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A New Estimate of the Mass of the Gravitational Scalar Field for Dark Energy

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At the Frontier of Spacetime

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 183))

Abstract

A new estimate of the mass of the pseudo dilaton is offered by following the fundamental nature that a massless Nambu-Goldstone boson, called a dilaton, in the Einstein frame acquires a nonzero mass through the loop effects which occur with the Higgs field in the relativistic quantum field theory as described by poles of D, spacetime dimensionality off the physical value \(D=4\). Naturally the technique of dimensional regularization is fully used to show this pole structure to be suppressed to be finite by what is called a Classical-Quantum-Interplay, to improve our previous attempt. Basically the same analysis is extended to derive also the coupling of a pseudo dilaton to two photons.

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Notes

  1. 1.

    We use the reduced Planckian unit system defined by \(c=\hbar =M_{\mathrm{P}}(=(8\pi G)^{-1/2})=1\). The units of length, time and energy in conventional units are given by \(8.10 \times 10^{-33}\mathrm{cm}, 2.70\times 10^{-43}\mathrm{s}, 2.44 \times 10^{18}\,\mathrm{GeV}\), respectively. As an example of the converse of the last entry, we find \(1\,\mathrm{GeV}=2.44^{-1}\times 10^{-18}\) in units of the Planck energy. In the same way, the present age of the universe \(t_0 \approx 1.37\times 10^{10}\mathrm{y}\) is \(10^{60.2}\) in units of the Planck time.

  2. 2.

    This mass corresponds approximately to the force-range \({\sim }100\,\mathrm{m}\), related to the suggested non-Newtonian force, as was discussed in [11], for example.

  3. 3.

    Symbols \(\varphi , \omega \) used in the original Refs. [3, 4] are now re-expressed by our more convenient ones; \(\varphi =(1/2)\xi \phi ^2, 4\omega =\epsilon \xi ^{-1}\).

  4. 4.

    \(\epsilon =0\) implies \(\zeta ^2=1/6\) corresponding to the coefficient \(a=1/3\) of the scalar component of the combined potential [11, 14]. We choose \(\eta _{00}=\eta ^{00}=-1\).

  5. 5.

    For details see [1, 2, 15].

  6. 6.

    The amplitude through the nonminimal coupling term observes the same tensor coupling as in General Relativity.

  7. 7.

    The special form of the first in the following, corresponding to \(g_{\mu \nu }\rightarrow \varOmega ^2 g_{\mu \nu }\), has been selected, because, unlike another type of the coordinate transformation \(x^\mu \rightarrow \varOmega x^\mu \) or \(\delta x^\mu = \ell x^\mu \), making it straightforward to be applied to the expanding universe with the 3-space simply uniform without any particular origin. See [16], for example, on introducing \(\delta x^\mu /x^2 = \beta ^\mu \).

  8. 8.

    The symbol m in the second equation is the mass in EF, to be better denoted by \(m_*\). To avoid too much notational complications in the following equations, however, we continue to use m without the subscript \(*\) for the observed mass for the Higgs mass in the whole subsequent part of the article.

  9. 9.

    This potential \(\mathcal V\) is shown to agree with the relevant part of the SM. See (87.3) of [18], for example. His \(V(\varphi )=(\lambda _r/4)(\varphi ^\dagger \varphi -(v^2/2))^2\) is reproduced precisely by our (3.15) by choosing \(\sigma =0, \,\,\lambda _r/4=\lambda /4!\), and \(\varphi =(1/\sqrt{2})(v+\tilde{\varPhi })\) only for the single component, with another component vanishing, corresponding to his (87.13). Notice also that a special relation chosen between the two terms in the parenthesis on RHS of \(V(\varphi )\) above has the same effect of the term linear in \(\tilde{\varPhi }\) removed, which we required in the sentence just prior to the foregoing footnote 8.

  10. 10.

    Some of the details of the required integrals will be found in Appendix N of [1].

  11. 11.

    Strictly speaking, the denominators should be \(((k+q/2)^2+m^2)((k-q/2)^2+m^2)\), where q is for the momentum of the size of \({\sim }\) \(\mu \). Since we finally find \(\mu \) negligibly smaller than m as in (3.49), we might justify the approximate computation as in (3.21). The same kind of approximation applies to almost any of the loop integrals to be encountered in the following of the present article.

  12. 12.

    Basically the same type of analysis can be applied to another example of a loop attached to the side of the left loop in Fig. 3.1b, for example.

  13. 13.

    For the self-masses in QED or QCD, divergences are removed simply to fit the observed values.

  14. 14.

    The effect of renormalized fields might have been ignored at this moment, given the crude approximation to be allowed in (3.49). For details see [18], for example.

  15. 15.

    It still appears that the two estimates above are more or less close to each other from a wider point of view, probably because the two approaches share the same concept on the pseudo dilaton in some way or the other.

  16. 16.

    See (6.174)–(6.181) of [1] for details simply for the scalar loop field. Extending to the Dirac field is tedious but straightforward.

  17. 17.

    This term contributes a term proportional to \(m^2\) due to a simple 1-loop of \(\langle 0|\varPhi {\bar{\varPhi }}|0\rangle \propto m^2\) in DR , thus cancelling the term of \(\sim \) \(m^2\) in the main loop term as in the upper line of Fig. 3.1.

  18. 18.

    The previous result Eq. (3.91) in [2], for example, can even be re-interpreted as our (3.56) multiplied by an “adjustable parameter” \(B/A=\zeta ^{-1}(2/3)\mathcal{Z}\).

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Acknowledgments

The author expresses his sincere thanks to K. Homma, H. Itoyama, C.S. Lim, K. Maeda, T. Tada and T. Yoneya for many useful discussions.

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Authors and Affiliations

  1. Advanced Research Institute for Science and Engineering, Waseda University, Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan

    Yasunori Fujii

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  1. Yasunori Fujii
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Correspondence to Yasunori Fujii .

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  1. PT-SW, DLR Projektträger, Berlin, Germany

    Torsten Asselmeyer-Maluga

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Fujii, Y. (2016). A New Estimate of the Mass of the Gravitational Scalar Field for Dark Energy. In: Asselmeyer-Maluga, T. (eds) At the Frontier of Spacetime. Fundamental Theories of Physics, vol 183. Springer, Cham. https://doi.org/10.1007/978-3-319-31299-6_3

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