Generalized geometrical coupling for vector field localization on thick brane in asymptotic anti–de Sitter spacetime

Sui, Tao-Tao; Guo, Wen-Di; Xie, Qun-Ying; Liu, Yu-Xiao

doi:10.1103/PhysRevD.101.055031

Generalized geometrical coupling for vector field localization on thick brane in asymptotic anti–de Sitter spacetime

Tao-Tao Sui (Institute of Theoretical Physics and Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China) ; Wen-Di Guo (Institute of Theoretical Physics and Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China; Centro de Astrofísica e Gravitação—CENTRA, Departamento de Física, Instituto Superior Técnico—IST, Universidade de Lisboa - UL, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal) ; Qun-Ying Xie (School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China) ; Yu-Xiao Liu (Institute of Theoretical Physics and Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China; Key Laboratory for Magnetism and Magnetic of the Ministry of Education, Lanzhou University, Lanzhou 730000, China)

It is known that a five-dimensional free vector field $A_{M}$ cannot be localized on Randall-Sundrum (RS)-like thick branes—namely, the thick branes embedded in asymptotic anti–de Sitter spacetime. To localize a vector field on the RS-like thick brane, an extra coupling term should be introduced. We generalize the geometrical coupling mechanism by adding two mass terms ( $α R g^{M N} A_{M} A_{N} + β R^{M N} A_{M} A_{N}$ ) to the action. We decompose the fundamental vector field $A_{M}$ into three parts: transverse vector part ${\hat{A}}_{μ}$ and scalar parts $ϕ$ and $A_{5}$ . Then we find that the transverse vector part ${\hat{A}}_{μ}$ decouples from the scalar parts. To eliminate the tachyonic modes of ${\hat{A}}_{μ}$ , the two coupling parameters $α$ and $β$ should satisfy a relation. Combining the restricted condition, we can get a combination parameter as $γ = \frac{3}{2} \pm \sqrt{1 + 12 α}$ . Only if $γ > 1 / 2$ can the zero mode of ${\hat{A}}_{μ}$ be localized on the RS-like thick brane. We also investigate the resonant character of the vector part ${\hat{A}}_{μ}$ for a general RS-like thick brane with a warp factor $A (z) = - \ln (1 + k^{2} z^{2}) / 2$ by choosing the relative probability method. The results show that the massive resonant Kaluza-Klein modes can exist only for $γ > 3$ . The number of resonant Kaluza-Klein states increases with the combination parameter $γ$ , and the lifetime of the first resonant state can be as long as our Universe’s. This indicates that the vector resonances might be considered one of the candidates of dark matter. Combining the conditions of experimental observations, the constraint shows that the parameter $k$ has a lower limit with $k ≳ 10^{- 17} eV$ , the combination parameter $γ$ should be greater than 57, and, accordingly, the mass of the first resonant state should satisfy $m_{1} ≳ 10^{- 15} eV$ .

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  "abstracts": [
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      "value": "It is known that a five-dimensional free vector field <math><msub><mi>A</mi><mi>M</mi></msub></math> cannot be localized on Randall-Sundrum (RS)-like thick branes\u2014namely, the thick branes embedded in asymptotic anti\u2013de Sitter spacetime. To localize a vector field on the RS-like thick brane, an extra coupling term should be introduced. We generalize the geometrical coupling mechanism by adding two mass terms (<math><mrow><mi>\u03b1</mi><mi>R</mi><msup><mrow><mi>g</mi></mrow><mrow><mi>M</mi><mi>N</mi></mrow></msup><msub><mrow><mi>A</mi></mrow><mrow><mi>M</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>+</mo><mi>\u03b2</mi><msup><mrow><mi>R</mi></mrow><mrow><mi>M</mi><mi>N</mi></mrow></msup><msub><mrow><mi>A</mi></mrow><mrow><mi>M</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math>) to the action. We decompose the fundamental vector field <math><msub><mi>A</mi><mi>M</mi></msub></math> into three parts: transverse vector part <math><msub><mover><mi>A</mi><mo>^</mo></mover><mi>\u03bc</mi></msub></math> and scalar parts <math><mi>\u03d5</mi></math> and <math><msub><mi>A</mi><mn>5</mn></msub></math>. Then we find that the transverse vector part <math><msub><mover><mi>A</mi><mo>^</mo></mover><mi>\u03bc</mi></msub></math> decouples from the scalar parts. To eliminate the tachyonic modes of <math><msub><mover><mi>A</mi><mo>^</mo></mover><mi>\u03bc</mi></msub></math>, the two coupling parameters <math><mi>\u03b1</mi></math> and <math><mi>\u03b2</mi></math> should satisfy a relation. Combining the restricted condition, we can get a combination parameter as <math><mrow><mi>\u03b3</mi><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>\u00b1</mo><msqrt><mrow><mn>1</mn><mo>+</mo><mn>12</mn><mi>\u03b1</mi></mrow></msqrt></mrow></math>. Only if <math><mi>\u03b3</mi><mo>&gt;</mo><mn>1</mn><mo>/</mo><mn>2</mn></math> can the zero mode of <math><msub><mover><mi>A</mi><mo>^</mo></mover><mi>\u03bc</mi></msub></math> be localized on the RS-like thick brane. We also investigate the resonant character of the vector part <math><msub><mover><mi>A</mi><mo>^</mo></mover><mi>\u03bc</mi></msub></math> for a general RS-like thick brane with a warp factor <math><mrow><mi>A</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mo>\u2212</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>/</mo><mn>2</mn></mrow></math> by choosing the relative probability method. The results show that the massive resonant Kaluza-Klein modes can exist only for <math><mi>\u03b3</mi><mo>&gt;</mo><mn>3</mn></math>. The number of resonant Kaluza-Klein states increases with the combination parameter <math><mi>\u03b3</mi></math>, and the lifetime of the first resonant state can be as long as our Universe\u2019s. This indicates that the vector resonances might be considered one of the candidates of dark matter. Combining the conditions of experimental observations, the constraint shows that the parameter <math><mi>k</mi></math> has a lower limit with <math><mi>k</mi><mo>\u2273</mo><mrow><msup><mn>10</mn><mrow><mo>\u2212</mo><mn>17</mn></mrow></msup></mrow><mtext> </mtext><mtext> </mtext><mi>eV</mi></math>, the combination parameter <math><mi>\u03b3</mi></math> should be greater than 57, and, accordingly, the mass of the first resonant state should satisfy <math><msub><mi>m</mi><mn>1</mn></msub><mo>\u2273</mo><mrow><msup><mn>10</mn><mrow><mo>\u2212</mo><mn>15</mn></mrow></msup></mrow><mtext> </mtext><mtext> </mtext><mi>eV</mi></math>."
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Published on:: 26 March 2020
Publisher:: APS
Published in:: Physical Review D , Volume 101 (2020)
Issue 5
DOI:: https://doi.org/10.1103/PhysRevD.101.055031
arXiv:: 2001.02154
Copyrights:: Published by the American Physical Society
Licence:: CC-BY-4.0

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