Zero Mass Case Research Articles

Anomalous magnetoresponse and the Stückelberg axion in holography

We study the magnetoresponse with nonconserved currents in holography. Nonconserved currents are dual to massive vector fields in anti--de Sitter (AdS). We introduce the mass in a gauge invariant way via the St\"uckelberg mechanism. In particular we find generalizations of the chiral magnetic effect, the chiral separation effect and the chiral magnetic wave. Since the associated charge is not conserved we need to source it explicitly by a coupling, the generalization of the chemical potential. In this setup we find that in general the anomalous transport phenomena are still realized. The values we find for nonzero mass connect continuously to the values of the anomalous conductivities of the consistent currents, i.e. the proper chiral magnetic effect vanishes for all masses (as it does for the consistent current in the zero mass case) whereas the chiral separation effect is fully present. The generalization of the chiral magnetic wave shows that for small momenta there is no propagating wave but two purely absorptive modes (one of them diffusive). At higher momenta we recover the chiral magnetic wave as a combination of the two absorptive modes. We also study the negative magneto resistivity and find that it grows quadratically with the magnetic field. The chiral magnetic wave and the negative magneto resistivity are manifestations of the chiral magnetic effect that takes place when the (nonconserved) charge is allowed to fluctuate freely in contrast to the case where the charge is fixed by an explicit source. Since the (classical) $U(1{)}_{A}$ symmetry of QCD is not at all a symmetry at the quantum level we also argue that using massive vectors in AdS to model the axial singlet current might result in a more realistic holographic model of QCD and should be a good starting point to investigate the dynamics of anomalous transport in the strongly coupled quark gluon plasma.

On the Positive Mass Theorem for Manifolds with Corners

We study the positive mass theorem for certain non-smooth metrics following P. Miao's work. Our approach is to smooth the metric using the Ricci flow. As well as improving some previous results on the behaviour of the ADM mass under the Ricci flow, we extend the analysis of the zero mass case to higher dimensions.

Improved estimates and a limit case for the electrostatic Klein–Gordon–Maxwell system

We study the class of nonlinear Klein–Gordon–Maxwell systems describing a standing wave (charged matter field) in equilibrium with a purely electrostatic field. We improve some previous existence results in the case of an homogeneous nonlinearity. Moreover, we deal with a limit case, namely when the frequency of the standing wave is equal to the mass of the charged field; this case shows analogous features of the well-known ‘zero-mass case’ for scalar field equations.

Singular perturbed problems in the zero mass case: asymptotic behavior of spikes

We discuss the asymptotic behavior of the least energy solution of a Dirichlet problem in the zero mass case. If Q is a uniformly positive potential having k isolated local minima, then we prove the existence of a positive multi-spike solutions having k peaks concentrating at each local minima of the potential.

Klein paradox and scattering theory for the semi-classical Dirac equation

We study the Klein paradox for the semi-classical Dirac operator on R with potentials having constant limits, possibly different at infinity. Using the complex WKB method, the time-independent scattering theory in terms of incoming and outgoing plane wave solutions is established. The corresponding scattering matrix is unitary. We obtain an asymptotic expansion, with respect to the semi-classical parameter h, of the scattering matrix in the cases of the Klein paradox, the total transmission and the total reflection. Finally, we treat the scattering problem in the zero mass case.

On a “Zero Mass” Nonlinear Schrödinger Equation

Abstract We look for positive solutions to the nonlinear Schrödinger equation −Ɛ2∆u − V (x)f′(u) = 0 in ℝN, where V is a continuous bounded positive potential and f satisfies particular growth conditions which make our problem fall in the so called “zero mass case”. We prove an existence result for any Ɛ > 0, and a multiplicity result for Ɛ sufficiently small.

A multiplicity result for the problem [formula omitted

In this paper we consider the nonlinear equation involving differential forms on a compact Riemannian manifold δ d ξ = f ′ ( 〈 ξ , ξ 〉 ) ξ . This equation is a generalization of the semilinear Maxwell equations recently introduced in a paper by Benci and Fortunato. We obtain a multiplicity result both in the positive mass case (i.e. f ′ ( t ) ≥ ε > 0 uniformly) and in the zero mass case ( f ′ ( t ) ≥ 0 and f ′ ( 0 ) = 0 ) where a strong convexity hypothesis on the nonlinearity is assumed.

Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator

An one-dimensional (1D) Dirac tight-binding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrödinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases.

Noncommutativity and Its Freedoms Due to the pp-Wave and Background Gauge Fields

In this paper we consider an open string attached to a Dp-brane, in the presence of the pp-wave and background gauge fields. The effects of the string mass on the open string propagator, open string metric and the noncommutativity parameter are studied. Some free matrices appear in the propagator and in the open string variables. The symmetries of the string propagator and consistency with the zero mass case put some restrictions on these matrices.

Duality for symmetric second rank tensors. II. The linearized gravitational field

The construction of dual theories for linearized gravity in four dimensions is considered. Our approach is based on the parent Lagrangian method previously developed for the massive spin-two case, but now considered for the zero mass case. This leads to a dual theory described in terms of a rank two symmetric tensor, analogous to the usual gravitational field, and an auxiliary antisymmetric field. This theory has an enlarged gauge symmetry, but with an adequate partial gauge fixing it can be reduced to a gauge symmetry similar to the standard one of linearized gravitation. We present examples illustrating the general procedure and the physical interpretation of the dual fields. The zero mass case of the massive theory dual to the massive spin-two theory is also examined, but we show that it only contains a spin-zero excitation.

Minimum Entropy Production of Neutrino Radiation in the Steady State

A thermodynamic minimum principle valid for photon radiation is shown to hold for arbitrary geometries. It is successfully extended to neutrinos, in the zero mass and chemical potential case, following a parallel development of photon and neutrino statistics. This minimum principle stems more from that of Planck than that of classical Onsager–Prigogine irreversible thermodynamics. Its extension from bosons to fermions suggests that it may have a still wider validity.

Foam diagram summation at finite temperature

We show that large- N φ 4 theory is not trivial if one accepts the presence of a tachyon with a truly huge mass, and that it allows exact calculation. We use it to illustrate how to calculate the exact resummed pressure at finite temperature and verify that it is infrared and ultraviolet finite even in the zero-mass case. In three dimensions a residual effect of the resummed infrared divergences is that at low temperature or strong coupling the leading term in the interaction pressure becomes independent of the coupling and is 4 5 of the free-field pressure. In four dimensions the pressure is well defined provided that the temperature is below the tachyon mass. We examine how rapidly this expansion converges and use our analysis to suggest how one might reorganise perturbation theory to improve the calculation of the pressure for the QCD plasma.

Radial Symmetry and Decay Rate of Variational Ground States in the Zero Mass Case

P.-L. Lions raised the question whether variational ground state solutions of the semilinear Dirichlet problem \begin{eqnarray*} - \Delta w &=& f(w) \mbox{in} {\Bbb R}^n, w(x) &\to& 0 \mbox{as} |x| \to \infty \end{eqnarray*} are radial with constant sign. We consider the zero mass case f(0)=f'(0)=0 without regularity assumptions for the nonlinearity. The celebrated symmetry result of Gidas, Ni, and Nirenberg and its refinements do not apply. Nevertheless we give an affirmative answer to the question of Lions. We prove that every variational ground state is either strictly positive or strictly negative. For positive nonlinearities positive solutions are radially symmetric with respect to some point and strictly decreasing in radial direction. For general nonlinearities we show that the same is true outside a compact set. This is a consequence of our main result, the second-order decay estimate w(r) = c,r2-n (1+O(r-2 )) \mbox{in the $C^1$-sense.} In addition we obtain an integral representation for t...

Relativistic point interaction with Coulomb potential in one dimension

The Dirac Hamiltonian in one space dimension is investigated under the influence of a potential of the form −γ/|x|. The corresponding (four-parameter) family of all self-adjoint extensions is given and described via the boundary form. The resolvent is calculated and the spectrum is studied. Furthermore, we examine the zero mass case. In the nonrelativistic limit we obtain the four-parameter family of Schrödinger operators with the Coulomb potential.

Perturbation theory of Wightman functions

A perturbative expansion of the Wightman functions, and more generally of vacuum expectation values of products of time-ordered and anti-time-ordered products, is derived for Ф 4 4 field theory. The result is expressed as a sum over generalized Feynman graphs. The derivation is based exclusively on the equation of motion and the Wightman axioms. Neither canonical commutation relations nor asymptotic conditions are needed at any point. In the zero-mass case the individual graphs are infrared divergent, but the sum over all graphs of a given order is convergent.

Irreducible Representations of the Poincaré Group Including Inversions

We classify completely all irreducible representations of the Poincaré Group (including inversions) corresponding to positive or zero mass systems, using a generalized version of the method of VARADARAJAN [9] which seems to be more convenient for practical computation than the method of PARTHASARATHY [5]. The case of zero mass — infinite helicity systems which was not considered previously in the literature is treated in a complete fashion.

Asymptotic Behaviour of Solutions of the Porous Medium Equation with Changing Sign

It is proved that solutions of the porous medium equation $u_t = \Delta (| u |^{m - 1} u),m > 1$, defined in $Q = \mathbb{R}^N \times (0,\infty )$ with initial data $u(x,0)$ integrable, compactly supported, and with changing sign, become nonnegative in finite time if $\int {u_0 (x)} dx > 0$. Precise asymptotic convergence rates then follow. The positivity result is related to finite propagation and is false for the heat equation, $m = 1$. Nevertheless, we obtain asymptotic convergence rates for ${{1 \geq m > (N - 2)_ + } / N}$ as $t \to \infty $. The same analysis works for the equation $u_t = {\operatorname{div}}(| {\nabla u} |^{p - 1} \nabla u)$ with $p > 1$. The case of zero mass, $\int {u_0 (x)} dx > 0$, is also studied for the porous medium equation (in $N = 1$) and the solution is shown to converge to an antisymmetric profile in that case.

Why the vacuum charge of a fermion-monopole system vanishes discontinuously as fermion mass vanishes

Using nonrelativistic scattering theory we attribute the vacuum charge induced by a fermion on a monopole to the branch cut introduced by the kinematic factor ( k 2+m 2) −1 2 in the momentum representation of the charge density. The absence of this cut in the zero mass case leads to the discontinuity in the charge.

Bound states, resonances, and symmetries of a neutral Dirac particle with anomalous magnetic moment, coupled to a fixed monopole

For the system consisting of a neutral Dirac particle with anomalous magnetic moment, interacting with a fixed magnetic monopole, zero-energy bound states are constructed for each possible value of the total angular momentum. Results of Kazama and Yang for the charge–monopole system are used to deduce the existence of other bound states for this system, when the mass of the bound particle is nonzero. In the zero-mass case, there are no other bound states, but there are resonant states, and these are determined exactly. A noncompact, so(3,2) symmetry algebra of the zero-energy bound states is given for the finite-mass case and for the zero-mass case. In each case the infinite number of such states is associated with an irreducible Majorana representation of the algebra.

A quantum field theory in momentum space of constant curvature

The model of a polarization operator in the “rainbow” diagram approximation in momentum space of constant curvature is studied. An integral equation is obtained for the absorptive component of the hadron polarization operator in a stereographic parametrization of the de Sitter space. The solution of the integral equation is found for the case of zero mass of the exchange particle. The sections and mean multiplicity of hadron generation in e+e− annihilation are calculated. The infrared divergences which develop in an exact summation of the class of diagrams considered here manifest themselves as a multiplicative renormalization constant.