2 - Kanazawa University, Institute for Theoretical Physics

SM criticality
&
Higgs inﬂation
Kin-ya Oda
(Osaka) !
with !
✦
Yuta Hamada
✦ YH, HK & Seong
✦
& Hikaru
Kawai
Chan Park
Masatoshi Yamada
✦ YH & Fuminobu
(Kyoto), PRD(2013), PTEP(2014) & JHEP(2014). (Sungkyunkwan), PRL(2014) & arXiv:1408. (Kanazawa), in progress. Takahashi
(Tohoku), appearing tomorrow.
Higgs discovery (2012)
[Picture from NY Times]
Higgs discovery (2012)
※ pictures from web
[Picture from NY Times]
Higgs discovery (2012)
※ pictures from web
[Picture from NY Times]
http://seiga.nicovideo.jp/seiga/im4010102
T IME AND M ATTER 2013 C ONFERENCE
Very much SM-ish
✦
If SM Higgs, ✴
Source for all particlesʼ’ mass: ✤
✴
✦
coupling/mass = VEV (const.) On dashed line. It is being so in two orders. ✴
Not only with gauge bosons. ✴
But also with quarks & lepton.
[CMS, 1309.0721]
e plane of reduced vector boson coupling
Great
Victory
of SM!
picture from web
Outline
1. Criticality: We live right on the edge of vacuum instability. 2. Indicates new principle beyond ordinary QFT? 3. Higgs, the only single sole ever observed elementary scalar ﬁeld, can serve as inﬂaton.
Outline
1. Already
Criticality: We live right on the by
edge covered
of vacuum instability. Nobuchika s review.
2. Indicates new principle beyond ordinary QFT? 3. Higgs, the only single sole ever observed elementary scalar ﬁeld, can serve as inﬂaton.
Outline
1. Already
Criticality: We live right on the by
edge covered
of vacuum instability. Nobuchika s review.
2. Indicates new principle beyond ordinary QFT? 3. Already
Higgs, the only covered
single sole ever by
observed elementary scalar ﬁeld, can Nobuchika
serve as inﬂaton. s review.
Main ﬁndings
1. Bare Higgs mass also points to a “criticality”. •
Hamada, Kawai, KO, PRD(2013). 3. SM criticality allows Higgs inﬂation with large r〜～0.1 with ξ〜～10−100. •
✦
Hamada, Kawai, KO, Park, PRL(2014). SM criticality may be explained by eternal topological Higgs inﬂation. (※Diﬀerent one from above.) •
Hamada, KO, Takahashi, appearing tomorrow.
Outline
1. Criticality: We live right on the edge of vacuum instability. 2. Indicates new principle beyond ordinary QFT? 3. Higgs, the only single sole ever observed elementary scalar ﬁeld, can serve as inﬂaton.
nd corresponds to 95% CL deviation of Mt ; see Eq. (10).
Vacuum (in)stability
Mt =171.39294 GeV
1¥1068
Mt =171.39314 GeV
4
V @GeV D
Mt =171.39334 GeV
5¥1067
0
-5¥10
17
Mt =171.39354 GeV
67
0
1¥10
18
18
j @GeVD
2¥10
(mt numbers given just to show amount of tuning)
3¥10
18
4¥10
18
[Hamada, Kawai, KO, 2014]
We are put on the edge
3.0
150
ty 10
i
l
i
tab 8 9
s
ta
e
7
M
6
5
4
LI =10 GeV
100
12
14
16
19
Stability
50
0
0
50
100
150
Higgs pole mass Mh in GeV
200
Planck-scale107
Instability
dominated
2.5
178
2.0
176
108
109
Meta-stability
Instability
Non-perturbativity
Top pole mass Mt in GeV
200
180
Yukawa coupling
yt HM
L och.
Pl
Cf. 1σ bTop
y ATop
lekhin, DM
jouadi &
M
pole mass
in
GeV
t
6 8 10
1010
1011
1012
1013
Instability
Stability
1.5
174
Meta-stability
1016
1,2,3 s
1.0
172
1019
0.5
170
SM
1018
1014
0.0
168
120
-2
17
10
No EW vacuum
122
-1 124
Stability
126
128
Higgs coupling lHMPl L
0
1 130
132
2
Higgs pole mass Mh in GeV
[Buttazzo et al. 1307.3536]
Figure 3: Left: SM phase diagram in terms of Higgs and top pole masses. The plane is
Figure 4: Left:
SM of
phase
diagram
in terms
of qu
divided into regions of absolute stability, meta-stability,
instability
the SM
vacuum,
and nonat the becomes
Planck non-perturbative
scale. The regio
yt renormalised
perturbativity of the Higgs quartic coupling. The
top Yukawa coupling
On the edge
※ Pictures from web.
On the edge
※ Pictures from web.
On the edge
※ Pictures from web.
On the edge
※ Pictures from web.
On the edge
※ Pictures from web.
On the edge
※ Pictures from web.
On the edge
※ Pictures from web.
On the edge
※ Pictures from web.
On the edge
※ Pictures from web.
On the edge
※ Pictures from web.
Criticality in bare mass
✦
Bare mass becomes small too for Planck scale cutoﬀ. !
!
!
!
✦
[Hamada, Kawai, KO, 2013]
Triple coincidence: λ, βλ & mB2 all become small at Planck scale!
Backup: Quadratic divergence
✦
2
2
2
2
mR = mB + (λ/16π +…) Λ + δm . renʼ’d
✦
2
bare
radiative corrections
Mass independent renormalization scheme: Skippable
I.
II.
Choose mB2 such that mR2 =0 for δm2 =0. ✤
Subtractive renormalization of Λ2. ✤
This step automatic in dim. reg. Include δm2 perturbatively. ✤
✦
Multiplicative renormalization of log Λ. 2
2
Dominant is not running mass δm , but bare mass mB .
Veltman condition
!
✦
2
2
2
2
mR = mB + (λ/16π +…) Λ + δm . renʼ’d
!
bare
radiative corrections
=0.
!
✦
2
”This mass-‐‑‒relation, implying a certain cancellation between bosonic and fermionic eﬀects, would in this view be due to an underlying supersymmetry. While quite speculative, the relation has the virtue of being veriﬁable in the not too distant future.” Veltman (1981) ✴
Diﬀerent from MSSM cancellation, in Veltmanʼ’s view. ✴
This is indeed veriﬁed after 30 years. ✴
Taken literally, indicates SUSY breaking at Planck scale.
Just to show how it looks
Skippable
igure 1: Nonvanishing two-loop Feynman diagrams. Arrows are omitted. The dash
olid, wavy, and dotted lines represent the scalar, fermion, gauge, and ghost propagat
spectively.
at ⇤ becomes9
m2B, 2-loop
Figure 1: Nonvanishing two-loop Feynman diagrams. Arrows are omitted. The dashed,
solid, wavy, and dotted lines represent the scalar, fermion, gauge, and ghost propagators,
respectively.
⇢
4
at ⇤ becomes
9ytB
=
9
m2B, 2-loop
2
+⇢ytB
=
4
9ytB
87 4
gY B
16
+
✓
2
ytB
87 4
g
16 Y B
+
B
7 2
9 2
2
gY B + g2B 16g3B
12
4
63 4
15 2 2
g2B
gY B g2B
16
8
✓
7 2
9 2
2
gY B + g2B
16g3B
12
4
63 4
15 2 2
g2B
g g
16
8 Y B 2B
2
2
18ytB
+ 3gY2 B + 9g2B
12
◆
2
B
I2 .
◆
(15)
This is one of our main results. Note that Eqs. (14) and (15) are minus the
9
As mentioned in Ref. [14], while at the one-loop level, only a restricted set of particles participates;
on the two-loop level, all kinds of particles up to the Planck mass enter in the discussion. We assume that
there appear only SM degrees of freedom up to the UV cuto↵ scale.
+
B
2
2
18ytB
+ 3gY2 B + 9g2B
7
12
2
B
I2 .
(15)
Note
✦
✦
We did it in Landau gauge in D=4 symmetric phase. Skippable
Coincides with Al-‐‑‒sarhi, Jack & Jone (1992). ✴
✦
Read oﬀ from d=3 residue in dimensional reduction in background Feynman gauge. A non-‐‑‒trivial consistency check.
All in all, there
MUST be
something at
Planck scale.
Outline
1. Criticality: We live right on the edge of vacuum instability. 2. Indicates new principle beyond ordinary QFT? 3. Higgs, the only single sole ever observed elementary scalar ﬁeld, can serve as inﬂaton.
A principle:
Something that you CANNOT explain but postulates, hoping to serve as a stepping stone for future theoretical progress.
Era of principles back!
✦
“Multiple point criticality principle” (MPP) [Nielsen et al. 1996,2001,2012] → next slides ✦
“Asymptotic safety” [Shaposhnikov & Wetterlich 2010; Weinberg originates?] ✴
✦
Rule of game: Find UV ﬁxed point including gravity. Our work in progress with Yamada. ”Classical scale invariance” [Meissner & Nicolai, 2007,2008; Iso, Orikasa & friends, 2009,2009,2012,2013; Kannike, Racioppi & Raaidal, 2014; Guo & Kang, 2014; …] ✴
Quadratic divergence can be subtracted once and for all. [Bardeen, 1995] 2
✴
✦
”Maximum entropy principle” [Hamada, Kawana & Kawai, 2014] ✴
✦
✦
Assume that running mass becomes zero at UV cutoﬀ: δm = 0. A version of MPP, assuming maximization of total entropy of Universe. “Hidden duality” [Kawamura, 2013,2013] ✴
Without much understanding, I said “something like S-‐‑‒duality” at PPP2014. ✴
He sent no: Rather point-‐‑‒particle version of modular invariance (on string world sheet). “Eternal topological Higgs inﬂation” [Hamada, KO & Takahashi, appearing tomorrow] ✴
Not a principle but more concrete explanation. Read tomorrow.
MPP review
C. Froggatt, H. B. Nielsen Phys.Lett. B368 (1996)
(This part based on Hikaruʼ’s slides)
QFT vs statistical mechanics
✦
In QFT, path integral is most fundamental concept: Z
!
✦
[d'] e
S[']
In statistical mechanics, ✴
Most fundamental concept is micro-‐‑‒canonical ensemble. ✴
Canonical ensemble follows from thermodynamic limit: !
✴
Z
[d'] (H[']
E)
=)
Z
[d'] e
H[']
Total energy E comes ﬁrst, then comes temperature T = β−1.
is determined as a result.
Example:
Water
under
Example: Water molecules in a cylinder with a
fixed pressure.
ﬁxed pressure
p
vapor
water
T
T*
vapor
water + vapor
water
E
✦
Under co-‐‑‒existing two phases, is wide
automatically T is automatically
tuned
to T* Tfor
range of E.
tuned to T* for wide range of E. T corresponds to coupling constants in field theory.
✦
T corresponds to coupling constant (mass) in QFT. Micro-canonical-like QFT
✦
God ﬁrst ﬁxes ﬁeld value to I0 (corresponding to ﬁxed E). !
Z=
iS[']
[d'] e
✓Z
4
d x |'|
2
I0
◆
Skippable
!
!
✦
Z
=
Z
iS[']
[d'] e
Z
2
im
2
(I0
dm e
R
d4 x |'|2 )
Analogy with statistical mechanics: ✴
2
After integrating out φ, one value of m dominates (corresponding to a chosen T*):
Z=
Z
2
2 iV F (m )
dm e
Origin of criticality
dm 2
Z
d
m2 d 4 x
exp i S
Z
dm
d exp i S
Assume that the
Veff
effective potential
for S
Assume that the
has two minima.
effective potential for S
✦ Assume Veﬀ has two minima: has two minima.
There is some
critical
There is some critical
value for
. value for .
2
✦
m2 d 4 x
†
m2 I0
†
m2 I0
2
Veff1
2
2
1
2
22
2
2
2
Critical value for m : m2
!
mc 2
m2
2
2
1
2
2
!
✴
m2
mc 2
m2
2
2
2
2
2
1
mc 2
m2
2
2
mc 2
2
2
2
mc 2
2
2
2
1
m2
mc 2
2
2
1
2
If m 〜～ mc , coexistence of two phases allow satisfying micro-‐‑‒canonical 2
2
constraint for a given I0 ∈ (Vφ1 , Vφ2 ) (corresponding to a ﬁxed E): !
✴
✦
2
⌧Z
4
d x |'|
2
= I0
2
m wants to take critical value mc . 2
To allow some natural value I0 〜～ MP , second minimum must be φ2 〜～ MP.
Generalization of MPP
The micro canonical like path integral can be
generalized to
4
✦
†
[arXiv:1212.5716] PREdicted
exp
i S the Higgs Mass
H. B. Nielsen
2
Z
d
d x
M
2
May even put arbitrary weight w(m ): Z
Z
2
2
dm w m
d exp i S
2
(I0
R
m
2
4
d x2
4
†
)
Skippable
!
✴
!
✴
✦
iS[']
Z=
[d'] e
2
2
2
2
m dominate: mc dominates
Some m =Again
mc would 2
2
Z=
Z
Z 2
2
22
dm 2 w
exp
m
2 miV
F (m2 )iV iIF
m
0
if
1
M
2
2
2
.
Coexisting phases would be preferred. 2
S(m ). in the RHS
dm w(m ) e
2
d x |'|
dm w(m
M ) ePlanck scale is natural
Can further assume extremizing any ✴
im
M
M 2m 2
2
F
2
m
,
2
2
slope IM
0/V
2
1
mc
2
→Maximum entropy principle [Hamada, Kawai & Kawana, 2014]
m
2
m2
potential (3) and from the 1-loop e↵ective potential (4), respectively. The dark red (upper) and blue (lower)
Note: Degenerate or ﬂat?
bands are the beta function times ten 10 ⇥ d
e↵ /d ln µ
evaluated at the tree and 1-loop levels, respectively.
We take MH = 125.9 GeV and ↵s = 0.1185. The band corresponds to 95% CL deviation of Mt ; see Eq. (10).
Mt =171.39294 GeV
1¥1068
Two c1.0¥10
riticality principles: Mt =171.34294 GeV
Mt =171.34314 GeV
Mt =171.34334 GeV
Mt =171.34354 GeV
65
5.0¥1064
Mt =171.39354 GeV
5¥1067
Skippable
✴
0
”Appearance of plateau” -1.0¥1065
✴
2¥1017
4¥1017
6¥1017
“Degenerate vacua” j @GeVD
0
0
-5¥1067
-5.0¥1064
✦
Mt =171.39314 GeV
Mt =171.39334 GeV
V @GeV4 D
1.5¥10
V @GeV4 D
✦
65
8¥1017
0
1¥1018
2¥1018
3¥1018
j @GeVD
4¥1018
FIG. 2: Left: The tree level Higgs potential as a function of Higgs field '. Right: The one-loop Higgs
Both are Here
parametrically the ame: potential.
we take M = 125.9 GeV and
↵ =s0.1185.
H
✴
s
Phenomenologically, assuming either one ives 10(17–18
almost) CMS value. Then, the tree and one-loop
Higgs potential
becomes
flatgaround
GeV as shown
in Fig.
2.
the same result as another, in constraining e.g. Mt . Let us expand the e↵ective potential of the Higgs field Ve↵ (') on the flat space-time background
✦
Sometimes they are used interchangeably.
✓
1
X
around its minimum:
V (') =
e↵ (µ
4
= ')
4
' ,
e↵ (µ)
=
min
+
n
(16⇡ 2 )n
ln
µ
µmin
◆2
,
(11)
Outline
1. Criticality: We live right on the edge of vacuum instability. 2. Indicates new principle beyond ordinary QFT? 3. Higgs, the only single sole ever observed elementary scalar ﬁeld, can serve as inﬂaton.
le
Higgs inﬂation from SM criticality
0.00
10¥dleff êd lnm
-0.05
Log10 m @GeVD
5
10
15
20
FIG. 1: The light red (lower) and blue (upper) bands are 2-loop RGE running of
e↵ (µ)
from the tree level
potential (3) and from the 1-loop e↵ective potential (4), respectively. The dark red (upper) and blue (lower)
✦
Already covered by Nobuchikaʼ’
s lecture and by Yutaʼ’s poster. We take M = 125.9 GeV and ↵ = 0.1185. The band corresponds to 95% CL deviation of M ; see Eq. (10).
✦
To summarize: bands are the beta function times ten 10 ⇥ d
H
s
t
1.5¥1065
1¥1068
Mt =171.34294 GeV
Mt =171.34314 GeV
Mt =171.34334 GeV
Mt =171.34354 GeV
V @GeV4 D
1.0¥1065
5.0¥1064
0
0
2¥1017
4¥1017
6¥1017
j @GeVD
Can earn e-‐‑‒folding at plateau. Mt =171.39354 GeV
5¥1067
0
-5¥1067
64
-1.0¥1065
Mt =171.39314 GeV
Mt =171.39334 GeV
Flatness required by (strong) principle. -5.0¥10
✤
evaluated at the tree and 1-loop levels, respectively.
Mt =171.39294 GeV
V @GeV4 D
✴
e↵ /d ln µ
8¥1017
1¥1018
0
2¥1018
3¥1018
j @GeVD
4¥1018
FIG. 2: Left: The tree level Higgs potential as a function of Higgs field '. Right: The one-loop Higgs
potential. Here we take MH = 125.9 GeV and
↵s =-90.1185.
6.¥10
✤
Only need milder ξ〜～10. 5.¥10-9
U @MP 4 D
CMS value. Then, the tree and one-loop4.¥10
Higgs -9
potential becomes flat around 1017–18 GeV as shown
in Fig. 2.
✴
Let ussexpand
the e↵ective potential of
the Higgs field V
2.¥10
Milder ξ〜～10 gives larger lope. -9
V (') =
e↵ (')
on the flat space-time background
1.¥10-9
around its minimum:
✴
3.¥10-9
e↵ (µ
= ')
4
' ,
Larger slope can give larger r=16ε 〜～ 0.2.
4
0
e↵ (µ) =
0.0
min
+
1
X
0.5n=2
✓
n
ln
(16⇡ 2 )n1.0
j @M D
µ
◆2
,
µmin 1.5
(11)
2.0
where the overall factor '4 is put to make the expansion well-bahaved. InPthe potential analysis
Furthermore,
Adding anything that
couples to Higgs
at intermediate scale
affects the result.
Higgs portal DM, right-‐‑‒handed neutrino, etc. etc.
→ Talk by Haba-‐‑‒san.
It s easy, fun,
important &
TESTABLE!
Come together.
Summary
1. Criticality: We live right on the edge of vacuum instability. 2. Indicates new principle beyond ordinary QFT? 3. Higgs, the only single sole ever observed elementary scalar ﬁeld, can serve as inﬂaton.
Thank you!
Backup
Table 5. Best-fit values and 68% confidence limits
tional nuisance parameters for “highL” data sets are
(Planck Collaboration 2013b).
インフレーション・パラダイムの確立
Planck Collaboration: Cosmological parameters
Planck+WP
Parameter
⌦b h2 . . . . . . . . . .
B
0.0
1.04119 1.04131 ± 0.00063
1.
0.0925
0.089+0.012
0.014
0
ns . . . . . . . . . . .
0.9619
0
ln(1010 As ) . . . . . . .
3.0980
0.9603 ± 0.0073
APS
100 . . . . . . . . . .
152
APS
143 . . . . . . . . . .
63.3
100✓MC . . . . . . . .
⌧ . . . . . . . . . . . .
Planck T T power spectrum. The points in the upper panel show the maximum-likelihood estimates of the primary CMB
m computed as described in the text for the best-fit foreground and nuisance parameters of the Planck+WP+highL fit listed
5. The red line shows the best-fit base ⇤CDM spectrum. The lower panel shows the residuals with respect to the theoretical
The error bars are computed from the full covariance matrix, appropriately weighted across each band (see Eqs. 36a and
d include beam uncertainties and uncertainties in the foreground model parameters.
✦
68% limits
0.022032 0.02205 ± 0.00028
⌦ c h2 . . . . . . . . . .
✦
Best fit
0.12038
0.1199 ± 0.0027
3.089+0.024
0.027
54 ± 10
標準宇宙論論の6個
ACIB
143 . . . . . . . . . .
で、70点ほどをばっちりフィット。 ACIB
217 . . . . . . . . . .
27.2
29+69
AtSZ
143 . . . . . . . . . .
6.80
...
のパラメタ
107+20
10
< 10.7
0.916
> 0.850
「インフレーションの他にこれができる理理論論を知らない」
CIB
r143⇥217
. . . . . . . .
0.406
0.42 ± 0.22
サトカツ先⽣生談@本研究会。 CIB . . . . . . . . . . 0.601
0.53+0.13
0.12
PS
r143⇥217
. . . . . . . .
Planck T E (left) and EE spectra (right) computed as described in the text. The red lines show the polarization spectra from
⇤CDM Planck+WP+highL model, which is fitted to the TT data only.
⇠ tSZ⇥CIB . . . . . . . .
0.03
3
171 ± 60
PS
A
117.0
217 . . . . . . . . . .
（うち2個がインフレーション由来）
0.0
0.
...
0
1
0
0
BICEP2前後
Outline
1. Higgs potential becomes ﬂat at string scale. ビフォー
2. Top mass & UV cutoﬀ scale constrained from Higgs ✦ φ2 chaotic 瀕死 inﬂation. ヒッグス・ 3. Playing with BICEP2
インフレーション ⼤大勝利利
✦
BICEP2前後
アフター
✦
✦
2
φ chaotic ⼤大勝利利ヒッグス・ インフレーション 瀕死
（Issues については⽻羽澄さんのトーク）
まずは普通の
ヒッグス・
インフレーション
[Bezrukov & Shaposhnikov (2008)…]
一般のヒッグス・重力作用
S=
Z
4
d x
p
g
"✓
2
◆
2
MP
'
R
1 + ⇠ 2 + ···
MP
2
#
◆
✓
2
4
(@')
' + ···
4
✦
R 〜～ g..g..g..∂.g..∂.g.. ∝ (g..)−1
✦
√−g ∝ (g..)2
S=
Z
4
d x
p
g
"✓
◆
2
'
R
1 + ⇠ 2 + ···
MP
2
◆#
✓
2
4
(@')
' + ···
4
✦
R 〜～ g..g..g..∂.g..∂.g.. ∝ (g..)−1
✦
よって次でアインシュタイン枠に移れる：
✓
2
2
MP
'
1 + ⇠ 2 + ···
MP
◆
✦
√−g ∝ (g..)2
gµ⌫ !
E
gµ⌫
S=
Z
✓
4
d x
p
2
g
'
1 + ⇠ 2 + ···
MP
✦
"✓
◆
ポテンシャルの変換：
✓
4
4
' + ···
◆
2
2
MP
'
R
1 + ⇠ 2 + ···
MP
2
◆#
✓
2
4
(@')
' + ···
4
gµ⌫ !
◆
✦
E
gµ⌫
√−g ∝ (g..)
4
!⇣
'
+
·
·
·
4
1+
'2
⇠ M2
P
+ ···
⌘2
2
平らなポテンシャル
✦
ポテンシャルの変換：
✓
4
4
' + ···
704
◆
4
'
+
·
·
·
4
!⇣
1+
+ ···
F. Bezrukov, M. Shaposhnikov / Physics Letters B 659 (2008) 703–706
MP = (8πGN )−1/2 = 2.4 × 1018 GeV. This model has “good”
particle physics phenomenology but gives “bad” inflation since
the self-coupling of the Higgs field is too large and matter
fluctuations are many orders of magnitude larger than those observed. Another extreme is to put M to zero and consider the
“induced” gravity [10–14], in which the electroweak symmetry√breaking √
generates the Planck mass [15–17]. This happens
if ξ ∼ 1/( GN MW ) ∼ 1017 , where MW ∼ 100 GeV is the
electroweak scale. This model may give “good” inflation [12–
14,18–20] even if the scalar self-coupling is of the order of
one, but most probably fails to describe particle physics experiments. Indeed, the Higgs field in this case almost completely
decouples from other fields of the SM2 [15–17], which corresponds formally to the infinite Higgs mass mH . This is in
conflict with the precision tests of the electroweak theory which
tell that m must be below 285 GeV [21] or even 200 GeV [22]
✦
φ ≫ MP/√ξ で定数ポテンシャルに。 ✦
ξ 〜～ 105 でちょうどいい揺らぎを与
える。
'2
⇠ M2
P
⌘2
Bezrukov & Shaposhnikov Fig.
1. Effective potential
in the Einstein frame.(2008)
(4), respectively. The dark red (upper) and blue (lower)
ま要は
n µ evaluated at the tree and 1-loop levels, respectively.
7
Mt =171.39294 GeV
1¥1068
Mt =171.39314 GeV
Mt =171.39334 GeV
5¥10
Mt =171.39354 GeV
67
6.¥10-9
5.¥10-9
0
U @MP 4 D
V @GeV4 D
nd corresponds to 95% CL deviation of Mt ; see Eq. (10).
4.¥10-9
3.¥10-9
2.¥10-9
1.¥10-9
-5¥10
0
0.0
67
j @MP D
0.5
1.0
1.5
2.0
参考: ξ=10 の絵。
FIG. 4: SM Higgs potential in the prescription I with ⇠ = 10 and c = 1, corresponding to
1017 GeV, and with
min
0
1¥1018
2¥1018
3¥1018
j @GeVD
4¥1018
= 2 c,
c,
and
2
= 0.56. The red (upper), green (center) and purple (lower) lines ar
c /2,
respectively. The values of
min
=2
c
and
c /2
are chosen just fo
Each line roughly corresponds to the one with the same color in Fig. 2.
B.
Prediction
We expanded the e↵ective potential of the Higgs field Ve↵ on the flat space-time
around its minimum as in Eq. (11):
V =
e↵ (µ)
4
'4 ,
1
X
✓
µ
◆2
✦ ポテンシャルの φ >
部分をニョーンと横に引き伸ばして平らに
function
of Higgs field
'. MRight:
one-loop Higgs
P/√ξ のThe
するようなかんじ。
0.1185.
e↵ (µ)
=
min
+
n=2
n
(16⇡ 2 )n
ln
µmin
.
The choice of scale (27) and (28) correspond to the prescription I and II, respect
Section II, we can safely neglect the higher order terms with n
3, and we wil
hereafter in this section. The higher order terms in Eq. (16) are ignored for the mom
C.
Prescription I
ありうる問題点
✦
ユニタリティ。 ✴
E > MP/ξ の散乱はマズイ。 ✴
が、インフレーション模型としては問題ない。 ✤
✴
散乱エネルギー E がでかいのと、場の値 φ がでかいのは別概念念。けど素粒粒⼦子屋としてはちょっちキモイよね。
インフレーションおさらい
（詳しくは、⾼高橋(史)トークとか）
予⾔言値は V(φ) が決まると完全に決まる。 ✴
与えられた l（k*）に対応する位置 φ* において、 10
Planck Collaboration: Constraints on inflation
⇤CDM + tensor
縦軸: テンソル・スカラー⽐比は傾きから、 r=16ε 〜～ 8(Vʼ’/V)2、ここで ε = (Vʼ’/V)2 /2。横軸: スペクトラル指数は凸具合から、 0.2
✤
Planck
0.960
Co
nv
Co ex
nca
ve
0.1
0.94
0.96
0.98
Primordial Tilt (ns )
0.94
0.96
ns
0.98
1.00
1.00
Fig. 1. Marginalized joint 68% and 95% CL regions for ns and r0.002 from Planck in com
the theoretical predictions of selected inflationary models.
reheating priors allowing N⇤ < 50 could reconcile this model
with the Planck data.
Exponential potential and power law inflation
Inflation with an exponential potential
V( ) = ⇤4 exp
2
Planck+WP+lensing
0.9653 ± 0.0069
< 0.13
0
0.3
0.0
✴
Planck+WP
0.9624 ± 0.0075
< 0.12
0
Table 4. Constraints on the primordial perturbation parameters in the ⇤CDM+r model f
The constraints are given at the pivot scale k⇤ = 0.002 Mpc 1 .
0.00
✤
Parameter
ns
r0.002
2 ln Lmax
0.25
✴
Planck+WP+highL
Planck+WP+highL+BICEP2
0.4
Model
r0.002
✦
この計算⾃自体はアホでもできる。 Tensor-to-Scalar Ratio (r0.002 )
0.05
0.10
0.15
0.20
✦
Mpl
!
(35)
is called power law inflation (Lucchin & Matarrese, 1985),
because the exact solution for the scale factor is given by
2
a(t) / t2/ . This model is incomplete, since inflation would
not end without an additional mechanism to stop it. Assuming
such a mechanism exists and leaves predictions for cosmological perturbations unmodified, this class of models predicts
r = 8(ns 1) and is now outside the joint 99.7% CL contour.
ns=1+2η−6ε = 1 +2Vʼ’ʼ’/V −3(Vʼ’/V) 、ここで η = Vʼ’ʼ’/V。
Inverse power law potential
Intermediate models (Barrow, 1990; Muslimov, 1990) with inverse power law potentials
4
!
lead to inflation with
where f = 4/(4 + )
is no natural end to
the inflationary pred
modified, this class o
(Barrow & Liddle, 1
joint 95% CL contou
Hill-top models
In another interesting
from an unstable equi
els (Albrecht & Stein
V
where the ellipsis ind
inflation, but needed
later on. An exponen
inflationary model a
2
r ⇡ 32 2⇤ Mpl
/µ4 . Th
ment with Planck+W
Planckian values of µ
Models with p
Backup: 観測対理論 Observation vs Theory
（高橋(史)スライドのコピペ）
Scalar mode
Tensor mode
V : the inﬂaton potential
10¥dleff êd lnm
-0.05
我々の立場（BICEP2前から書いてる）
Log10 m @GeVD
5
10
15
20
FIG. 1: The light red (lower) and blue (upper) bands are 2-loop RGE running of
e↵ (µ)
from the tree level
[Hamada, Kawai, KO (2014)]
potential (3) and from the 1-loop e↵ective potential (4), respectively. The dark red (upper) and blue (lower)
✦
bands are the beta function times ten 10 ⇥ d
e↵ /d ln µ
evaluated at the tree and 1-loop levels, respectively.
実験事実の外挿: ヒッグス・ポテンシャルはプランク・スケールで平坦。 We take MH = 125.9 GeV and ↵s = 0.1185. The band corresponds to 95% CL deviation of Mt ; see Eq. (10).
Mt =171.39294 GeV
!
✦
Mt =171.34294 GeV
Mt =171.34314 GeV
Mt =171.34334 GeV
Mt =171.34354 GeV
1.0¥1065
5.0¥1064
0
1¥10
Mt =171.39354 GeV
5¥1067
0
-5¥1067
-5.0¥1064
-1.0¥1065
Mt =171.39314 GeV
Mt =171.39334 GeV
V @GeV4 D
ただし第0近似ではこの図のどれでも「平坦」→ V @GeV4 D
✴
1.5¥1065
68
0
2¥1017
4¥1017
6¥1017
j @GeVD
8¥1017
1¥1018
0
2¥1018
3¥1018
j @GeVD
4¥1018
この「平坦」性は、前述のような（場の理理論論をちょっと越えた）原理理により、
要請される。 FIG. 2: Left: The tree level Higgs potential as a function of Higgs field '. Right: The one-loop Higgs
potential. Here we take MH = 125.9 GeV and ↵s = 0.1185.
CMS value. Then, the tree and one-loop Higgs potential becomes flat around 1017–18 GeV as shown
in Fig. 2.
✴
us expand the e↵ective potential of the Higgs field V (') on the flat space-time background
ので標準模型の紫外切切断 Λ よLetり上ではポテンシャルは平坦になっているであろう。 e↵
around its minimum:
V (') =
✴
e↵ (µ
なお緑の場合をそのままは使えない。 4
= ')
4
' ,
e↵ (µ)
=
min
+
1
X
n=2
n
(16⇡ 2 )n
✓
ln
µ
µmin
◆2
,
(11)
where the overall factor '4 is put to make the expansion well-bahaved. In the potential analysis
around the minimum, we can safely neglect the higher order terms with n
✤
N 〜～ 50 を稼ごうとすると ε <<< 1、 ✤
V は決まっているので、 ✤
ゆらぎの⼤大きさ ∝ V/ε がでかくなりすぎる。
3, and we will omit
Const
この立場からくる制限
Constraint
拡大
[Hamada, Kawai, KO (2014)]
拡大すると
✦
✦
17GeV
Λは最大でΛ
5
10
17
Figure 3: Left: Excluded region by Eq. (27)
(red,
and by GeV。 Eq. (26) (blue, right) in
標準模型の紫外切切断に上限: Λ <
5left)
×10
future scale?
exclusion
log10 (⇤/GeV) vs Mt plane. Right: Enlarged view for ⇤ vs Mt . Expected
string
limits within 95%CL: r < 10 2 and 10 3 are also presented by dashed and dot-dashed lines,
respectively.
弦スケールがでてくるのはちょっと⾯面⽩白い。
Mt=173.3 2.8GeV[Djouadi et. al. 2012]
17
leff
0.00
この立場の、r=0.2 への応用
10¥dleff êd lnm
-0.05
5
Log10 m @GeVD
10
15
20
(µ) from the tree level
[Hamada, Kawai, KO, Park (2014)]
FIG. 1: The light red (lower) and blue (upper) bands are 2-loop RGE running of
e↵
potential (3) and from the 1-loop e↵ective potential (4), respectively. The dark red (upper) and blue (lower)
✦
普通のヒッグス・インフレーションは、我々の議論論では bands are the beta function times ten 10 ⇥ d
e↵ /d ln µ
evaluated at the tree and 1-loop levels, respectively.
We take MH = 125.9 GeV and ↵s = 0.1185. The band corresponds to 95% CL deviation of Mt ; see Eq. (10).
Λ=MP/√ξ に対応する。 1.5¥10
1.0¥10
65
Mt =171.34294 GeV
Mt =171.34314 GeV
Mt =171.34334 GeV
Mt =171.34354 GeV
Mt =171.39314 GeV
Mt =171.39334 GeV
5.0¥1064
0
5¥10
Mt =171.39354 GeV
67
0
-5¥1067
-5.0¥1064
-1.0¥1065
0
2¥1017
4¥1017
6¥1017
j @GeVD
8¥1017
1¥1018
0
2¥1018
3¥1018
4¥1018
j @GeVD
強い意味の「原理理」により緑付近が選ばれたとせよ。 FIG. 2: Left: The tree level Higgs potential as a function of Higgs field '. Right: The one-loop Higgs
potential. Here we take MH = 125.9 GeV and ↵s = 0.1185.
CMS value. Then, the tree and one-loop Higgs potential becomes flat around 1017–18 GeV as shown
✴
平らな領領域で e-‐‑‒folding を稼げば、ξ を⼤大きくしないでもよい。 in Fig. 2.
Let us expand the e↵ective potential of the Higgs field Ve↵ (') on the flat space-time background
around its minimum:
-9
V (') =
✤
たとえば ξ=7 とかでもよい。 e↵ (µ
= ')
4
4
' ,
e↵ (µ)
=
min
+
6.¥10
1
X
-9
n
2 )n
(16⇡
4.¥10-9
n=2
3.¥10-9
5.¥10
U @MP 4 D
✦
1¥1068
V @GeV4 D
✴
V @GeV4 D
Mt =171.39294 GeV
65
✓
ln
µ
µmin
◆2
,
(11)
where the overall factor '4 is put to make the expansion well-bahaved.
In the potential analysis
-9
2.¥10
1.¥10-9 with n
around the minimum, we can safely neglect the higher order terms
0
0.0
3, and we will omit
j @MP D
0.5
1.0
1.5
2.0
ξ が⼩小さいので、引き伸ばしが少なく、傾き ε が⼩小さすぎない。 FIG. 4: SM Higgs potential in the prescription I with ⇠ = 10 and c = 1, correspondin
✴
1017 GeV, and with
min
= 2 c,
c,
and
2
= 0.56. The red (upper), green (center) and purple (lower) lin
c /2,
respectively. The values of
min
=2
c
and
c /2
are chosen j
Each line roughly corresponds to the one with the same color in Fig. 2.
B.
✴
結果じゅうぶん⼤大きな r=16ε を実現可能。
Prediction
We expanded the e↵ective potential of the Higgs field Ve↵ on the flat spacearound its minimum as in Eq. (11):
V =
e↵ (µ)
=
e↵ (µ)
4
min
+
'4 ,
1
X
n
2 n
✓
ln
µ
◆2
.
Backup
[Hamada, Kawai, KO, Park (2014)]
Lower V by ξ to get proper εV=10−3
1 ¥ 10
65
8 ¥ 10
4
V @GeV D
64
6 ¥ 1064
Earn N* at this plateau
5
Original ξ〜～10 gives too small εV
4 ¥ 10
64
ξ=7 is OK!
2 ¥ 10
64
0.0
0.2
0.4
0.6
h @MPD
0.8
1.0
詳細な予言
[Hamada, Kawai, KO, Park, to appear]
★ c = μmin / (MP/√ξ) としてこんなかんじ。 14
0.20
0.05
x=6
x=6
c=0.98
x=7
x=8
0.15
dns
r
0.05
c=1
0.03
c=1.01
0.95
ns
x=8
c=1.03
c=1.04
x=9
c=1.05
0.00
c=1.05 x=50
0.90
c=1.02
0.02
0.01
c=1.02
c=1.03
c=1.04
x=20
0.00
0.85
c=1
x=10
c=1.01
x=15
c=0.99
x=7
x=9
0.10
c=0.98
0.04
c=0.99
dlnk
0.25
1.00
x=50
x=10 x=15
-0.01
0.85
x=20
0.90
0.95
ns
1.00
途中でなんか入ったら？
gauge singlet real scalar S to the SM. We further imp
under which the SM fields are even and S is odd. This Z2
he decay of S into the SM particles, making it stable. The
例）Higgs portal DM
[Hamada, Kawai, KO (2014)]
1
2
L = LSM + (@µ S)
2
1 2 2
mS S
2
r0=0, L=1017GeV
175
⇢ 4
S
4!
 2 †
S H H.
2
r0=0.6, L=1017GeV
175
first see the behavior of S as the DM. The mass eigenva
174 monotonicity
monotonicity
173
value of potential
172
2
mDM
171
170
200
400
600
800
mDM @GeVD
=
1000
Mt @GeVD
Mt @GeVD
174
173
172
2
mS
171
170
2
value of potential
v
,
+
2
200
400
600
800
mDM @GeVD
1000
246
GeV
is
the
Higgs
vacuum
expectation
value
(VEV).
★強い意味の「原理理」から平坦性が要請されるとす
rt in the thermal bath of the SM sector through the coupl
ると、m
=
4
00-‐‑‒470 G
eV、M
=171.2GeV。
DM
t
er this interaction is frozen out, the abundance of S is fi
Figure 6: The excluded regions in the Mt vs mDM plane from the monotonicity (upper-left,
blue) and from the value of the potential (lower-right, red) in the Z2 scalar DM model to
achieve the flat potential Higgs inflation. Left and right panels are for ⇢0 = 0 and 0.6,
respectively.
1017 GeV.4 When we impose this flatness condition, the coupling , the DM
mass mDM , and the top mass Mt are completely fixed as functions of ⇢0 :
DM見えちゃう！
2
WIMP−nucleon cross section (cm )
for the
interpre
calibrat
−44
10
6
8
−40
10
−45
10
−42
10
−44
10
1
10
2
m
10
WIMP
3
(GeV/c2)
10
10
12
LUX
and 201
will est
operatio
electric
backgro
further
Subsequ
conduct
improvi
of WIM
This
Departm
[from LUX (2014)]
等々
どんどん参入しましょう
★DM以外にも右巻きニュートリノ等々、ヒッグス
の４点結合の⾛走りを変えるものがなんでも途中
にあれば、予⾔言が変わる。（Cf. ⾼高橋亮亮トーク） ★RGE も V, Vʼ’, Vʼ’ʼ’ → As, ns, r も計算は簡単。 ✴ dns/dlnk、dnt/dlnk 等々も同様に簡単に⾏行行ける。 ★こんごバンバン進歩する宇宙の観測で検証でき
る（重要）。

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