On the Nature of Muon

Nakamura, Seitaro; Itami, K.; Ugai, Hiroyuki

doi:10.1143/ptp.34.256

Cited by 41 publications

(118 citation statements)

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“…[46]). For the RD stage and the scalar-scalar coupling, ξ (2)µ is given in (C. 35), (C. 36), and (C.37). The 2nd-order transformations of the 2nd-order metric perturbations for a general RW spacetime are given by (C.31), (C.32), (C.33), and (C.34), which, for the RD stage and the scalar-scalar coupling, reduce tō…”

Section: Nd-order Residual Gauge Modesmentioning

confidence: 99%

Second-order cosmological perturbations. III. Produced by scalar-scalar coupling during radiation-dominated stage

Wang

Zhang

2018

Phys. Rev. D

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We study the 2nd-order scalar, vector and tensor metric perturbations in Robertson-Walker (RW) spacetime in synchronous coordinates during the radiation dominated (RD) stage. The dominant radiation is modeled by a relativistic fluid described by a stress tensor T µν = (ρ + p)U µ U ν + g µν p with p = c 2 s ρ, and the 1st-order velocity is assumed to be curlless. We analyze the solutions of 1st-order perturbations, upon which the solutions of 2nd-order perturbation are based. We show that the 1st-order tensor modes propagate at the speed of light and are truly radiative, but the scalar and vector modes do not. The 2nd-order perturbed Einstein equation contains various couplings of 1st-order metric perturbations, and the scalar-scalar coupling is considered in this paper. We decompose the 2nd-order Einstein equation into the evolution equations of 2nd-order scalar, vector, and tensor perturbations, and the energy and momentum constraints. The coupling terms and the stress tensor of the fluid together serve as the effective source for the 2nd-order metric perturbations. The equation of covariant conservation of stress tensor is also needed to determine ρ and U µ . By solving this set of equations up to 2nd order analytically, we obtain the 2nd-order integral solutions of all the metric perturbations, density contrast and velocity. To use these solutions in applications, one needs to carry out seven types of the numerical integrals. We perform the residual gauge transformations between synchronous coordinates up to 2nd order, and identify the gauge-invariant modes of 2nd-order solutions.

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Section: Nd-order Residual Gauge Modesmentioning

confidence: 99%

Second-order cosmological perturbations. III. Produced by scalar-scalar coupling during radiation-dominated stage

Wang

Zhang

2018

Phys. Rev. D

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“…All of their results already reported are of the order 10~8, which cannot explain the difference AfM at all. On the latter problem, much work [26][27][28][29][30][31][32][33][34] has been done. Even if we are concerned only with particles existing at present other than e, /*, and 7, we must consider all the hadrons.…”

Section: Muon Magnetic Momentmentioning

confidence: 99%

Spectral Function of the Photon Propagator—Mass Spectrum and Timelike Form Factors of Particles

Terazawa

1969

Phys. Rev.

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We consider contributions to the spectral function of the photon propagator from the states which contain a particle (charge e and mass mj) of arbitrary spin /, and its antiparticle. We express the contributions in terms of timelike form factors $j(a) (where a--P 2 >0, and P is the momentum of the photon) with the normalization 5v(0)= (2/-f-l)e 2 . The unitarity limit of the spectral function can be transformed into the asymptotically bounded condition $j{a)<0{a). The experimental information about the anomalous magnetic moment of the muon gives a restriction on the sum of all the contributions. Using the restriction, we examine various mass spectra of charged particles and obtain simple results. For example: If there is an infinitely rising mass spectrum mj=f(J), then the asymptotic form of the mass formula must be bounded by condition mj>0{J) (case I or II) or m/>0(/ 1/(2m) ) (case III), where / is a parameter in the form factors, assumed to be $j(a)== (2J+l)e 2 (a/4vij*-l)\l-a/y?\-* 1 for / = 0, 1, 2, •-, and $j (a) = (27+ l)e 2 \ 1-a/fi 2 \ ~2 1 for / = J, §, • • •. For the purpose of experimental observations of the timelike form factors and the spectral function of the photon, the colliding-beam experiments e + +e~ (or JI + +M~) -* X+X (where X is a particle of arbitrary spin) are discussed in some detail.

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“…So three types of nonlinear couplings are more interesting in cosmological studies: scalarscalar, scalar-tensor, and tensor-tensor. In the literature [46][47][48][49][50][51][52][53], the studies of the 2nd-order perturbations mainly consider the scalar-scalar coupling, whereas the scalar-tensor and the tensor-tensor couplings have not been sufficiently investigated. The authors of Ref.…”

Section: Introductionmentioning

confidence: 99%

Second-order cosmological perturbations. IV. Produced by scalar-tensor and tensor-tensor couplings during the radiation dominated stage

Wang

Zhang

2019

Phys. Rev. D

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We continue to study the 2nd-order cosmological perturbations in synchronous coordinates in the framework of the general relativity (GR) during the radiation dominated (RD) stage, and to focus on the scalar-tensor and tensor-tensor couplings. The 1st-order curl velocity and the associated 1st-order vector metric perturbations are assumed to be vanishing. By analytically solving the 2nd-order Einstein equation and the energy-momentum conservation equations, we obtain the 2nd-order formal solutions (in the integral form) of all the metric perturbations, density contrast and velocity; perform the transformation between the synchronous coordinates; and identify the residual gauge modes in the 2nd-order solutions. In addition, we present the 2nd-order gauge transformations of the solutions from synchronous to Poisson coordinates. To apply these formal solutions to concrete cosmological study, one needs to choose proper initial conditions and do several numerical integrals. * √ 3 , so that they are waves propagating at the sound speed 1 √ 3 of the relativistic fluid. On the other hand, the tensor modes (2.28) are waves propagating at the speed of light. In the MD stage [60-62], the scalar, and density contrast are not waves and do not propagate; only the tensor modes still propagate at the speed of light. Therefore,whether or not the scalar modes propagate actually depends on the background matter; nevertheless, the tensor modes always propagate at speed of light, regardless of the background matter. Thus, the tensor modes are radiative as dynamic degrees of freedom, differing from the scalar and vector modes.3 The 2nd-order perturbed equations Equations with scalar-tensor couplingsThe 2nd-order perturbed Einstein equations are listed in Appendix A for a general RW spacetime. As said earlier in (2.19), the transverse vector mode and the curl

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On the Nature of Muon

Cited by 41 publications

References 0 publications

Second-order cosmological perturbations. III. Produced by scalar-scalar coupling during radiation-dominated stage

Second-order cosmological perturbations. III. Produced by scalar-scalar coupling during radiation-dominated stage

Spectral Function of the Photon Propagator—Mass Spectrum and Timelike Form Factors of Particles

Second-order cosmological perturbations. IV. Produced by scalar-tensor and tensor-tensor couplings during the radiation dominated stage

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