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Gravitational Relaxation of Electroweak Hierarchy Problem

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Abstract

In the present paper, we discuss gravitational relaxation models for the electroweak hierarchy problem. We show that modified gravity can naturally relax the electroweak hierarchy problem where conformal transformation provides a crucial rule about what modified gravity theories are favored to relax the electroweak hierarchy. The conformal transformation connects different gravitational theories and rescaling the metric changes the dimensional parameters like the Higgs mass or the cosmological constant in different frames drastically. When the electroweak scale is naturally realized by dynamical and running behavior of dilatonic scalar field or scaling parameter, the modified gravity theories can relax the electroweak hierarchy problem. We discuss the theoretical and phenomenological validity of the gravitational relaxation models.

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Notes

  1. On the other hand, the cosmological constant problem is more serious from the viewpoint of the naturalness or hierarchy problem. The quantum radiative corrections to the vacuum energy density ρvacuum which is dubbed zero-point vacuum energy enlarges up to the cut-off scale MUV as follows:

    $$\delta{ \rho }_{vacuum } =\frac{1}{2}{\int}^{{ M }_{\text{ UV }} } { \frac { { d }^{3 }k }{ { \left( 2\pi \right) }^{3 } } \sqrt { { k }^{2 }+{ m }^{2 } } } =\frac { { M }^{4}_{\text{ UV }}}{ 16{ \pi }^{2 } } +\frac{ { m }^{2 }{ M }^{2}_{\text{ UV } } }{ 16{ \pi }^{2 } } +\frac { { m }^{4 } }{ 64{ \pi }^{2 } } \log \left( \frac { { m }^{2 } } { M }^{2}_{\text{ UV } } \right) +{\cdots} ,$$

    which is much larger than the dark energy 2.4 × 10− 3 eV in the current Universe.

  2. The five-dimensional metric in the Randall-Sundrum model takes the form

    $$\begin{array}{@{}rcl@{}} { ds }^{2 }={ e }^{-2k{ r }_{c }\phi }{ \eta }_{\mu \nu }{ d }x^{\mu }{ d }x^{\nu }+{ r }_{c }^{2 }d{ \phi }^{2 } , \end{array}$$

    where ημν is the 4D Minkowski metric.

  3. In the Randall-Sundrum model, the four-dimensional action for the gravity sector is given by

    $$\begin{array}{@{}rcl@{}} { S }_{\text{gravity }}\supset \int { { d }^{4 }x } \int { d\phi }\ 2{ M }^{3 }{ r }_{c }{ e }^{-2k{ r }_{c }\left| \phi \right| }\sqrt { -\overline { g } }\ \overline { R }, \end{array}$$

    where M is the five-dimensional Planck scale and \(\overline { R }\) is constructed by the rescaling metric \(\overline { g }_{\mu \nu }\). Ther 4D Planck scale Mpl can be determined as follows:

    $$\begin{array}{@{}rcl@{}} M_{\text{pl}}^{2}= { M }^{3 }{ r }_{c } \int { d\phi }\ { e }^{-2k{ r }_{c }\left| \phi \right| }= \frac { { M }^{3 } }{ k } \left\{ 1-{ e }^{-2k{ r }_{c }\pi } \right\}. \end{array}$$

  4. Polyakov proposed that the cosmological constant could be screened by the IR behavior of quantum gravity and the behavior can be translated by the RG running of the auxiliary gravitational field [18, 19]. The electroweak hierarchy is also discussed by Ref. [23].

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  1. KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki, 305-0801, Japan

    Hiroki Matsui

  2. The Graduate University of Advanced Studies (Sokendai), Tsukuba, Ibaraki, 305-0801, Japan

    Hiroki Matsui & Yoshio Matsumoto

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  1. Hiroki Matsui
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Matsui, H., Matsumoto, Y. Gravitational Relaxation of Electroweak Hierarchy Problem. Int J Theor Phys 61, 182 (2022). https://doi.org/10.1007/s10773-022-05168-w

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