism-are obtained for a circular loop in the xy plane, with radius defined in the inertial range. Fluctuations of the velocity field around the saddle-point solutions are relevant, leading to the lorentzian asymptotic behavior Z(λ) ∼ 1/λ 2. The O(1/λ 4) subleading correction and the asymmetry between right and left PDF tails due to parity breaking mechanisms are also investigated.
Physical Review E, 2001
We study the nonequilibrium dynamics of a charge interacting with its own radiation, which origin... more We study the nonequilibrium dynamics of a charge interacting with its own radiation, which originates the radiation damping. The real-time equation of motion for the charge and the associated Langevin equation is found in classical limit. The equation of motion for the charge allows one to obtain the frequency-dependent coefficient of friction. In the lowest order we find that although the coefficient of static friction vanishes, there is dynamical dissipation represented by a non-Markovian dissipative kernel.
Physics Letters A, 2010
In this work we study Lagrangian formulations for the noncommutative versions of the irrotational... more In this work we study Lagrangian formulations for the noncommutative versions of the irrotational and rotational fluid models. These formulations will be obtained by using the Faddeev-Jackiw symplectic formalism. In this context, interesting results will be revealed, for example, a chiral behavior into both fluid models comes up. In fact, distinct and non-dynamically equivalent Lagrangian descriptions of the noncommutative versions of the fluid models can be proposed.
Physical Review D, 1992
ABSTRACT Quantum corrections to the tachyon amplitude are computed perturbatively using Polyakov&... more ABSTRACT Quantum corrections to the tachyon amplitude are computed perturbatively using Polyakov's method of quantization for the bosonic string. The tachyonic pole is eliminated for dimension {ital D}{ge}13.
Physical Review D, 1998
We study the non-equilibrium dynamics of domain walls in real time for φ 4 and Sine Gordon models... more We study the non-equilibrium dynamics of domain walls in real time for φ 4 and Sine Gordon models in 1 + 1 dimensions in the dilute regime. The equation of motion for the collective coordinate is obtained by integrating out the meson excitations around the domain wall to oneloop order. The real-time non-equilibrium relaxation is studied analytically and numerically to this order. The constant friction coefficient vanishes but there is dynamical friction and relaxation caused by off-shell non-Markovian effects. The validity of a Markovian description is studied in detail. The proper Langevin equation is obtained to this order, the noise is Gaussian and additive but colored. We analyze the classical and hard thermal loop contributions to the self-energy and noise kernels and show that at temperatures larger than the meson mass the hard contributions are negligible and the finite temperature contribution to the dynamics is governed by the classical soft modes of the meson bath. The long time relaxational dynamics is completely dominated by classical Landau damping resulting in that the corresponding time scales are not set by the temperature but by the meson mass. The noise correlation function and the dissipative kernel obey a generalized form of the Fluctuation-Dissipation relation.
Optics Communications, 1999
We consider the coupling of the electromagnetic vacuum field with an oscillating perfectlyreflect... more We consider the coupling of the electromagnetic vacuum field with an oscillating perfectlyreflecting mirror in the nonrelativistic approximation. As a consequence of the frequency modulation associated to the motion of the mirror, low frequency photons are generated. We calculate the photon emission rate by following a nonperturbative approach, in which the coupling between the field sidebands is taken into account. We show that the usual perturbation theory fails to account correctly for the contribution of TM-polarized vacuum fluctuations that propagate along directions nearly parallel to the plane surface of the mirror. As a result of the modification of the field eigenfunctions, the resonance frequency for photon emission is shifted from its unperturbed value.
Nuclear Physics B - Proceedings Supplements, 2004
We discuss the stability of the anomaly-induced inflation (modified Starobinsky model) with respe... more We discuss the stability of the anomaly-induced inflation (modified Starobinsky model) with respect to the arbitrary choice of initial data and with respect to the small perturbations of the conformal factor and tensor modes of the metric in the later period of inflation and, partially, in the present Universe.
Nuclear Physics B - Proceedings Supplements, 2004
The behaviour of gravitational waves in the anomaly-induced inflationary phase is studied. The me... more The behaviour of gravitational waves in the anomaly-induced inflationary phase is studied. The metric perturbations exhibit a stable behaviour, with a very moderate growth in the amplitude of the waves. The spectral indice is computed, revealing an almost flat spectrum.
Nuclear Physics B, 2003
We analyze various phases of inflation based on the anomaly-induced effective action of gravity (... more We analyze various phases of inflation based on the anomaly-induced effective action of gravity (modified Starobinsky model), taking the cosmological constant Λ and k = 0, ±1 topologies into account. The total number of the inflationary e-folds may be enormous, but at the last 65 of them the inflation greatly slows down due to the contributions of the massive particles. For the supersymmetric particle content, the stability of inflation holds from the initial point at the sub-Planck scale until the supersymmetry breaks down. After that the universe enters into the unstable regime with the eventual transition into the stable FRW-like evolution with small positive cosmological constant. It is remarkable, that all this follows automatically, without fine-tuning of any sort, independent on the values of Λ and k. Finally, we consider the stability under the metric perturbations during the last 65 e-folds of inflation and find that the amplitude of the ones with the wavenumber below a certain cutoff have an acceptable range.
Modern Physics Letters A, 2008
Improving the beginning steps of a previous work, we settle the dual embedding method (DEM) as an... more Improving the beginning steps of a previous work, we settle the dual embedding method (DEM) as an alternative and efficient method for obtaining dual equivalent actions also in D = 3. We show that we can obtain dual equivalent actions which were previously obtained in the literature using the gauging iterative Noether dualization method (NDM). We believe that, with the arbitrariness property of the zero mode, the DEM is more profound since it can reveal a whole family of dual equivalent actions. The result confirms the one obtained previously which is important since it has the same structure that appears in the Abelian Higgs model with an anomalous magnetic interaction.
Journal of Physics A: Mathematical and General, 2005
We construct a supersymmetrized version of the model to the radiation damping [1] introduced by t... more We construct a supersymmetrized version of the model to the radiation damping [1] introduced by the present authors [2]. We dicuss its symmetries and the corresponding conserved Noether charges. It is shown this supersymmetric version provides a supersymmetric generalization of the Galilei algebra obtained in [2]. We have shown that the supersymmetric action can be splited into dynamically independent external and internal sectors.
Journal of Physics A: Mathematical and General, 2005
... 0 = ˜ν(0) ˜θ ∂ ˜V (0)(ai,pi, η) ∂ ˜ξ(0) ˜θ = µθ ∂V (0)(ai,pi) ∂ξ(0)θ + µθ ∂G(1)(ai,pi, η) ∂ξ(... more ... 0 = ˜ν(0) ˜θ ∂ ˜V (0)(ai,pi, η) ∂ ˜ξ(0) ˜θ = µθ ∂V (0)(ai,pi) ∂ξ(0)θ + µθ ∂G(1)(ai,pi, η) ∂ξ(0)θ + µθ ∂G(2)(ai,pi, η) ∂ξ(0)θ + ··· + G(1)(ai,pi, η) ∂η + ∂G(2)(ai,pi, η) ∂η + ··· , 21) ... For a quadratic correction term, we get µθ ∂G(1)(ai,pi, η) ∂ξ(0)θ + ∂G(2)(ai,pi, η) ∂η = 0. (23) ...