真空崩壊確率の完全な1-LOOP ORDER での計算とその応用について

素粒子物理学の進展2015, 9/14-18
真空崩壊確率の完全な1-LOOP ORDER
での計算とその応用について
∼スケール不定性を減らせるか∼
庄司裕太郎
東京大学
Now ongoing…
Collaborators: Motoi Endo, Takeo Moroi, Mihoko Nojiri (KEK)
目次
•
導入 - 核生成確率とスケール不定性 -
•
1-loop orderでの計算 - Toy model -
•
SM + STAU SYSTEM - Top loop -
•
まとめ
導入
核生成確率とスケール不定性
準安定真空の崩壊
Standard Model
V
180
6 8 10
200
107
108
109
Instability
Instability
lity 10
abi
9
t
s
8
tae
7
M
6
5
LI =104 GeV
100
Stability
50
12
14
16
Φ
Top pole mass Mt in GeV
150
1010
19
Non-perturbativity
Top pole mass Mt in GeV
178
1011
1012
1013
176
1016
174
1,2,3 s
Meta-stability
1019
172
170
1018
14
False Vacuum
Stability
10
0
0
50
True Vacuum
100
150
200
Higgs pole mass Mh in GeV
168
120
17
10
122
124
126
128
130
132
Higgs pole mass Mh in GeV
D. Buttazzo, et. al. 1307.3536/hep-ph
Figure 3: Left: SM phase diagram in terms of Higgs and top pole masses. The plane is
divided into regions of absolute stability, meta-stability, instability of the SM vacuum, and nonperturbativity of the Higgs quartic coupling. The top Yukawa coupling becomes non-perturbative
for Mt > 230 GeV. The dotted contour-lines show the instability scale ⇤I in GeV assuming
Higgs
mass,
hgg, hγγ,
…Mh and Mt
↵3 (MZ ) = 0.1184. Right: Zoom in the region of
the preferred
experimental
range of
(the grey areas denote the allowed region at 1, 2, and 3 ). The three boundary lines correspond
to 1- variations of ↵3 (MZ ) = 0.1184 ± 0.0007, and the grading of the colours indicates the size
muon g-2, hγγ, …
of the theoretical error.
Minimal Supersymmetric Standard Model
Top quark
Leptons
SUSY
Stop
ht̃L t̃R
Sleptons h`˜L `˜R
The quantity e↵ can be extracted from the e↵ective potential at two loops [112] and is explicitly
given in appendix C.
Charge and/or Color Breaking Minima
4.3
The SM phase diagram in terms of Higgs and top masses
The two most important parameters that determine the various EW phases of the SM are the
真空の崩壊確率
Bubble Nucleation Rate
Tunneling FV
FV
TV
E
At the Leading order,
Bounce action
= Ae
B = SE (
B
E
B : Bounce solution
B)
V
@2 = V 0( )
dim: [1/(time*volume)]
Pre-exponential factor
A'm
4
r = ±1
r=0
m : “typical” mass scale
SO(4) symmetric classical solution
TOY MODEL
Potential
Bounce
Action
Nucleation Rate
2
m
V =
2
@
2
A
2
2
3
↵
+
8
4
0
=V ( )
B = SE (
B) =
4
'm e
B
Z

1
d x (@
2
4
2
)
+V(
B
B)
TOY MODEL
In fact, the potential is scale-dependent.
m̄2 (Q)
t̄(Q) +
2
Potential V =
Bounce
Action
Nucleation Rate
@
2
Ā(Q)
2
2
3
↵(Q)
¯
+
8
4
0
=V ( )
B = SE (
B) =
4
'm e
Z

1
d x (@
2
4
2
)
+V(
B
B)
B
Maybe, the best Q is a “typical” scale…
繰り込み点
V
Momentum of the bounce
r = ±1
V
r=0
But, we do not know what is the best scale
Height of the barrier
Depth of TV
Distance btw. FV and TV
Particle mass @ FV
Particle mass @ TV
Φ
スケール依存性の大きさ
V =
m̄2 (Q)
t̄(Q) +
2
2
Ā(Q)
2
3
↵
¯ (Q)
+
8
4
Beta functions
t
3Am2
=
16⇡ 2
A
9↵A
=
16⇡ 2
m2
↵
3
2
2
=
(↵m
+
3A
)
2
16⇡
9↵2
=
16⇡ 2
Renormalization conditions
@Q = m
m̄2 (m) = m2 , Ā(m) = m, t̄(m) = 0, ↵
¯ (m) = ↵
スケール依存性の大きさ
405
●
●
●
400
h iT V
VT V
●
●
395
kB
↵ = 0.6
●
●
Vtop
B
390
●
m
385
,F V
m
●
,T V
●
Preliminary
●
●
380
●
●
375
●
●
0
1
2
3
Q/m
4
5
~10% uncertainty
Much larger uncertainty in a realistic model (w/ top loop)
1-LOOP ORDERでの計算
さて、どうしましょうか。
ちゃんと読み返してみると
= Ae
B
2
B
A=
4⇡ 2
Expectation
✓
0
00
det S |Bounce
det S 00 |False
◆
B
Cancellation of
the scale dependence
@1-loop
cf.) RGEs are related to
1/2
B
B
B
B
B
divergent part
もっと読み返してみると
= Ae
B
2
B
A=
4⇡ 2
✓
0
00
det S |Bounce
det S 00 |False
◆
1/2
SOLVE corresponding Ordinary Differential Equations
Since 1928
Many mathematical proofs,
but not so many pheno. results
I. M. Gelfand, A. M. Yaglom;
S. Coleman;
J. H. van Vleck;
R. H. Cameron, W. T. Martin;
R. Dashen, B. Hasslacher, A. Neveu;
R. Forman;
K. Kirsten, A. J. McKane;
…
method, proof, renormalization, zero modes, fermions, implementation, …
Please invite me to your LAB!!
結果
= Ae
405
●
●
●●●
●● ● ● ● ● ●
●
●
●
●
●
●
400
395
B, B+δB
B
390
4
⌘m e
●
●
1-loop
●
●
B
●
●
●
↵ = 0.6
●
385
B
●
Classical
●
380
●
●
375
●
●
0
1
2
3
Q/m
4
5
Preliminary
結果
↵ = 0.3
7300
7200
●
●
●
1-loop
●
●
●
●
●
152
B, B+δB
●●● ●
●● ● ●
● ● ●
●
●
154
●●●
●● ● ● ● ●
●
●
●
●
●
●
●
●
●
7400
●
●
●
●
●
156
B, B+δB
↵ = 0.9
7100
7000
●
●
●
●
●
●
1-loop
●
●
●
●
Classical
●
●
6900
●
150
●
Classical
6800
●
●
●
●
0
2
●
4
6
Q/m
8
10
●
0.0
0.5
Preliminary
V
Φ
1.0
1.5
2.0
2.5
Q/m
Preliminary
3.0
V
Φ
SM+STAU SYSTEM
TOP LOOP
軽いSTAU
Staus can be light
m⌧˜ > 103.5GeV (LEP)
hγγ coupling, co-annihilation with Bino, …
But, the potential may become unstable toward the stau direction
V
2
2
1
m
m
= + p2 y⌧ X⌧ ⌧˜L ⌧˜R h + L ⌧˜L2 + R ⌧˜R2 + · · ·
2
2
Stable
EW vacuum is
the global minimum
Meta-Stable
tdec & 13.8Gyr
Unstable
tdec . 13.8Gyr
X ⌧ = A⌧
tan
µ tan
= hHu0 i/hHd0 i
No EW vac.
Tachyonic stau
|X⌧ |
考えるスペクトラム
For simplicity,
we consider the case where only the staus are light
O(10TeV)
O(100GeV)-O(1TeV)
Other superparticles, heavy Higgs bosons
Staus
173GeV
Top quark
125GeV
SM Higgs boson
Effective theory
有効理論
h
SM+Staus
3yt2
=
4⇡ 2
h
t
3yt4
8⇡ 2
h dominant contrib.
h
yt
p ht̄R tL
2
4
L= + h
4
1
+ p y⌧ X⌧ ⌧˜L ⌧˜R h
2
X ⌧ = A⌧
1
+
4
✓
y⌧2
g 02
+ (2˜
⌧R2
32
g 02
g2
4
⌧˜L2 )2
cos 2
◆
1
2 2
⌧˜L h +
4
= hHu0 i/hHd0 i
tan
Mh2 2 m2L 2 m2R 2
⌧˜L +
⌧˜R
+
h +
2
2
2
✓
y⌧2
g 02
cos 2
2
µ tan
Inputs
◆
⌧˜R2 h2
g2 4
+ ⌧˜L
32
RGE, Det: Calculated up to leading in y_t
=> Overall factor of Det is NOT determined!
結果其の壱
mL = mR = 600GeV, X⌧ = 95TeV, tan
Preliminary
●
500
= 15
Classical
●
450
●
B
●
●
400
●
●
●
350
●
●
●
●
●
●
300
mt
2
200
400
600
Q [GeV]
800
1000
●
●
1200
2m⌧˜
結果其の壱
mL = mR = 600GeV, X⌧ = 95TeV, tan
Preliminary
●
Classical
500
= 15
B, B+δB
●
450
Partial 1-loop (y_t only)
●
●
●
●
400
●
●
●
●
●
●
●
●
●
●
●
●
●
350
●
●
●
●
●
●
●
●
●
300
mt
2
200
400
600
Q [GeV]
800
1000
●
●
●
●
1200
2m⌧˜
Caveat: Overall factor is NOT determined!
結果其の弐
= 15
mL = mR = m⌧˜
tdec & 13.8Gyr , B(+ B) & 400
Tachyonic stau
mt
< Q < 2m⌧˜
2
Partial 1-loop (B+δB=400)
Classical (B=400)
200
Xτ [TeV]
tan
Unstable
150
e
l
b
ta
s
a
et
M
100
600
Preliminary
800
1000
m ∼τ [GeV]
1200
1400
Stable
Caveat: The position of the green lines can be changed!
まとめ
•
The bubble nucleation rate has often been estimated
without calculating the pre-exponential factor.
•
This estimate involves uncertainty in the renormalizaiton
scale, which, we showed, results in O(10%) uncertainty in
the exponent of the bubble nucleation rate.
•
To reduce the uncertainty, we explicitly calculated the preexponential factor and showed that it is greatly reduced.
•
Scalars and fermions have already been implemented, but
the gauge bosons are now ongoing.
405
●
●
●● ● ● ● ● ●
●●●
●
●
●
●
●
●
400
B, B+δB
395
390
●
●
●
●
●
●
●
Boson
●
385
●
●
380
●
●
375
●
●
0
1
2
3
Q/m
●
500
B, B+δB
●
450
●
●
●
●
400
●
●
●
●
Fermion
●
●
●
●
●
●
●
●
●
350
●
●
●
●
●
●
●
●
●
300
200
400
600
Q [GeV]
800
1000
●
●
●
●
1200
4
5
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