Showing papers in "Physical Review D in 1983"

Wave Function of the Universe

Abstract: The quantum state of a spatially closed universe can be described by a wave function which is a functional on the geometries of compact three-manifolds and on the values of the matter fields on these manifolds. The wave function obeys the Wheeler-DeWitt second-order functional differential equation. We put forward a proposal for the wave function of the "ground state" or state of minimum excitation: the ground-state amplitude for a three-geometry is given by a path integral over all compact positive-definite four-geometries which have the three-geometry as a boundary. The requirement that the Hamiltonian be Hermitian then defines the boundary conditions for the Wheeler-DeWitt equation and the spectrum of possible excited states. To illustrate the above, we calculate the ground and excited states in a simple minisuperspace model in which the scale factor is the only gravitational degree of freedom, a conformally invariant scalar field is the only matter degree of freedom and $\ensuremath{\Lambda}g0$. The ground state corresponds to de Sitter space in the classical limit. There are excited states which represent universes which expand from zero volume, reach a maximum size, and then recollapse but which have a finite (though very small) probability of tunneling through a potential barrier to a de Sitter-type state of continual expansion. The path-integral approach allows us to handle situations in which the topology of the three-manifold changes. We estimate the probability that the ground state in our minisuperspace model contains more than one connected component of the spacelike surface.

Highly relativistic nucleus-nucleus collisions: The central rapidity region

Abstract: The space-time evolution of the hadronic matter produced in the central rapidity region in extreme relativistic nucleus-nucleus collisions is described. We find, in agreement with previous studies, that quark-gluon plasma is produced at a temperature &200— 300 MeV, and that it should survive over a time scale & 5 fm/c. Our description relies on the existence of a flat central plateau and on the applicability of hydrodynamics.

Spontaneous Creation of Almost Scale - Free Density Perturbations in an Inflationary Universe

Abstract: The creation and evolution of energy-density perturbations are analyzed for the "new inflationary universe" scenario proposed by Linde, and Albrecht and Steinhardt. According to the scenario, the Universe underwent a strongly first-order phase transition and entered a "de Sitter phase" of exponential expansion during which all previously existing energy-density perturbations expanded to distance scales very large compared to the size of our observable Universe. The existence of an event horizon during the de Sitter phase gives rise to zero-point fluctuations in the scalar field $\ensuremath{\varphi}$, whose slowly growing expectation value signals the transition to the spontaneous-symmetry-breaking (SSB) phase of a grand unified theory (GUT). The fluctuations in $\ensuremath{\varphi}$ are created on small distance scales and expanded to large scales, eventually giving rise to an almost scale-free spectrum of adiabatic density perturbations (the so-called Zel'dovich spectrum). When a fluctuation reenters the horizon ($\mathrm{radius}\ensuremath{\simeq}{H}^{\ensuremath{-}1}$) during the Friedmann-Robertson-Walker (FRW) phase that follows the exponential expansion, it has a perturbation amplitude ${\frac{\ensuremath{\delta}\ensuremath{\rho}}{\ensuremath{\rho}}|}_{H}=(4 or \frac{2}{5})H\frac{\ensuremath{\Delta}\ensuremath{\varphi}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})}$, where $H$ is the Hubble constant during the de Sitter phase (${H}^{\ensuremath{-}1}$ is the radius of the event horizon), $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})$ is the mean value of $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}$ at the time (${t}_{1}$) that the wavelength of the perturbation expanded beyond the Hubble radius during the de Sitter epoch, $\ensuremath{\Delta}\ensuremath{\varphi}$ is the fluctuation in $\ensuremath{\varphi}$ at time ${t}_{1}$ on the same scale, and $4(\frac{2}{5})$ applies if the Universe is radiation (matter) dominated when the scale in question reenters the horizon. Scales larger than about ${10}^{15}\ensuremath{-}{10}^{16}{M}_{\ensuremath{\bigodot}}$ reenter the horizon when the Universe is matter dominated. Owing to the Sachs-Wolfe effect, these density perturbations give rise to temperature fluctuations in the microwave background which, on all angular scales \ensuremath{\gg}1\ifmmode^\circ\else\textdegree\fi{}, are $\frac{\ensuremath{\delta}T}{T}\ensuremath{\simeq}(\frac{1}{5})H\frac{\ensuremath{\Delta}\ensuremath{\varphi}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})}$. The value of $\ensuremath{\Delta}\ensuremath{\varphi}$ expected from de Sitter fluctuations is $O(\frac{H}{2\ensuremath{\pi}})$. For the simplest model of "new inflation," that based on an SU(5) GUT with Coleman-Weinberg SSB, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})\ensuremath{\ll}{H}^{2}$ so that $\frac{\ensuremath{\delta}T}{T}\ensuremath{\gg}1$---in obvious conflict with the large-scale isotropy of the microwave background. One remedy for this is a model in which the inflation occurs when $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\varphi}}({t}_{1})\ensuremath{\gg}{H}^{2}$. We analyze a supersymmetric model which has this feature, and show that a value of ${\frac{\ensuremath{\delta}\ensuremath{\rho}}{\ensuremath{\rho}}|}_{H}\ensuremath{\simeq}{10}^{\ensuremath{-}4}\ensuremath{-}{10}^{\ensuremath{-}3}$ on all observable scales is not implausible.

Quantization of Gauge Theories with Linearly Dependent Generators

Abstract: The quantization rules for gauge theories with open algebras are generalized to the case of linearly dependent generators. The given zero-eigenvalue eigenvectors of the generators may also be linearly dependent and possess zero-eigenvalue eigenvectors which may also be linearly dependent and so on. We give the solution for the general case of such a hierarchy.

Scaling violations in inclusive e + e - annihilation spectra

Abstract: The origin of the observed scaling violations in inclusive ${e}^{+}{e}^{\ensuremath{-}}$ annihilation is investigated. Perturbative jet evolution is not necessarily the only reason for scale breaking in the hadron spectra at present energies. Remnants of finite-transverse-momentum and mass effects are still important in nonperturbative, cascade-type, jet formation in the \ensuremath{\sim}10 GeV range. Heavy-quark fragmentation has a strong impact on hadronic inclusive spectra. A simple parametrization for the heavy-quark fragmentation function is given which describes well recently measured charmed-meson spectra. Taking these effects into account, good agreement with the observed scaling violations is obtained in cascade-type jet models with hard-gluon bremsstrahlung.

Supergravity as the messenger of supersymmetry breaking

Abstract: A systematic study is made of theories in which supergravity is spontaneously broken in a ''hidden'' sector of superfields that interact with ordinary matter only through supergravity. General rules are given for calculating the low-energy effective potential in such theories. This potential is given as the sum of ordinary supersymmetric terms involving a low-energy effective superpotential whose mass terms arise from integrating out the heavy particles associated with grand unification, plus supersymmetry-breaking terms that depend on the details of the hidden sector and the Kaehler potential only through the values of four small complex mass parameters. The result is not the same as would be obtained by ignoring grand unification and inserting small mass parameters into the superpotential from the beginning. The general results are applied to a class of models with a pair of Higgs doublets.

Birth of inflationary universes

Abstract: A cosmological model is proposed in which the Universe is created by quantum tunneling from "nothing" into a de Sitter space The tunneling is described by a de Sitter---Hawking---Moss instanton After the tunneling, the model evolves along the lines of the inflationary scenario It is argued that at any time there exist parts of the Universe which are still in the de Sitter phase, while other parts have already recollapsed This model does not have a big-bang singularity and does not require any initial or boundary conditions

Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant

Abstract: We examine the late-time behavior of intially expanding homogeneous cosmological models satisfying Einstein's equation with a positive cosmological constant $\ensuremath{\Lambda}$. It is shown that such models of all Bianchi types except IX exponentially evolve toward the de Sitter solution, with time scale ${(\frac{3}{\ensuremath{\Lambda}})}^{\frac{1}{2}}$. The behavior of Bianchi type-IX universes is similar, provided that $\ensuremath{\Lambda}$ is sufficiently large compared with spatial-curvature terms. Thus, a positive cosmological constant provides an effective means of isotropizing homogeneous universes.

Coherent scalar-field oscillations in an expanding universe

Abstract: Motivated by the cosmological importance of coherent (classical), scalar-field oscillations in the context of the invisible axion and the new inflationary-universe scenario, we analyze, in general, the classical evolution of a scalar field in an isotropic and homogeneous cosmology. For a scalar potential of the form $V(\ensuremath{\varphi})=a{\ensuremath{\varphi}}^{n}$, the energy density of the scalar-field oscillations decreases as ${R}^{\ensuremath{-}\frac{6n}{(n+2)}}$ when the oscillations are rapid compared to the expansion rate ($R=\mathrm{cosmic}\mathrm{scale}\mathrm{factor}$). We also investigate the effect of higher-order terms in the potential perturbatively, and analyze the decay of the coherent field oscillations due to quantum particle creation.

On the Elimination of Scale Ambiguities in Perturbative Quantum Chromodynamics

Abstract: We present a new method for resolving the scheme-scale ambiguity that has plagued perturbative analyses in quantum chromodynamics (QCD) and other gauge theories. For Abelian theories the method reduces to the standard criterion that only vacuum-polarization insertions contribute to the effective coupling constant. Given a scheme, our procedure automatically determines the coupling-constant scale appropriate to a particular process. This leads to a new criterion for the convergence of perturbative expansions in QCD. We examine a number of well known reactions in QCD, and find that perturbation theory converges well for all processes other than the gluonic width of the $\ensuremath{\Upsilon}$. Our analysis calls into question recent determinations of the QCD coupling constant based upon $\ensuremath{\Upsilon}$ decay.

Topology in the Weinberg-Salam theory

Abstract: We show that the configuration space of the classical, bosonic Weinberg-Salam theory has a non-contractible loop. This probably implies that there is an unstable, static, finite-energy solution of the field equations. Its energy is the height of the barrier for tunneling between "topologically distinct" vacuums. We establish an upper bound on this energy of order 10 TeV.

Evolution without evolution: Dynamics described by stationary observables

Abstract: Because the time parameter in the Schr\"odinger equation is not observable, energy apparently obeys a superselection rule in the same sense that charge does. That is, observables must all commute with the Hamiltonian and hence be stationary. This means that it is consistent with all observations to assume that any closed system such as the Universe is in a stationary state. We show how the observed dynamic evolution of a system can be described entirely in terms of stationary observables as a dependence upon internal clock readings.

Simple Rules for Discontinuities in Finite Temperature Field Theory

Abstract: The discontinuity, or imaginary part, of the self-energy function at $T\ensuremath{ e}=0$ is found to be of the form $\ensuremath{\Gamma}={\ensuremath{\Gamma}}_{d}\ensuremath{\mp}{\ensuremath{\Gamma}}_{i}$ for bosons and fermions, respectively. The generalized decay rate ${\ensuremath{\Gamma}}_{d}$ and inverse decay rate ${\ensuremath{\Gamma}}_{i}$ are recognizable as integrals over phase space of amplitudes squared, weighted with certain statistical factors that account for the possibility of particle absorption from the medium or particle emission into the medium. Nonequilibrium statistical mechanics shows that $\ensuremath{\Gamma}$ gives precisely the rate at which the single-particle distribution function approaches the equilibrium form.

Geodesic motion and confinement in Gödel's universe

Abstract: We present a complete study of geodesic motion in G\"odel's universe, using the method of the effective potential. A clear physical picture of free motion and its stability in this universe emerges. A large class of geodesics have finite intervals in which the particle moves back in time ($\frac{\mathrm{dt}}{\mathrm{ds}}l0$) without violation of causality. G\"odel's geometry produces the important property of confinement for a large class of geodesics. We use this property to discuss the construction of a gravitational container. This structure is highly stable, since there is no singularity in its interior, and is independent of the energy of the particles contained in it.

Twisted-Eguchi-Kawai model: A reduced model for large-N lattice gauge theory

Abstract: We study the large-$N$ reduced model recently proposed by the present authors. This model is a modified version of the Eguchi-Kawai model incorporating twisted boundary conditions. It is shown that the Schwinger-Dyson equations of our model are the same as in the infinite-lattice theory provided ${[\mathrm{U}(1)]}^{4}$ symmetry is not spontaneously broken. We study the model at strong coupling, weak coupling, and intermediate coupling using analytical and Monte Carlo techniques. At weak coupling, it is shown that for a particular choice of twist, ${[\mathrm{U}(1)]}^{4}$ symmetry is not broken and we prove how one recovers usual planar perturbation theory. Monte Carlo data for $\ensuremath{\chi}$ ratios show striking agreement with Wilson-theory results.

qq qq ― system in a potential model

Abstract: We have examined the $\mathrm{qq}\stackrel{-}{\mathrm{qq}}$ system in a nonrelativistic potential model with color-dependent confinement forces and hyperfine interactions by solving the four-particle Schr\"odinger equation variationally. We find that normally the ground state of this system consists of two free mesons, but that exceptions to this rule probably occur for $K\overline{K}$ systems, where we find weakly bound ${0}^{++}$ states with a meson-meson structure reminiscent of the nucleon-nucleon structure of the deuteron. We show that these states may be identified with the ${S}^{*}$ and $\ensuremath{\delta}$ just below $K\overline{K}$ threshold. We further argue that the $\mathrm{qq}\stackrel{-}{\mathrm{qq}}$ system is not only nearly barren of bound states, but that it may not support any resonances. Finally, independent of their identification with observed states, we note that the $\mathrm{qq}\stackrel{-}{\mathrm{qq}}$ bound states are a model for the weak binding and color-singlet clustering observed in nuclei.

Homogeneity of Riemannian space-times of Gödel type

Abstract: The conditions for space-time homogeneity of a Riemannian manifold with a G\"odel-type metric are examined. The Raychaudhuri-Thakurta necessary conditions for space-time homogeneity are shown to be also sufficient and to lead to five linearly independent Killing vectors. These vector fields are exhibited for the most general case and their algebra is examined. The irreducible set of isometrically independent space-time-homogeneous G\"odel-type metrics is shown to be given, in cylindrical coordinates, by $d{s}^{2}={[dt+(\frac{4\ensuremath{\Omega}}{{m}^{2}})sin{\mathrm{h}}^{2}(\frac{\mathrm{mr}}{2})d\ensuremath{\varphi}]}^{2}\ensuremath{-}(\frac{1}{{m}^{2}})sin{\mathrm{h}}^{2}(\mathrm{mr})d{\ensuremath{\varphi}}^{2}\ensuremath{-}d{r}^{2}\ensuremath{-}d{z}^{2}$, where $\ensuremath{\Omega}$ is the vorticity and $\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{m}^{2}\ensuremath{\le}+\ensuremath{\infty}$, ${m}^{2}=2{\ensuremath{\Omega}}^{2}$ corresponding to the G\"odel metric. Sources of Einstein's equations leading to these metrics as solutions are examined, and it is shown that the inclusion of a scalar field extends the previously known region of solutions $\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{m}^{2}\ensuremath{\le}2{\ensuremath{\Omega}}^{2}$ to $\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{m}^{2}\ensuremath{\le}4{\ensuremath{\Omega}}^{2}$. The problem of ambiguity of physical sources of the same metric and that of violation of causality in G\"odel-type space-time-homogeneous universes are examined. In the case ${m}^{2}=4{\ensuremath{\Omega}}^{2}$, we obtain the first exact G\"odel-type solution of Einstein's equations describing a completely causal space-time-homogeneous rotating universe.

Quantization of Dynamical Systems and Stochastic Control Theory

Abstract: In the general framework of stochastic control theory we introduce a suitable form of stochastic action associated to the controlled process. A variational principle gives all the main features of Nelson's stochastic mechanics. In particular, we derive the expression for the current velocity field as the gradient of the phase action. Moreover, the stochastic corrections to the Hamilton-Jacobi equation are in agreement with the quantum-mechanical form of the Madelung fluid (equivalent to the Schr\"odinger equation). Therefore, stochastic control theory can provide a very simple model simulating quantum-mechanical behavior.

New analytical stellar model in general relativity

Abstract: A new analytical solution has been obtained for stellar models by solving Einstein's field equation for the spherically symmetric and static case. The variation of density is smooth and gradual. The density remains positive under all conditions. For all finite pressures the configurations are stable under radial perturbations. For $\frac{\mathrm{dP}}{d\ensuremath{\rho}}\ensuremath{\le}1$, the maximum mass of a neutron-star model is $4.56{M}_{\ensuremath{\bigodot}}$, and the surface and the central red-shifts are 0.787 and 2.673, respectively. For an infinite central pressure the surface red-shift is 1.575 which is greater than that for any other analytical solution with varying density.

Quantizing fourth-order gravity theories: The functional integral

Abstract: The first part of a detailed analysis of the one-loop divergences of fourth-order gravity theories is presented The functional-integral structure is determined and the three kinds of ghosts which appear are discussed Field operators and gauge-fixing terms are derived and are found to be far more complicated than those of standard second-order theory, whose details are presented for comparison to the fourth-order case We outline how, in future papers, we will approach the computation of the one-loop counterterms using pseudo-differential operator techniques and related methods We also discuss briefly what other workers have done and are doing in studies of the problems inherent in fourth-order gravity and its extensions Finally, we emphasize the need to investigate the full non-linear theory with coordinate-space methods and mention some of the problems brought about by linearization Details of calculations and the multilevel-mass-spectrum theory are presented in several appendices

Quantum dynamics of Kaluza-Klein theories

Abstract: Some of the quantum properties of Kaluza-Klein theories are studied. The classical features of these theories are reviewed, and the quantization of the gravitational field in an arbitrary number of dimensions is described. These results are then applied to a detailed analysis of the five-dimensional Kaluza-Klein model. The fifth dimension is taken to be compact and a quantum effective potential, as a function of the five-five component of the metric, is constructed. It is argued that the one-loop computation is reliable as long as the distance around the fifth dimension is large compared to the Planck length. The effective potential separates into two pieces: an induced cosmological constant, independent of the size of the fifth dimension, and a distance-dependent "Casimir" energy. The cosmological term is subtracted, leaving an attractive Casimir potential which will contract the fifth dimension to a size on the order of the Planck length. Consequences of this result are discussed and some of the ways in which it can be generalized are outlined.

Dynamical origin of black-hole radiance

Abstract: An explanation of the thermal quantum radiance of black holes discovered by Hawking is offered in terms of a black-hole metric undergoing quantum zero-point fluctuations of zero mean in its gravitational quasinormal modes. It is shown that such zero-temperature fluctuations, governed by the uncertainty principle, lead to the formation of a quantum ergosphere that enables matter of all species to tunnel out of the hole. The results confirm that a black hole cannot be in equilibrium at zero temperature. A dynamical temperature is calculated by equating the mean irreducible mass associated with the quantum ergosphere to the mean thermal energy of a quantum oscillator with the lowest quasinormal frequency. The result agrees with the Hawking temperature to within two per cent. The nature of the dynamical equilibrium and the higher modes are discussed, and it is calculated that the thermal excitations of the resonant modes have the canonical distribution to within several per cent. A calculation of the black-hole entropy using the statistical mechanics of the quasinormal modes yields a value $(0.27654){\ensuremath{\hbar}}^{\ensuremath{-}1}(16\ensuremath{\pi}{M}^{2})$, which is near the value usually assumed, $(0.25){\ensuremath{\hbar}}^{\ensuremath{-}1}(16\ensuremath{\pi}{M}^{2})$. Characteristic fluctuation scales are derived. The rms energy fluctuation of the physical ("dressed") event horizon is about ${10}^{17}$ GeV, independent of $M$. The physical metric fluctuations near the hole are of order unity when the hole has mass \ensuremath{\approx} (0.15) (Planck mass) \ensuremath{\approx} 1.8\ifmmode\times\else\texttimes\fi{}${10}^{18}$ GeV.

Experimental study of the A dependence of inclusive hadron fragmentation

Abstract: Data are presented on the inclusive production of ..pi../sup + -/, K/sup + -/, p, and p-bar for ..pi../sup +/, K/sup +/, and protons incident on nuclear targets at 100 GeV. The results cover the kinematic range 30< or =P< or =88 GeV/c for P/sub t/ = 0.3 and 0.5 GeV/c. The observed A dependence of the invariant cross sections exhibits remarkable simplicity, which does not naturally follow from current models of particle production. The results show that the hypothesis of limiting fragmentation can be extended to include collisions with nuclei.

Nucleon-nucleon partial-wave analysis to 1100 MeV.

Abstract: Comprehensive analyses of nucleon-nucleon elastic-scattering data below 1100 MeV laboratory kinetic energy are presented. The data base from which an energy-dependent solution and 22 single-energy solutions are obtained consists of 7223 pp and 5474 np data. A resonancelike structure is found to occur in the $^{1}\mathrm{D}_{2}$, $^{3}\mathrm{F}_{3}$, $^{3}\mathrm{P}_{2}$${\mathrm{\ensuremath{-}}}^{3}$${\mathrm{F}}_{2}$, and $^{3}\mathrm{F}_{4}$${\mathrm{\ensuremath{-}}}^{3}$${\mathrm{H}}_{4}$ partial waves; this behavior is associated with poles in the complex energy plane. The pole positions and residues are obtained by analytic continuation of the ``production'' piece of the T matrix obtained in the energy-dependent solution. The new phases differ somewhat from previously published VPI solutions, especially in I=0 waves above 500 MeV, where np data are very sparse. The partial waves are, however, based upon a significantly larger data base and reflect correspondingly smaller errors. The full data base and solution files can be obtained through a computer scattering analysis interactive dial-in (SAID) system at VPI, which also exists at many institutions around the world and which can be transferred to any site with a suitable computer system. The SAID system can be used to modify solutions, plan experiments, and obtain any of the multitude of predictions which derive from partial-wave analyses of the world data base.

New view of quark and lepton mass hierarchy

Abstract: The hierarchical structure of realistic quark and lepton masses is investigated on the basis of a three-family model, where masses and mixings of the first- and second-generation quarks are successfully described by lepton masses only. An excellent prediction ${m}_{\ensuremath{\tau}}=1.7866$ GeV is obtained from the input data ${m}_{e}$ and ${m}_{\ensuremath{\mu}}$ only.

Implications of supersymmetric SO(10) grand unification

Abstract: We investigate the properties of a class of supersymmetric SO(10) grand unified models with an intermediate mass scale M/sub R/ corresponding to the breaking of left-right symmetry. We consider the possibility that the supersymmetry-breaking scale M/sub S/ coincides with M/sub R/ and study the constraints imposed on the value of M/sub R/ by low-energy parameters such as sin/sup 2/theta(M/sub W/) and ..cap alpha../sub strong/(M/sub W/) for different plausible scenarios for the mass spectrum of the supermultiplet fields. We find that the allowed values of M/sub R/ are greater than 10/sup 10/ GeV. If, however, the supersymmetry-breaking scale M/sub S/>M/sub R/ and M/sub S/ is chosen as a free parameter, lower intermediate mass scales are possible. Some implications of our model are noted.

Lepton-number violation with quasi-Dirac neutrinos

Abstract: We investigate lepton-number violation in weak interactions with massive Dirac neutrinos. In the framework of a simple $\mathrm{SU}{(3)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ electroweak model we find that neutrinos naturally acquire only Dirac masses at tree level, and for an odd number of lepton families, one neutrino remains massless. After a spontaneous breakdown of symmetry we find that the SU(2)\ifmmode\times\else\texttimes\fi{}U(1) effective theory has lepton-number-violating currents which couple to the standard gauge bosons. Flavorconserving $|\ensuremath{\Delta}l|=2$ processes such as neutrinoless double-$\ensuremath{\beta}$ decay are forbidden in the simplest model, but processes such as ${\ensuremath{\mu}}^{\ensuremath{-}}+(A, Z)\ensuremath{\rightarrow}{e}^{+}+(A, Z\ensuremath{-}2)$ and ${K}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{e}^{\ensuremath{-}}{\ensuremath{\mu}}^{\ensuremath{-}}$ are allowed.