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Bubble of nothing decays of unstable theories

Draper, Patrick, Garcia, Isabel Garcia, Lillard, Benjamin·2021·Physical Review D
DOI 10.1103/physrevd.104.l121701
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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
CC-BY · From the PDF
Image 15
CC-BY · From the PDF
Image 16
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Metadata

Authors
Draper, P., Garcia, I. G., & Lillard, B.
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2021-12-02
Volume
104
Issue
12
ISSN
  • 2470-0010
  • 2470-0029
DOI
10.1103/physrevd.104.l121701
Cited by
15
References
3
Language
en
Licence
creativecommons.org/licenses/by/4.0/ (vor)
Funders
  • U.S. Department of Energy (DE-SC0015655)
  • Gordon and Betty Moore Foundation (GBMF7392)
  • National Science Foundation (PHY-1748958)

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Draper, P., Garcia, I. G., & Lillard, B. (2021). Bubble of nothing decays of unstable theories. Physical Review D. https://doi.org/10.1103/physrevd.104.l121701

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