An lowerbound on the bounce action - KEK...2018. 12. 4 @ KEK-PH 2018 winter An lowerbound on the...
Transcript of An lowerbound on the bounce action - KEK...2018. 12. 4 @ KEK-PH 2018 winter An lowerbound on the...
2018. 12. 4 @ KEK-PH 2018 winter
An lowerbound on the bounce action.
Ryosuke Sato, Masahiro Takimoto
[arXiv:1707.01099]
Phys. Rev. Lett. 120 (2018) 091802
[ 1 / 10 ]
Ryosuke Sato (DESY)
ฮ โ exp(โ๐๐ธ[๐])
In general, it is tough to solve EOM explicitly
> Lifetime of metastable
[Coleman (1977), Callan Coleman (1977)]
๐๐ธ ๐ = โซ ๐4๐ฅ1
2๐ป๐ 2 + ๐ ๐
[ 2 / 10 ]
๐ป2๐๐ โ๐๐
๐๐๐= 0,
Lifetime of metastable vacuum.
> Bounce solution
[ 3 / 10 ]
๐๐ธ ๐ โฅ24
๐๐๐ ๐๐๐ โก max
๐โ
4๐ ๐
|๐|4
๐
๐(๐)
โ๐๐๐
4๐ 4
๐ป2๐๐ โ๐๐
๐๐๐= 0,
๐๐ธ ๐ = โซ ๐4๐ฅ1
2๐ป๐ 2 + ๐ ๐
lim๐ฅ โโ
๐๐ ๐ฅ = 0 (false vacuum) Bounce sol. :
Bounce action :
[Sato, Takimoto (2017)]
Lowerbound on the bounce action.
> Setup
> Result
๐๐ ๐ฅ โก ๐(๐๐ฅ) ๐ ๐๐ = ๐2๐ + ๐4๐
๐๐[๐๐]
๐๐ ๐=1
= 0 2๐ = โ4๐, ๐ =1
2๐
๐ = 1
2๐ =
1
2 ๐๐ ๐2๐3
๐๐
๐๐
2โ
0
Bounce sol. is spherical.
[Callan Glaser Martin (1977)]
[Blum Honda Sato Takimoto Tobioka (2016)]
[ 4 / 10 ]
Bounce = sol. of EOM โ ๐ฟ๐ = 0 for any perturbation ๐ฟ๐
๐ = โซ ๐4๐ฅ1
2๐ป๐ 2
๐ = โซ ๐4๐ฅ ๐(๐)
Let us assume the existence of the bounce sol.
Derivation (1/3) : preparation.
๐ฟ๐ = ๐ 0 โ ๐ โ = โ ๐๐๐๐
๐๐
โ
0
๐ฟ๐ = ๐ โ โ ๐ 0 = ๐๐3
๐
๐๐
๐๐
2โ
0
Let us fix ๐ฟ๐ and ๐ฟ๐ > 0 by hand for the moment
๐ = ๐ + 2๐ผ ๐ฟ๐ + ๐๐โ
0
๐๐
๐๐โ ๐ฝ ๐ฟ๐ โ ๐๐
โ
0
3
๐
๐๐
๐๐
2
๐๐๐๐ ๐ฟ๐, ๐ฟ๐ = 12 ๐ฟ๐ 4
๐ฟ๐
๐ =1
2๐ = ๐๐ ๐2๐3
๐๐
๐๐
2โ
0
๐ฟ๐
๐ฟ๐
[ 5 / 10 ]
โ ๐ can be minimized by Lagrange multiplier method.
Derivation (2/3) : Lagrange multiplier.
๐ฟ๐
๐ฟ๐
[ 6 / 10 ]
๐ =๐
2 โฅ min
๐
๐๐๐๐ ๐ฟ๐, ๐ฟ๐
2 = min
๐
6 ๐ฟ๐ 4
๐ฟ๐
We do not know ๐ฟ๐ and ๐ฟ๐
Derivation (3/3) : bound on S.
๐๐๐ โก max๐
โ4๐ ๐
๐ 4 ๐ โฅ24
๐๐๐,
We do not know ๐ฟ๐ and ๐ฟ๐
The minimum of lowerbound is smaller than ๐ ๐ฟ๐
๐ฟ๐
[ 7 / 10 ]
๐ =๐
2 โฅ min
๐
๐๐๐๐ ๐ฟ๐, ๐ฟ๐
2 = min
๐
6 ๐ฟ๐ 4
๐ฟ๐
โ๐๐๐
4๐ 4
Derivation (3/3) : bound on S.
๐ =1
2๐2๐2 โ
1
3๐ด๐3 +
1
4๐4๐4, ๐ โก
9๐4๐2
8๐ด2
Ratio between the actual value of lowerbound
~ Double-well
Thin-wall
~ negative quartic
Fubini instanton like
[ 8 / 10 ]
๐ =9๐2
2๐ด2๐(๐ )
Example 1 : single scalar field.
[ 9 / 10 ]
Red line is given by Hisano-Sugiyama (2011)
(๐ = 700 GeV, ๐๐๐ = ๐๐ฟ + 200 GeV)
๐ฟ โ ๐๐
๐ฃ๐tan๐ฝ โ๐ ๐ฟ๐ ๐
Example 2 : MSSM with large tanbeta.
> In general, it is technically tough to calculate lifetime of metastable vacuum.
> We derived a lowerbound without solving EOM
> It is useful to analyze multiscalar models.
[ 10 / 10 ]
๐๐ธ ๐ โฅ24
๐๐๐ ๐๐๐ โก max
๐โ
4๐ ๐
|๐|4
๐
๐(๐)
โ๐๐๐
4๐ 4
Summary.
Backup
Lagrange multiplier
๐ฟ๐ = ๐ 0 โ ๐ โ = โ ๐๐๐๐
๐๐,
โ
0
๐ฟ๐ = ๐ โ โ ๐ 0 = ๐๐3
๐
๐๐
๐๐
2โ
0
๐ = ๐ + 2๐ผ ๐ฟ๐ + ๐๐โ
0
๐๐
๐๐โ ๐ฝ ๐ฟ๐ โ ๐๐
โ
0
3
๐
๐๐
๐๐
2
๐๐
๐๐= โ
๐ผ๐
๐2๐4 + 3๐ฝ
๐ = ๐๐ ๐2๐3๐๐
๐๐
2โ
0
๐ฟ๐
๐ฟ๐
๐ฟ๐
๐ฟ๐ = 0, ๐ = โ
๐ผ๐
๐2๐4 + 3๐ฝ,
๐๐
๐๐ผ=
๐๐
๐๐ฝ= 0 ๐ผ =
24 ๐ฟ๐ 3
๐ฟ๐, ๐ฝ =
12 ๐ฟ๐ 4
๐ฟ๐ 2
๐ ๐ผ =24 ๐ฟ๐ 3
๐ฟ๐, ๐ฝ =
12 ๐ฟ๐ 4
๐ฟ๐ 2 =
12 ๐ฟ๐ 4
๐ฟ๐+ ๐๐
๐2๐4 + 3๐ฝ
๐๐ +
๐ผ๐
๐2๐4 + 3๐ฝ
2โ
0
Let us determine lowerbound of ๐ for fixed ๐ฟ๐ and ๐ฟ๐
Minimize ๐ (Lagrange multiplier method)
ใใใซใ
ๆฅตๅคใฎๆกไปถใฏใ
For fixed ๐ฟ๐ and ๐ฟ๐, ๐ โฅ12 ๐ฟ๐ 4
๐ฟ๐