Covariant Expansion Research Articles

Aspects of nonlocality in quantum field theory, quantum gravity and cosmology

This paper contains a collection of essays on nonlocal phenomena in quantum field theory, gravity and cosmology. Mechanisms of nonlocal contributions to the quantum effective action are discussed within the covariant perturbation expansion in field strengths and spacetime curvatures. Euclidean version of the Schwinger–Keldysh technique for quantum expectation values is presented as a special rule of obtaining the nonlocal effective equations of motion for the mean quantum field from the Euclidean effective action. This rule is applied to a new model of ghost free nonlocal cosmology which can generate the de Sitter (dS) cosmological evolution at an arbitrary value of Λ — a model of dark energy with the dynamical scale selected by a kind of a scaling symmetry breaking mechanism. This model is shown to interpolate between the superhorizon phase of a scalar mediated gravity and the short distance general relativistic limit in a special metric frame related by a nonlocal conformal transformation to the original metric.

Computing the temperature dependence of effective CP violation in the standard model

CP violation in the standard model originates from the Cabibbo-Kobayashi-Maskawa mixing matrix. Upon integrating all fermions out of the theory, its effects are captured by a series of effective nonrenormalizable operators for the bosonic gauge and Higgs fields. We compute the CP-violating part of the effective action to the leading nontrivial, sixth order in the covariant gradient expansion as a function of temperature. In the limit of zero temperature, our result addresses the discrepancy between two independent calculations existing in the literature [1,2]. We find that CP violation in the standard model is strongly suppressed at high temperature, but that at T<1GeV it may be relevant for certain scenarios of baryogenesis. We also identify a selected class of operators at the next, eighth order and discuss the convergence of the covariant gradient expansion.

Temperature Dependence of Standard ModelCPViolation

We analyze the temperature dependence of CP violation effects in the standard model by determining the effective action of its bosonic fields, obtained after integrating out the fermions from the theory and performing a covariant gradient expansion. We find nonvanishing CP violating terms starting at the sixth order of the expansion, albeit only in the C-odd-P-even sector, with coefficients that depend on quark masses, Cabibbo-Kobayashi-Maskawa matrix elements, temperature and the magnitude of the Higgs field. The CP violating effects are observed to decrease rapidly with temperature, which has important implications for the generation of a matter-antimatter asymmetry in the early Universe. Our results suggest that the cold electroweak baryogenesis scenario may be viable within the standard model, provided the electroweak transition temperature is at most of order 1 GeV.

Transport equation approach to calculations of Hadamard Green functions and non-coincident DeWitt coefficients

Building on an insight due to Avramidi, we provide a system of transport equations for determining key fundamental bi-tensors, including derivatives of the world-function, \sigma(x,x'), the square root of the Van Vleck determinant, \Delta^{1/2}(x,x'), and the tail-term, V(x,x'), appearing in the Hadamard form of the Green function. These bi-tensors are central to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity. Their transport equations may be used either in a semi-recursive approach to determining their covariant Taylor series expansions, or as the basis of numerical calculations. To illustrate the power of the semi-recursive approach, we present an implementation in \textsl{Mathematica} which computes very high order covariant series expansions of these objects. Using this code, a moderate laptop can, for example, calculate the coincidence limit a_7(x,x) and V(x,x') to order (\sigma^a)^{20} in a matter of minutes. Results may be output in either a compact notation or in xTensor form. In a second application of the approach, we present a scheme for numerically integrating the transport equations as a system of coupled ordinary differential equations. As an example application of the scheme, we integrate along null geodesics to solve for V(x,x') in Nariai and Schwarzschild spacetimes.

Standard modelCPviolation and cold electroweak baryogenesis

Using large scale real-time lattice simulations, we calculate the baryon asymmetry generated at a fast, cold electroweak symmetry breaking transition. CP-violation is provided by the leading effective bosonic term resulting from integrating out the fermions in the Minimal Standard Model at zero temperature, and performing a covariant gradient expansion [1]. This is an extension of the work presented in [2]. The numerical implementation is described in detail, and we address issues specifically related to using this CP-violating term in the context of Cold Electroweak Baryogenesis. The results support the conclusion of [2], that Standard Model CP-violation may be able to reproduce the observed baryon asymmetry in the Universe in the context of Cold Electroweak Baryogenesis.

Schwinger-DeWitt technique for quantum effective action in brane induced gravity models

We develop the Schwinger-DeWitt technique for the covariant curvature expansion of the quantum effective action for brane induced gravity models in curved spacetime. This expansion has a part nonanalytic in DGP type scale parameter, leading to the cutoff scale which is given by the geometric average of the mass of the quantum field in the bulk and DGP scale. This cutoff is much higher than the analogous strong coupling scale of the DGP model treated by weak field expansion in the tree-level approximation. The lowest orders of this curvature expansion are calculated for the case of the scalar field in the $(d+1)$-dimensional bulk with the brane carrying the $d$-dimensional kinetic term of this field. The ultraviolet divergences in this model are obtained for a particular case of $d=4$.

THE ROLE OF TRANSLATIONAL INVARIANCE IN NONLINEAR GAUGE THEORIES OF GRAVITY

The internal structure of the tetrads in a Poincaré nonlinear gauge theory of gravity is considered. Minkowskian coordinates becomes dynamical degrees of freedom playing the role of Goldstone bosons of the translations. A critical length allowing a covariant expansion similar to the weak field approach is deduced, the zeroth order metric being maximally symmetric (Minkowskian in some cases).

Covariant generalization of cosmological perturbation theory

We present an approach to cosmological perturbations based on a covariant perturbative expansion between two worldlines in the real inhomogeneous universe. As an application, at an arbitrary order we define an exact scalar quantity which describes the inhomogeneities in the number of e-folds on uniform density hypersurfaces and which is conserved on all scales for a barotropic ideal fluid. We derive a compact form for its conservation equation at all orders and assign it a simple physical interpretation. To make a comparison with the standard perturbation theory, we develop a method to construct gauge-invariant quantities in a coordinate system at arbitrary order, which we apply to derive the form of the $n$th order perturbation in the number of e-folds on uniform density hypersurfaces and its exact evolution equation. On large scales, this provides the gauge-invariant expression for the curvature perturbation on uniform density hypersurfaces and its evolution equation at any order.

Nonlocal action for long-distance modifications of gravity theory

We construct the covariant nonlocal action for recently suggested long-distance modifications of gravity theory motivated by the cosmological constant and cosmological acceleration problems. This construction is based on the special nonlocal form of the Einstein–Hilbert action explicitly revealing the fact that this action within the covariant curvature expansion begins with curvature-squared terms. Nonlocal form factors in the action of both quantum and brane-induced nature are briefly discussed. In particular, it is emphasized that for certain class of quantum initial value problems nonlocal nature of the Euclidean action does not contradict the causality of effective equations of motion.

Noncommutative chiral anomaly and the Dirac-Ginsparg-Wilson operator

It is shown that the local axial anomaly in $2-$dimensions emerges naturally if one postulates an underlying noncommutative fuzzy structure of spacetime . In particular the Dirac-Ginsparg-Wilson relation on ${\bf S}^2_F$ is shown to contain an edge effect which corresponds precisely to the ``fuzzy'' $U(1)_A$ axial anomaly on the fuzzy sphere . We also derive a novel gauge-covariant expansion of the quark propagator in the form $\frac{1}{{\cal D}_{AF}}=\frac{a\hat{\Gamma}^L}{2}+\frac{1}{{\cal D}_{Aa}}$ where $a=\frac{2}{2l+1}$ is the lattice spacing on ${\bf S}^2_F$, $\hat{\Gamma}^L$ is the covariant noncommutative chirality and ${\cal D}_{Aa}$ is an effective Dirac operator which has essentially the same IR spectrum as ${\cal D}_{AF}$ but differes from it on the UV modes. Most remarkably is the fact that both operators share the same limit and thus the above covariant expansion is not available in the continuum theory . The first bit in this expansion $\frac{a\hat{\Gamma}^L}{2}$ although it vanishes as it stands in the continuum limit, its contribution to the anomaly is exactly the canonical theta term. The contribution of the propagator $\frac{1}{{\cal D}_{Aa}}$ is on the other hand equal to the toplogical Chern-Simons action which in two dimensions vanishes identically .

The 3He( e, e′ d) p reaction in ( q, ω)-constant kinematics

The cross section for the 3He( e, e′ d) p reaction has been measured as a function of the missing momentum p m in ( q, ω)-constant kinematics at beam energies of 370 and 576 MeV for values of the three-momentum transfer q of 412, 504 and 604 MeV/ c. The L( + TT), T and LT structure functions have been separated for q=412 and 504 MeV/ c. The data are compared to three-body Faddeev calculations, including meson-exchange currents (MEC), and to calculations based on a covariant diagrammatic expansion. The influence of final-state interactions and meson-exchange currents is discussed. The p m -dependence of the data is reasonably well described by all calculations. However, the most advanced Faddeev calculations, which employ the AV18 nucleon–nucleon interaction and include MEC, overestimate the measured cross sections, especially the longitudinal part, and at the larger values of q. The diagrammatic approach gives a fair description of the cross section, but under(over)estimates the longitudinal (transverse) structure function.

1+3 Covariant Cosmic Microwave Background Anisotropies I: Algebraic Relations for Mode and Multipole Expansions

This is the first of a series of papers systematically extending a 1+3 covariant and gauge-invariant treatment of kinetic theory in curved space-times to a treatment of cosmic microwave background temperature anisotropies arising from inhomogeneities in the early universe. The present paper deals with algebraic issues, both generically and in the context of models linearised about Robertson–Walker geometries. The approach represents radiation anisotropies by projected symmetric and trace-free tensors. The angular correlation functions for the mode coefficients are found in terms of these quantities, following the Wilson–Silk approach, but derived and dealt with in 1+3 covariant and gauge-invariant form. The covariant multipole and mode-expanded angular correlation functions are related to the usual treatments in the literature. The 1+3 covariant and gauge-invariant mode expansion is related to the coordinate approach by linking the Legendre functions to the projected symmetric trace-free representation, using a covariant addition theorem for the tensors to generate the Legendre polynomial recursion relation. This paper lays the foundation for further papers in the series, which use this formalism in a covariant and gauge-invariant approach to developing solutions of the Boltzmann and Liouville equations for the cosmic microwave background before and after decoupling, thus providing a unified covariant and gauge-invariant derivation of the variety of approaches to cosmic microwave background anisotropies in the current literature, as well as a basis for extension of the theory to include nonlinearities.

Nucleon generalized polarizabilities within a relativistic constituent quark model

Nucleon generalized polarizabilities are investigated within a relativistic framework, defining such quantities through a Lorentz covariant multipole expansion of the amplitude for virtual Compton scattering. The key physical ingredients in the calculation of the nucleon polarizabilities are the Lorentz invariant reduced matrix elements of the electromagnetic transition current, which can be evaluated from off-energy-shell helicity amplitudes. The evolution of the proton paramagnetic polarizability, $\beta_{para}(q)$, as a function of the virtual-photon three-momentum transfer $ q,$ is explicitly evaluated within a relativistic constituent quark model by adopting transition form factors obtained in the light-front formalism. The discussion is focussed on the role played by the effects due to the relativistic approach and to the transition form factors, derived within different models.

Einstein-Langevin equations from running coupling constants

The Einstein-Langevin equations take into account the backreaction of quantum matter fields on the background geometry. We present a derivation of these equations to lowest order in a covariant expansion in powers of the curvature. For massless fields, the equations are completely determined by the running coupling constants of the theory.

Relation between quasirigidity andL-rigidity in space-times of constant curvature and weak fields

The relation between quasirigidity andL-rigidity in space-times of constant nonzero curvature and in space-times with small curvature (weak fields) is studied. The covariant expansion of bitensors about a point is considered. We obtain an increase in the order of magnitude, underL-rigidity conditions, of the rate of change with respect to a comoving orthonormal frame of the linear momentum, angular momentum, and reduced multipole moments of the energy-momentum tensor. Thus,L-rigidity leads to quasirigidity in such space-times.

Strong Higgs binding of heavy-fermion systems.

We study the fermion-antifermion bound-state spectrum for heavy quarks and leptons by including the effect of Higgs-boson exchange. We use the Bethe-Salpeter equation in the ladder approximation and the $\frac{N}{D}$ method for our calculation. The ladder Bethe-Salpeter equation is solved by using several well-known approximation schemes: the instantaneous approximation, the Blankenbecler-Sugar approximation, and covariant expansion in terms of Tschebeshev polynomials. Our results show that for quark-antiquark systems the Higgs-boson exchange generally dominates over QCD for quark masses larger than 500 GeV, and leads to very deep binding for quark masses in the TeV region. In the absence of the QCD force, as is the case with leptons, we determine the minimum value of the fermion mass needed to form a bound state as a function of the Higgs-boson mass. We further find that tightly bound states where the bound-state masses drop to zero appear for fermion masses between 1 and 1.8 TeV for all the approximations to the Bethe-Salpeter equation that we employed. The $\frac{N}{D}$ method also leads to tightly bound states for fermion masses larger than about 1 TeV but does not in general yield zero-mass bound states. The meaning of the tightly bound states is discussed.

Evaluation of finite lowest-order self-energies

We show that the various lowest-order self-energies can all be made finite on account of the proposal by Danos and Rafelski that the space-time integrations in the covariant perturbation expansion of theS-matrix are to be open instead of closed. We also discuss how we can evaluate these finite lowest-order self-energies.

Derivative expansions in quantum electrodynamics.

A new method for calculating the covariant derivative expansions is presented, particularly in Abelian gauge theories, which can be used to find derivative expansions around nonvanishing gauge field-strength tensors. We apply this method to find the $O({({\ensuremath{\partial}}_{\ensuremath{\lambda}}{F}_{\ensuremath{\mu}\ensuremath{\nu}})}^{2})$ terms of electron determinants in quantum electrodynamics.

The wave function and the effective action in quantum cosmology: Covariant loop expansion

The wave function in quantum cosmology is presented in the form of a rigorously defined covariant functional integral involving the relativistic gauge-breaking and ghost contributions and calculable beyond the semiclassical approximation. The functional integral is formulated both in the lorentzian and euclidean spacetimes and includes the case of the Hartle-Hawking prescription for the cosmological wave function. The relationship between this function and the on-shell effective action in quantum cosmology is established. The loop diagrammatic technique for the proposed integral is derived by setting the boundary-value problems for the gravitational and ghost-field propagators and the one-loop approximation is presented, useful for application to concrete cosmological problems.

Superprojectors

We present a simple algorithm for constructing the N-extended superfield projection operators for irreducible representations of supersymmetry, and explicitly perform all simplifications due to spinor derivative algebra. The method is based on covariant expansion of a general superfield in terms of chiral superfields, and requires no knowledge of Casimir operators. We list these superprojectors for various N = 1, 2, and 4 superfields, and apply our results to quantized the linearized N = 2 vector multiplet in a supersymmetric gauge.