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De Sitter decays to infinity

Draper, Patrick, Garcia, Isabel Garcia, Lillard, Benjamin·2021·Journal of High Energy Physics
DOI 10.1007/jhep12(2021)154
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Abstract

Bubbles of nothing are a class of vacuum decay processes present in some theories with compactified extra dimensions. We investigate the existence and properties of bubbles of nothing in models where the scalar pseudomoduli controlling the size of the extra dimensions are stabilized at positive vacuum energy, which is a necessary feature of any realistic model. We map the construction of bubbles of nothing to a four-dimensional Coleman-De Luccia problem and establish necessary conditions on the asymptotic behavior of the scalar potential for the existence of suitable solutions. We perform detailed analyses in the context of five-dimensional theories with metastable dS 4× S1 vacua, using analytic approximations and numerical methods to calculate the decay rate. We find that bubbles of nothing sometimes exist in potentials with no ordinary Coleman-De Luccia decay process, and that in the examples we study, when both processes exist, the bubble of nothing decay rate is typically faster. Our methods can be generalized to other stabilizing potentials and internal manifolds.

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Image 1
CC-BY · From the PDF
Figure 3 Diagram R5 Figure 11: Just L ( ) 5 Figure 2 . Left: diagram showing the geometry of the KK false vacuum ('vacuum') and BON ('instanton') solutions. The r direction lies along the symmetry axis, with the periodic y coordinate along the azimuth. In the false vacuum, φ ≡ 0 , and the KK radius is independent of r . For Witten's bubble, the KK radius approaches zero near r ≳ R 5 , whereas far away from the bubble wall the geometry asymptotically approaches that of the false vacuum. Right: after dimensional reduction, the scalar modulus φ describes the proper length of the KK circle, L . As a function of the CDL coordinate, defined in eq. (2.20), L vanishes at ξ = 0 , corresponding to the location of bubble wall, whereas L ≃ 2 πR 5 in the asymptotic region ξ ≫ R 5 .
Figure 3 Diagram R5 Figure 11: Just L ( ) 5 Figure 2 . Left: diagram showing the geometry of the KK false vacuum ('vacuum') and BON ('instanton') solutions. The r direction lies along the symmetry axis, with the periodic y coordinate along the azimuth. In the false vacuum, φ ≡ 0 , and the KK radius is independent of r . For Witten's bubble, the KK radius approaches zero near r ≳ R 5 , whereas far away from the bubble wall the geometry asymptotically approaches that of the false vacuum. Right: after dimensional reduction, the scalar modulus φ describes the proper length of the KK circle, L . As a function of the CDL coordinate, defined in eq. (2.20), L vanishes at ξ = 0 , corresponding to the location of bubble wall, whereas L ≃ 2 πR 5 in the asymptotic region ξ ≫ R 5 .CC-BY · From the PDF
Figure 6 Cartoon 3 Figure 3 . The five-dimensional Euclidean Schwarzschild gravitational instanton, or Witten's BON, expressed in terms of the degrees of freedom φ bon ( ξ ) ( left ) and ρ bon ( ξ ) ( right ) of the corresponding CDL problem. The behavior of the solutions in the near-horizon ( ξ ≪ R 5 ) and asymptotic ( ξ ≫ R 5 ) regimes is indicated in the figure, following eq. (2.21) and eq. (2.22), respectively.
Figure 6 Cartoon 3 Figure 3 . The five-dimensional Euclidean Schwarzschild gravitational instanton, or Witten's BON, expressed in terms of the degrees of freedom φ bon ( ξ ) ( left ) and ρ bon ( ξ ) ( right ) of the corresponding CDL problem. The behavior of the solutions in the near-horizon ( ξ ≪ R 5 ) and asymptotic ( ξ ≫ R 5 ) regimes is indicated in the figure, following eq. (2.21) and eq. (2.22), respectively.CC-BY · From the PDF
Figure 6 Cartoon 3 Figure 4 . A diagram comparing the BON and CDL boundary conditions. The BON initial conditions φ (0) → -∞ , φ ′ → + ∞ ensure that φ ( ξ ) quickly approaches the false vacuum. Its approach to φ ( ξ ) → φ fv is determined by the details of the potential, which is approximately quadratic in the neighborhood of φ ≈ φ fv . This direct approach to φ fv is indicated by the straight blue line in the figure. The CDL solution, on the other hand, begins at finite φ (0) = φ CDL with φ ′ (0) = 0 . It subsequently 'rolls' in the inverted potential, as indicated by the dashed red line, until it approaches φ fv . In the dashed blue line we show a second BON example, one in which the potential U ( φ → -∞ ) diverges (with -U ( φ ) shown in dark gray). The φ ′ (0) → ∞ BON initial conditions can compensate for the divergence in U ( φ ) , so that there is a BON solution even if there is no corresponding CDL solution for that potential.
Figure 6 Cartoon 3 Figure 4 . A diagram comparing the BON and CDL boundary conditions. The BON initial conditions φ (0) → -∞ , φ ′ → + ∞ ensure that φ ( ξ ) quickly approaches the false vacuum. Its approach to φ ( ξ ) → φ fv is determined by the details of the potential, which is approximately quadratic in the neighborhood of φ ≈ φ fv . This direct approach to φ fv is indicated by the straight blue line in the figure. The CDL solution, on the other hand, begins at finite φ (0) = φ CDL with φ ′ (0) = 0 . It subsequently 'rolls' in the inverted potential, as indicated by the dashed red line, until it approaches φ fv . In the dashed blue line we show a second BON example, one in which the potential U ( φ → -∞ ) diverges (with -U ( φ ) shown in dark gray). The φ ′ (0) → ∞ BON initial conditions can compensate for the divergence in U ( φ ) , so that there is a BON solution even if there is no corresponding CDL solution for that potential.CC-BY · From the PDF
antipodal points of S 3 Figure 4: Diagram R5 2 1.5 Figure 9: Piecewise toy potential U ( ), together with more realistic examples 4 Figure 5 . Left: the piecewise potential U toy ( φ ) from eq. (3.17), with φ 1 = -0 . 2 M p and U fv = 0 . 2 U 0 . Right: the quantity φ ⋆ = -m 2 R 2 5 2 √ 3 2 M p indicates the point where the potential U ( φ ) begins to have a noticeable effect on the equation of motion for φ (see the discussion around eq. (3.45)). For φ ≪ φ ⋆ , φ ( ξ ) is well approximated by the Witten BON solution, even if the potential U ( φ ) is large. This feature greatly enhances the utility of the piecewise quadratic toy model. All three of the 'realistic' smooth potentials will have similar solutions for φ ( ξ ) and ρ ( ξ ) , despite the significant differences in their asymptotic φ →-∞ limits, as long as mR 5 ≪ 1 is small enough that U ( | φ | ≤ | φ ⋆ | ) remains approximately quadratic.
antipodal points of S 3 Figure 4: Diagram R5 2 1.5 Figure 9: Piecewise toy potential U ( ), together with more realistic examples 4 Figure 5 . Left: the piecewise potential U toy ( φ ) from eq. (3.17), with φ 1 = -0 . 2 M p and U fv = 0 . 2 U 0 . Right: the quantity φ ⋆ = -m 2 R 2 5 2 √ 3 2 M p indicates the point where the potential U ( φ ) begins to have a noticeable effect on the equation of motion for φ (see the discussion around eq. (3.45)). For φ ≪ φ ⋆ , φ ( ξ ) is well approximated by the Witten BON solution, even if the potential U ( φ ) is large. This feature greatly enhances the utility of the piecewise quadratic toy model. All three of the 'realistic' smooth potentials will have similar solutions for φ ( ξ ) and ρ ( ξ ) , despite the significant differences in their asymptotic φ →-∞ limits, as long as mR 5 ≪ 1 is small enough that U ( | φ | ≤ | φ ⋆ | ) remains approximately quadratic.CC-BY · From the PDF
Image 6
CC-BY · From the PDF
Figure 20 Potential U ( ) Figure 7 . Left: the scalar potential ˆ U ( φ ) , as given in eq. (4.1), for several values of δ ≡ U fv /U 0 , as indicated in the figure. A potential of this kind, exhibiting a de Sitter vacuum that is separated by a finite barrier from the compactification limit, admits hybrid BON-CDL and BON-HM instanton solutions. Right: to illustrate the physical meaning of a , we show ˆ U with fixed δ = 0 . 2 as a function of the proper length of the KK direction, L = 2 πR 5 exp (√ 2 3 φ/M p ) . Larger values of a create a steeper potential near the false vacuum at L = 2 πR 5 , while also flattening U ( L ) at L ≪ πR 5 .
Figure 20 Potential U ( ) Figure 7 . Left: the scalar potential ˆ U ( φ ) , as given in eq. (4.1), for several values of δ ≡ U fv /U 0 , as indicated in the figure. A potential of this kind, exhibiting a de Sitter vacuum that is separated by a finite barrier from the compactification limit, admits hybrid BON-CDL and BON-HM instanton solutions. Right: to illustrate the physical meaning of a , we show ˆ U with fixed δ = 0 . 2 as a function of the proper length of the KK direction, L = 2 πR 5 exp (√ 2 3 φ/M p ) . Larger values of a create a steeper potential near the false vacuum at L = 2 πR 5 , while also flattening U ( L ) at L ≪ πR 5 .CC-BY · From the PDF
Image 8
CC-BY · From the PDF
Figure 14 Parameter space R 3 /R 5 Figure 9 . The trajectories of the BON bounce solutions in the ( η, R 5 ) plane are shown for a = 5 and various 0 < δ < 1 . Each value of δ has a BON-FV branch of solutions, with R 3 ≈ R 5 ( φ 0 ≈ 0 ) in the u 0 ≪ 1 limit where the potential becomes unimportant. The BON-CDL branch of solutions has R 3 < R 5 ( φ 0 ≈ φ cdl ) in the u 0 ≪ 1 limit, and an action ∆ S ≈ S CDL that is large compared to π 2 M 2 p R 2 5 . When u 0 ≳ 1 , the potential U ( φ ) is no longer a small perturbation to the Witten bubble, and we find that the BON-FV and BON-CDL branches merge together, with u 0 for the bounce solution bounded from above by some u max 0 . For u 0 > u max 0 , all solutions to the equations of motion are of the overshoot type. As δ → 0 the CDL action diverges as δ -1 , and the unification of the BON-FV and BON-CDL branches is pushed to ever-larger values of u 0 ≫ 1 .
Figure 14 Parameter space R 3 /R 5 Figure 9 . The trajectories of the BON bounce solutions in the ( η, R 5 ) plane are shown for a = 5 and various 0 < δ < 1 . Each value of δ has a BON-FV branch of solutions, with R 3 ≈ R 5 ( φ 0 ≈ 0 ) in the u 0 ≪ 1 limit where the potential becomes unimportant. The BON-CDL branch of solutions has R 3 < R 5 ( φ 0 ≈ φ cdl ) in the u 0 ≪ 1 limit, and an action ∆ S ≈ S CDL that is large compared to π 2 M 2 p R 2 5 . When u 0 ≳ 1 , the potential U ( φ ) is no longer a small perturbation to the Witten bubble, and we find that the BON-FV and BON-CDL branches merge together, with u 0 for the bounce solution bounded from above by some u max 0 . For u 0 > u max 0 , all solutions to the equations of motion are of the overshoot type. As δ → 0 the CDL action diverges as δ -1 , and the unification of the BON-FV and BON-CDL branches is pushed to ever-larger values of u 0 ≫ 1 .CC-BY · From the PDF
Image 10
CC-BY · From the PDF
Figure 31 ⇢ ( ⇠ ) with inset Left: BON-FV Right: BON-CdL 15 Figure 11 . Profiles for ρ ( ξ ) for U 0 R 2 5 /M 2 p = 0 . 01 on the BON-FV ( left ) and BON-CDL ( right ) branches. For the BON-FV branch the solutions for ρ ( ξ ) are nearly identical to the false vacuum solution ρ fv ( ξ ) for ξ ≳ R 5 . This encompasses the majority of the domain 0 ≤ ξ ≤ ξ max ≈ π Λ , which is why we only show δ = 0 . 001 . . . 0 . 2 . For the BON-CDL solutions, the profiles for ρ ( ξ ) follow the ξ ≫ R 5 linear growth of eq. (2.22) out to ξ ≳ 10 R 5 or larger, with a relatively sharp transition from ρ ′ ≈ 1 to ρ ′ ≈ -1 in the case of small δ . As δ → 1 , the BON-CDL solutions for ρ ( ξ ) more closely match the Hawking-Moss behavior ρ ( ξ ) ≈ ρ HM ( ξ ) for all ξ ≳ R 5 , even though the a = 5 CDL and HM solutions for φ ( ξ ) are distinct.
Figure 31 ⇢ ( ⇠ ) with inset Left: BON-FV Right: BON-CdL 15 Figure 11 . Profiles for ρ ( ξ ) for U 0 R 2 5 /M 2 p = 0 . 01 on the BON-FV ( left ) and BON-CDL ( right ) branches. For the BON-FV branch the solutions for ρ ( ξ ) are nearly identical to the false vacuum solution ρ fv ( ξ ) for ξ ≳ R 5 . This encompasses the majority of the domain 0 ≤ ξ ≤ ξ max ≈ π Λ , which is why we only show δ = 0 . 001 . . . 0 . 2 . For the BON-CDL solutions, the profiles for ρ ( ξ ) follow the ξ ≫ R 5 linear growth of eq. (2.22) out to ξ ≳ 10 R 5 or larger, with a relatively sharp transition from ρ ′ ≈ 1 to ρ ′ ≈ -1 in the case of small δ . As δ → 1 , the BON-CDL solutions for ρ ( ξ ) more closely match the Hawking-Moss behavior ρ ( ξ ) ≈ ρ HM ( ξ ) for all ξ ≳ R 5 , even though the a = 5 CDL and HM solutions for φ ( ξ ) are distinct.CC-BY · From the PDF
Figure 24 Action as function of R 5 Figure 12 . The de Sitter-subtracted Euclidean action ∆ S is shown in units of M 4 p /U 0 as a function of R 5 , for a = 5 and various 0 < δ < 1 . The dashed gray line shows the ∆ S ≈ π 2 M 2 p R 2 5 expectation from the small u 0 ≪ 1 limit of the BON-FV solutions. It tracks the numericallycomputed action closely on this branch for all u 0 ≪ 1 . This branch is always of lowest action and determines the tunneling rate. On the BON-CDL branch the action approaches a constant, the ordinary CDL action ∆ S ≈ S CDL , which is large compared to M 4 p /U 0 for δ ≲ 0 . 5 . For δ ≪ 1 , S CDL ≈ 24 π 2 M 4 p / ( δ · U 0 ) , grows as 1 /δ . At larger values of u 0 , where π 2 M 2 p R 2 5 ∼ ∆ S CDL , the BON-FV and BON-CDL branches of bounce solutions meet. (This method of estimating the value of u 0 = U 0 R 2 5 /M 2 p at which the two branches meet becomes imprecise once u 0 ≳ 1 , where π 2 M 2 p R 5 is no longer a good approximation to the BON-FV action.)
Figure 24 Action as function of R 5 Figure 12 . The de Sitter-subtracted Euclidean action ∆ S is shown in units of M 4 p /U 0 as a function of R 5 , for a = 5 and various 0 < δ < 1 . The dashed gray line shows the ∆ S ≈ π 2 M 2 p R 2 5 expectation from the small u 0 ≪ 1 limit of the BON-FV solutions. It tracks the numericallycomputed action closely on this branch for all u 0 ≪ 1 . This branch is always of lowest action and determines the tunneling rate. On the BON-CDL branch the action approaches a constant, the ordinary CDL action ∆ S ≈ S CDL , which is large compared to M 4 p /U 0 for δ ≲ 0 . 5 . For δ ≪ 1 , S CDL ≈ 24 π 2 M 4 p / ( δ · U 0 ) , grows as 1 /δ . At larger values of u 0 , where π 2 M 2 p R 2 5 ∼ ∆ S CDL , the BON-FV and BON-CDL branches of bounce solutions meet. (This method of estimating the value of u 0 = U 0 R 2 5 /M 2 p at which the two branches meet becomes imprecise once u 0 ≳ 1 , where π 2 M 2 p R 5 is no longer a good approximation to the BON-FV action.)CC-BY · From the PDF
Figure 16 Potential U ( , ⇤ ) Figure 13 . The scalar potential ˆ U λ ( φ ) of eq. (5.4), for several values of λ ≡ Λ CC 5 M 2 p /U 0 .
Figure 16 Potential U ( , ⇤ ) Figure 13 . The scalar potential ˆ U λ ( φ ) of eq. (5.4), for several values of λ ≡ Λ CC 5 M 2 p /U 0 .CC-BY · From the PDF
Image 14
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Image 15
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Image 16
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Metadata

Authors
Draper, P., Garcia, I. G., & Lillard, B.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2021-12-21
Volume
2021
Issue
12
ISSN
1029-8479
DOI
10.1007/jhep12(2021)154
Cited by
14
References
9
Language
en
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