Higher Order Lagrangians Research Articles

Direct CP violation in K-->3 pi decay.

In light of the fact that the presence of the ${Z}^{0}$ penguin diagram suppresses strongly $\frac{{\ensuremath{\epsilon}}_{2\ensuremath{\pi}}^{\ensuremath{'}}}{\ensuremath{\epsilon}}$ in ${K}^{0}\ensuremath{\rightarrow}2\ensuremath{\pi}$ decay for large values of ${m}_{t}$, we reanalyze the direct decay-amplitude $\mathrm{CP}$ violation in ${K}^{0}\ensuremath{\rightarrow}3\ensuremath{\pi}$ decays in the Kobayashi-Maskawa model. The effects due to electroweak penguins, isospin breaking, and higher-order weak chiral Lagrangians are studied in the large-${N}_{c}$ approach. We find that the Li-Wolfenstein relation between ${\ensuremath{\epsilon}}_{3\ensuremath{\pi}}^{\ensuremath{'}}$ and ${\ensuremath{\epsilon}}_{2\ensuremath{\pi}}^{\ensuremath{'}}$ is modified dramatically: the former receives very large contributions from the higher-derivative chiral terms and sizable contributions from the isospin-breaking correction due to ${\ensuremath{\pi}}^{0}\ensuremath{-}\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}^{\ensuremath{'}}$ mixing and $\ensuremath{\eta}$, ${\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\rightarrow}3\ensuremath{\pi}$ transitions. When the top quark becomes very heavy, effects of the electroweak penguin terms are enhanced. Unlike $\frac{{\ensuremath{\epsilon}}_{2\ensuremath{\pi}}^{\ensuremath{'}}}{\ensuremath{\epsilon}}$, which decreases as ${m}_{t}$ increases, the $\mathrm{CP}$-violating parameter $\frac{{\ensuremath{\epsilon}}_{3\ensuremath{\pi}}^{\ensuremath{'}}}{\ensuremath{\epsilon}}$ is of order ${10}^{\ensuremath{-}2}$ and increases with the heavy top-quark mass.

Treatment of higher-order Lagrangians via the construction of dynamically equivalent first-order Lagrangians

For a given, in general, singular Lagrangian containing higher-order time derivatives, a dynamically equivalent Lagrangian with only first-order time derivatives is constructed. A Hamiltonian structure for this first-order Lagrangian is then found with the use of the Dirac theory of constraints. It is shown that in the case of a nonsingular higher-order Lagrangian, the Ostrogradsky dynamics is derived in this way. Further, it is shown that ambiguities characteristic of higher-order Lagrangian systems do not appear when using this construction. In particular, it is shown that the addition of a total time derivative term to the higher-order Lagrangian can only induce a time-independent canonical transformation, even in the case of a singular Lagrangian.

Redefinition of position variables and the reduction of higher-order Lagrangians

Single-time Lagrangians are treated in this paper, describing the dynamics of systems of point particles, which are given as formal power series in some ordering parameter and which may contain higher time derivatives in all terms but the leading one. An efficient method for eliminating the higher time derivatives directly on the Lagrangian level is presented. This method clarifies the meaning of using the lower-order equations of motion in higher-order terms in a Lagrangian. The method consists of an iterative use of ‘‘contact’’ transformations in the jet prolongation of the extended configurations space and is called ‘‘the method of redefinition of position variables.’’ Several examples from electrodynamics and relativistic gravity are treated explicitly.

Large-Nc higher-order weak chiral Lagrangians coupled to external electromagnetic fields: Applications to radiative kaon decays.

Using factorization and bosonization of quark currents derived from the anomalous Wess-Zumino-Witten terms and from the electroweak perturbations to a QCD-motivated model for nonanomalous ${p}^{4}$ chiral Lagrangians, we obtain leading $\frac{1}{{N}_{c}}$ fourth-order $\ensuremath{\Delta}S=1$ weak chiral Lagrangians coupled to external electromagnetic fields, valid to the zeroth order of gluonic corrections. Applications to radiative $K$ decays, e.g., $K\ensuremath{\rightarrow}\ensuremath{\pi}{\ensuremath{\gamma}}^{*}\ensuremath{\rightarrow}\ensuremath{\pi}{l}^{+}{l}^{\ensuremath{-}}$, $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\gamma}\ensuremath{\gamma}$, $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\gamma}$, and $K\ensuremath{\rightarrow}\ensuremath{\pi}l\ensuremath{\nu}\ensuremath{\gamma}$, are studied. The twofold ambiguity for the predictions of $K\ensuremath{\rightarrow}\ensuremath{\pi}{l}^{+}{l}^{\ensuremath{-}}$ is suggestively resolved. The branching ratio of ${K}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\gamma}\ensuremath{\gamma}$ is predicted to be 5.1 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}7}$, while the current upper bound based on a pion phase-space spectrum is ${10}^{\ensuremath{-}6}$. The direct ${K}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{0}\ensuremath{\gamma}$ decays offer an excellent test on both anomalous and nonanomalous weak chiral Lagrangians; the agreement between theory and experiment is marvelous. The branching ratio of the interference between innerbremsstrahlung and direct-emission amplitudes in the process ${K}_{L}\ensuremath{\rightarrow}\ensuremath{\pi}e\ensuremath{\nu}\ensuremath{\gamma}$ is found to be of order ${10}^{\ensuremath{-}5}$.

K → πππ decays in large Nc chiral perturbation theory

Corrections to the current-algebra analysis of ΔI = 1 2 and 3 2 K → πππ decay amplitudes including K s→ π + π − π 0 are calculated using the large N c higher-order effective chiral lagrangians for strong and weak interactions. Neglecting gluonic corrections to the coupling constants of the higher-derivative chiral lagrangians, we find an agreement between theory and experiment within 30% for the constant and linear terms of ΔI = 1 2 transitions. Quadratic coefficients of various ΔI = 3 2 amplitudes in the Dalitz plot are predicted. The branching ratio and the slope parameter of K s → π + π − π 0 are found to be 3.9 × 10 −7 and 4.4 × 10 −2, respectively.

Analysis ofK?3? decays in chiral perturbation theory

Using the recently proposed higher-order chiral Lagrangians determined from the integration of nontopological chiral anomalies, we calculate corrections to the current-algebra analysis ofK→3Π decay amplitudes expanded in powers of the Dalitz variables. Effects of quartic-derivative weak chiral Lagrangians are determined through the use of short-distance effective weak Hamiltonian and the factorization method. We find that (1) the constant and linear terms in the amplitude for ΔI=1/2K→3Π are in excellent agreement with experiment; the previous discrepancy of (20–35)% between current algebra and data is thus accounted for by the higher-order effective Lagrangians, (2) the penguin interaction does not play an essential role in the ΔI=1/2 rule, for otherwise it will lead to a large disagreement for the constant and linear terms, (3) one of the two quadratic terms in the ΔI=1/2 process, which arise from the quartic chiral Lagrangians, is in accord with data within experimental errors, while the other is off by four standard deviations, (4) the linear term in the ΔI=3/2 transitions is in good agreement with experiment and contributions from quadratic terms are sizable.

Equivalence of higher-order Lagrangians. II. The Cartan form for particle Lagrangians

It is shown how Cartan’s method of equivalence may be used to obtain the Cartan form for an r th-order particle Lagrangian on the line by solving the standard equivalence problem under contact transformations on the jet bundle J r+k for k≥r−1.

Ostrogradski's theorem for higher-order singular Lagrangians

A demonstration is given of the equivalence of Euler-Lagrange and Hamilton-Dirac equations for constrained systems derived from singular Lagrangians of higher order in the derivatives.

Lagrangian and Hamiltonian constraints for second-order singular Lagrangians

The authors study the different kinds of constraints which appear when one deals with singular Lagrangians depending on second-order derivatives. We characterise Ker FL* and deduce the generalised Hamilton-Dirac equations of motion. The operators relating the Hamiltonian and the Lagrangian constraints are displayed. They extend their results to higher-order singular Lagrangians.

Characteristic functions and higher order Lagrangians

In the context of the extension of the Hamilton–Jacobi theory to include Lagrangians involving higher derivatives the characteristic function seems not to have been considered. This omission is here rectified.

On the problem of stability for higher-order derivative Lagrangian systems

The problem of stability for dynamical systems whose Lagrangian function depends on the derivatives of a higher order than one is studied. The difficulty of this analysis arises from the indefiniteness of the Hamiltonian, so that the well-known Lagrange-Dirichlet theorem cannot be used and the methods of the canonical perturbation theory (KAM theory) must be employed. We show, with an example, that the indefiniteness of the energy does not forbid the stability.

On a Weyl-type theorem for higher-order Lagrangians

Weyl’s theorem is extended making use of the theory of concomitants to obtain a Lagrangian density for the massless bosonic fields without dimensional constants. It turns out to be quadratic in the gravitational field and encompasses all the theories that usually appear in the literature. It is shown that the gauge invariance of the Lagrangian follows from the invariance of the field equations.

Lagrange multipliers and gravitational theory

The Lagrange multiplier version of the Palatini variational principle is extended to nonlinear Lagrangians, where it is shown in the case of the quadratic Lagrangians, as expected, that this version of the Palatini approach is equivalent to the Hilbert variational method. The (nonvanishing) Lagrange multipliers for the quadratic Lagrangians are then explicitly obtained in covariant form. It is then pointed out how the Lagrange multiplier approach in the language of the (3+1) -formalism developed by Arnowitt, Deser, and Misner permits the recasting of the equations of motion for quadratic and general higher-order Lagrangians into the ADM canonical formalism. In general without the Lagrange multiplier approach, the higher order ADM problem could not be solved. This is done explicitly for the simplest quadratic Lagrangian (g1/2R2) as an example.

Cosmological singularities and higher-order gravitational lagrangians

General relativity is modified by adding terms proportional to R 2 and R μν R μν to the Lagrangian. One class of solutions of the modified field equations is free of singularities but does not lead to asymptotic behaviour (for large time) of the Friedmann type. A second class, which shows the correct asymptotic behaviour, does contain the usual singularities of Friedmann universes, collapse being modulated by small oscillations only. The quantum effects considered here are thus unable to prevent the occurrence of cosmological singularities under physically reasonable conditions.

Gravitational collapse and higher-order gravitational lagrangians

A general form of higher-order contributions (in R ij ) to the Einstein field equations is displayed. The additional terms may either stabilize or destabilize self-gravitating objects in gravitational collapse depending on the sign of the coefficient introducing the quadratic term. If the quadratic term is stabilizing, intertial mass can be converted to radiation with an efficiency approaching 100%, and arbitrarily large masses can be stabilized. On the other hand, the resultant field equations are pathological in that they admit gravitons with negative mass-squared (i.e., tachyons). A nonsingular class of vacuum solutions exist in general for the quadratic case (“grey dimples”).

A note on the relativistic invariance of quantized field theories

In an earlier paper (1), which will be referred to as A, the present author has demonstrated the relativistic invariance, for general transformations of coordinates, of the Einstein-Bose and Fermi-Dirac quantizations of linear field equations derived from higher order Lagrangians. The proof consisted of the identification of the commutation relations with the generalized Poisson brackets introduced by Weiss (2) and proving the invariance of the latter.

On the relativistic invariance of quantized field theories

Heisenberg & Pauli (1929) have shown how to quantize field theories derived from Lagrangians containing first-order derivatives of the field quantities. They showed their quantization to be Lorentz invariant. Fuchs (1939) subsequently showed that the quantized theory was in fact invariant under general transformations of co-ordinates. The present author in another paper has shown how the theory of Heisenberg & Pauli can be extended to field equations derived from higher order Lagrangians, i. e. Lagrangians containing higher deri­vatives than the first of the field quantities. In the present paper the general relativistic invariance of the higher order quantized theories is established, making use of the generalized Poisson brackets introduced by Weiss.

On the quantization of field theories derived from higher order lagrangians

Heisenberg and Pauli (1) have shown how to quantize field theories derived from a Lagrangian containing first derivatives of the field quantities only. The present paper extends the theory of quantization of fields to the case of higher order Lagrangians, i.e. Lagrangians in which higher derivatives than the first appear. It is shown how such field equations can be put into Hamiltonian form and how the quantization can subsequently be carried out. Both the cases of Einstein-Bose and Fermi-Dirac quantization are discussed. It is established that the quantization is relativistically invariant and consistent with the field equations. An interesting feature of the present theory is that the Hamiltonian proves to be different, in general, from the integral of the 4–4 component of the energy momentum tensor.