Transcript Slide

Standard Model and the Planck Scale
Dec 12 2015
at NCTS
Hikaru Kawai
(Kyoto Univ.)
In collaboration with
K. Kawana, Y. Hamada, K. Oda, and S. C. Park.
Outline
The observed values of the Higgs and top masses
indicate that SM is valid up to the string/Planck
scale and no new physics is forced to appear.
In string theory, non-supersymmetric tachyon-free
vacua appear more frequently than
supersymmetric ones.
The simplest guess is that SM is directly connected
to string theory at the string scale.
Only minor modifications come in between the SM
and the string scale.
However we need some mechanisms by which
the fine tunings of the Higgs mass and the
cosmological constant are performed.
The naturalness problem
Suppose the underlying fundamental theory,
such as string theory, has the momentum
scale mS and the coupling constant gS .
Then, by dimensional analysis and the power
counting of the couplings, the parameters of
the low energy effective theory are given as
follows:
naturalness problem (cont.’d)
2
gS
GN ∼ 2 .
mS
dimension -2 (Newton constant)
g1 , g2 , g3 ∼ g S ,
dimension 0
(gauge and Higgs couplings)
H
∼
2
gS .
tree
dimension 2 (Higgs mass)
2
mH
∼
0  g S mS .
2
2
unnatural ! → mH ∼ 100GeV  ≪ g S mS ∼ 10 GeV 
2
2
2
2
18
2
dimension 4
(vacuum energy or cosmological constant)
 ∼ 0  g s  mS .
2
unnatural ! !→
 ∼  2 ~ 3meV 
4
≪
mS ∼ 10 GeV 
4
18
4
4
SUSY as a solution to the naturalness problem
Bosons and fermions cancel the radiative
corrections:
+
bosons
fermions
+
⇒ mH  0.
2
2
⇒ mH ∼ M SUSY .
2
⇒   0.
⇒ 
∼
4
M SUSY .
However, SUSY must be spontaneously
broken at some momentum scale MSUSY ,
below which the cancellation does not work.
(cont’d)
Therefore, if MSUSY is close to mH , the
Higgs mass is naturally understood,
although the cosmological constant is still
a big problem.
However, no signal of the SUSY partners
is observed in the LHC below 1 TeV.
It is better to think about the other
possibilities.
Possibility of desert
The first thing we should know is whether the
SM is valid to the string/Planck scale or it breaks
down below the scale.
If it is the former case, there is a possibility that
the SM is directly connected to the Planck scale
physics.
Is the SM valid to the Planck/string scale?
Y. Hamada, K. Oda and HK: arXiv:1210.2358 ,
arXiv:1305.7055 , 1308.6651
In order to answer the question, we consider the
bare Lagrangian of SM with cutoff momentum Λ,

.
and determine the bare parameters in such a way
that the observed parameters are recovered.
If no inconsistency arises, it means that the SM can
be valid to the energy scale Λ.
Bare parameters as a
function of the cutoff
[1405.4781]
2-loop RGE
mHiggs =125.6 GeV
The MC mass should
be distinguished from
the QCD pole mass.
The latter has a rather
large error.
1
2
I1 =

2
16
[Hamada, Oda, HK ,1210.2538, 1308.6651]
Higgs self coupling λ
λ takes a small
value at high
energy scale.
[1405.4781]
If Mt=171GeV,
λ=βλ=0
at 1017-18GeV.
[Hamada, Oda, HK,1210.2538, 1308.6651]
Bare mass
If Mt=170GeV, the bare mass becomes 0 at MP.
[Hamada, Oda, HK,1210.2538]
Triple coincidence
Three quantities,
B ,  B  , mB
become close to zero around the Planck/string scale.
This means that the Higgs potential becomes zero
around the Planck/string scale.
V
𝑚𝑠
𝜙
Such situation was predicted by Froggatt and Nielsen ’95.
Multiple Point Criticality Principle (MPP)
Review of MPP
In the ordinary quantum theory, we start with the
path integral of the form of canonical ensemble:
  d  exp   S   .
On the other hand in the statistical mechanics,
the most fundamental concept is the micro
canonical ensemble
Z 
 d   H    E  ,
and the canonical ensemble follows in the
thermodynamic limit:
  d    H    E 

  d  exp   H   / T  .
Important point: In the micro canonical ensemble
we control extensive variables, while in the
canonical ensemble we control intensive variables.
Example: Consider a given number of water molecules
put in a box with a given volume.
In the micro canonical ensemble the energy is controlled.
p
E
L
vapor
S
G
𝑬𝒄
vapor
water
T
If 𝑬 < 𝑬𝒄 , the system is automatically on the critical line.
The coexisting phase is obtained without fine tuning.
Intensive variables are tuned “miraculously”.
Let’s imagine that quantum theory is defined by a
path integral of the micro canonical ensemble type
  d    S    C  .
It is equivalent to the ordinary quantum mechanics, if
the space-time volume is sufficiently large.
If C is in some region, the coupling constants are
automatically tuned in such a way that two vacua
degenerates and coexist.
In other words, it is natural to have degenerate vacua.
V
𝑚𝑠 𝜙
Working hypotheses
It is not clear how much we should trust, but let’s
assume MPP as a working hypothesis.
MPP:
Coupling constants are tuned so that the
vacuum is on the boundary of various phases.
We can also imagine another working hypothesis.
Maximum Entropy Principle (MEP):
Coupling constants are tuned so that the
entropy of the universe becomes maximum.
Probably many authors have made similar
statements, and there would be many possible
arguments.
For example, suppose that we pic up a universe
randomly from the multiverse. Then the most
probable universe is expected to be the one that
has the maximum entropy. (T. Okada and HK)
Higgs inflation
Question: Can the Higgs field play the roll of inflaton?
Hamada, Oda, HK: arXiv:1308.6651
Hamada, Oda, Park and HK: arXiv: 1403.5043
Higgs inflation
(1) modest approach
We trust the effective potential only below the string
scale, and try to make bounds on the parameters.
Hamada, Oda and HK: arXiv: 1308.6651
(2) optimistic approach
We trust the potential including the string scale.
We assume that nature does fine tunings if they are
necessary.
We introduce a non-minimal coupling.
Hamada, Oda, Park and HK, arXiv: 1403.5043
Cook. Krauss, Lawrence , Long and Sabharwal, arXiv: 1403.4971
Bezrukov and Shaposhnikov, arXiv: 1403.6078
Higgs potential
V   
  
4
4
mH = 125.6 GeV
𝜑[GeV]
Modest approach
We trust the above effective potential only below some
scale Λ .
Above Λ it is not described by SM, but we need string
theory.
string
SM
𝚲
Necessary conditions
At present we do not have enough knowledge above Λ.
But there is a possibility that the Higgs potential in the
stringy region has a proper shape, and the inflation
occurs in this region.
In order for this scenario to work, the necessary
conditions are
d
VSM  0 for    ,
d
VSM     V*.
where
3 AS
 r* 
4
65
4
V* 
r* M P  1.3  10  
 GeV .
2
 0.11 
2
mH  125.6 GeV
r  0.1
𝑟 ∼ 0.01 is possible if the cutoff scale is ∼ 1017 GeV .
Optimistic approach
We assume that the Higgs potential can be trusted up
to the inflection point region.
The naïve guess is that inflation occurs if the
parameters are tuned such that the inflection point
almost becomes a saddle point.
𝜑[GeV]
However, this inflection point can not produce a
realistic inflation:
sufficient 𝑁∗ in this region ⇒ 𝜑∗ ∼ saddle point
⇒ small 𝜀 ⇒ too large 𝐴𝑆 ∼ 𝑉∗ /𝜀
𝜑[GeV]
Non minimal Higgs inflation with flat potential
We introduce a non-minimal coupling  R 2
as the original Bezrukov-Shaposhnikov’s.
Then a realistic Higgs inflation is possible.
In the Einstein frame the
effective potential becomes
V  h  ,  h 
𝜑[GeV]
h
1 h / MP
2
2
.
𝝃 need not be very large
 r* 
4
V*  1.3  10  
 GeV .
 0.11 
65
𝑉 ∼ 𝜆 𝜑ℎ 𝜑ℎ4 ∼ 𝜆 𝜑ℎ 𝑀𝑃4 /𝜉 2 .
What is new here is that 𝜆 𝜑 becomes small
around the string scale: 𝜆 𝜑ℎ ∗ ∼ 10−6 .
This allows rather small values for 𝜉 ( 𝜉 ∼ 10) .
realistic values of 𝒓 and 𝒏𝑺
𝜇𝑚𝑖𝑛
𝑐 = 𝜉
𝑀𝑃
𝜆′ 𝜇𝑚𝑖𝑛 = 0
For example,
𝑚𝐻 = 125.6GeV
⇒ 𝜇𝑚𝑖𝑛 ∼ 5.5 × 1017 GeV
Hamada, Oda, Park, HK: arXiv:1408.4864
What we have seen so far.
• All the coupling constants are small at the string scale.
It may suggest that our vacuum is close to one of the
perturbative vacua of string theory.
• Higgs potential becomes almost flat around the string
scale.
−6 4
−6 4
V ∼ 10 𝜑 ∼ 10 𝑚𝑠
• This suggests that the Higgs potential is zero in the
string tree level and is generated by the 1-loop
correction, because the string 1-loop correction is
proportional to
𝐶𝑙𝑜𝑜𝑝 =
𝑆9
2 2𝜋 10
∼ 10−7 .
• Higgs inflation is possible if one of the parameters is
tuned so that the potential nearly has a saddle point.
• Such fine tuning might be understood by MEP.
inflation ⇒ large entropy
Can we say something
beyond 𝑀𝑃 by string
theory?
Higgs potential in string theory
Higgs potential for large field values can be
estimated, and turns out to be compatible
with MPP.
Hamada, Oda , HK: arXiv: 1501.04455
Higgs potential in string theory
To be concrete we take heterotic string, and consider a
4 dimensional perturbative vacuum that is nonsupersymmetric and tachyon-free.
Because the Higgs particle is a real scalar and
massless at the tree level, its emission vertex is
written as
𝑉 𝑘 = 𝑂(𝑧, 𝑧)𝑒 𝑖𝑘⋅𝑋 ,
where 𝑂(𝑧, 𝑧) is a real (1,1) operator.
The Higgs potential for the field value 𝜙 can be
obtained by evaluating the partition function for the
world sheet action
.
flat potential in tree level ⇔ exactly marginal
A typical exactly marginal operator is given as
𝑂 𝑧, 𝑧 = 𝑂
where 𝑂
1,0
and 𝑂
1,0
𝑧 ×𝑂
0,1
are (1,0) and (0,1) operators.
0,1
𝑧 ,
Then by using bosonization
𝑂
1,0
𝑧 = 𝜕𝑌(𝑧) , 𝑂
0,1
𝑧 = 𝜕𝑍 𝑧 ,
we can evaluate the partition function.
It is not clear whether this is completely general or
not, but we consider this form.
Examples of factorized 𝑶(𝒛, 𝒛)
★ Higgs comes from gauge-Higgs unification.
Emission vertex of gauge field 𝐴𝜇𝑎 is
𝜕𝑧 𝑋𝜇 𝑗 𝑎 𝑧 𝑒 𝑖𝑘⋅𝑋 .
Then the emission vertex of
𝑎
𝐴5
is
𝜕𝑧 𝑋 5 𝑗 𝑎 𝑧 𝑒 𝑖𝑘⋅𝑋 .
★ Generic cases in the Fermionic construction.
★ Higgs comes from an untwisted sector in the orbifold
construction.
This is rather generic.
Simplest example - radion potentialAs the simplest example we consider the case
𝑂 𝑧, 𝑧 = 𝜕𝑧 𝑋 5 𝜕𝑧 𝑋 5 ,
where 𝑋 5 is a compactified dimension with the compactification
radius 𝐿.
Then the world sheet action to consider is
𝑆 = 𝑆0 + 𝜙
𝑆0 =
𝑑2 𝑧 𝑂(𝑧, 𝑧) =
𝑑2 𝑧(1 + 𝜙)𝜕𝑧 𝑋 5 𝜕𝑧 𝑋 5 + ⋯ .
𝑑2 𝑧𝜕𝑧 𝑋 5 𝜕𝑧 𝑋 5
This comes back to the original form 𝑆0 by the redefinition
𝑋′ 5 = 1 + 𝜙 𝑋5.
But the compactification radius is changed from 𝐿 to
𝐿′= 1 + 𝜙 𝐿.
The radion potential for large field can be evaluated
by considering the dimensional reduction of 5D
gravity to 4D:
𝐶: 5D cosmological constant
𝑆𝑒𝑓𝑓 =
5
𝑔MN
𝑑 5 𝑥 −𝑔
𝑔𝜇𝜈
=
0
5
(ℛ
5
− 𝒞) + ⋯ .
0
′
,
𝐿
=
′2
𝐿
1 + 𝜙𝐿 .
⇒
𝜇𝜈
𝑆𝑒𝑓𝑓 =
1𝑔
′
′
𝑑 𝑥 −𝑔𝐿′(ℛ −
𝜕
𝐿
𝜕
𝐿
−
𝒞)
+
⋯
.
𝜇
𝜈
2 𝐿′2
4
Converting it to the Einstein frame
𝑔𝜇𝜈 =
′ 𝐸
𝐿 𝑔𝜇𝜈
,
we have
𝑆𝑒𝑓𝑓
=
𝐿′= 1 + 𝜙(𝑥) 𝐿
𝐸𝜇𝜈
1
𝑔
𝒞
4
𝐸
′
′
𝐸
𝑑 𝑥 −𝑔 (ℛ −
𝜕
𝐿
𝜕
𝐿
−
)
+
⋯
.
𝜇
𝜈
2 𝐿′2
𝐿′
The Einstein frame potential is runaway for large
radius if cosmological constant in 5D is positive.
This is true for all orders. It follows only from the
low energy effective action in 5D.
General 𝟏, 𝟎 × 𝟎, 𝟏 case
Background for 𝜕𝑧 𝑌𝜕𝑧 𝑍 corresponds to the
Lorentz boost for the momentum lattice for
Y and Z.
3cases: runaway, periodic, chaotic
In the runaway case, a new space dimension is
generated and opened up from the internal degree
of freedom.
In string theory there is no essential difference
between internal degrees of freedom and the spacetime coordinates.
𝑉
4D
5D
𝜙
A toy model
We consider 9D toy model instead of 4D theory.
9D string obtained by compactifying one direction
(the ninth direction) of the10D 𝑺𝑶 𝟏𝟔 × 𝑺𝑶(𝟏𝟔)
non-supersymmetric heterotic string on S1 with
radius 𝑟.
The ninth component of the 10D gauge field
becomes a scalar, whose emission vertex is Example
of GHU
𝑎
9 𝑎
𝑖𝑘⋅𝑋
𝑉 = 𝜕𝑧 𝑋 𝑗 𝑧 𝑒
,
where 𝑗 𝑎 (𝑧) is the 𝑆𝑂 16 × 𝑆𝑂(16) current.
We then introduce a background 𝐴 for one
component 𝑎.
One-loop potential in Jordan frame
periodic in A
increasing in r
One-loop potential in Einstein frame
Runaway for
large r
True for all
orders
We have seen that we have a runaway
vacuum in addition to our vacuum.
Eternal inflation at domain wall
Domain wall between two vacua:
✴ For a given random initial condition.
If the curvature at the maximum is of order one in the
Planck unit, the domain wall is wide enough that
✴ DW supports eternal inflation.
✴ A solution to horizon problem.
Eternal inflation at false vacuum is also possible.
Higgs inflation works.
Need another inflaton.
Solution to CC problem
If we assume MPP, our vacuum should degenerate
with the runaway vacuum, which has vanishing CC.
So the CC of our vacuum should be zero.
𝑉
𝜙
This might be too good.
CC of our universe is not exactly zero ~ 2.2 𝑚𝑒𝑉 .
4
Quantum fluctuation of action
In general a system follows quantum mechanics.
In the classical limit, it is described by the least
action principle.
However the value of the action is not completely
determined because of the quantum fluctuation.
Classical action is defined only within this error.
The fluctuation is evaluated as
2
(Δ𝑆) ∼
4
𝑑 𝑥ℒ 𝑥
4
𝑑 𝑦ℒ 𝑦
(𝜕𝜙)2
∼𝑉×
4
𝑀𝑃
(𝜕𝜙)2
.
𝑉: space-time volume
The classical Lagrangian density is determined
up to an ambiguity Δℒ ∼
2
𝑀𝑃
𝑉
.
If 𝑉 = ∞, Δℒ = 0.
For a de Sitter space with Hubble parameter 𝐻,
−4
it is natural to take 𝑉 ∼ 𝐻 .
So we have Δℒ ∼
2
𝑀𝑃
⋅𝐻 .
2
In particular the CC is determined up to the
2
2
4
error ∼ 𝑀𝑃 ⋅ 𝐻 ∼ 𝑚𝑒𝑉 .
Inflation as the string scale physics
Question: If everything is in string scale,
can the inflation occur?
It is possible if some number of fine tunings
are allowed.
Hamada, Kawana, HK: arXiv: 1507.03106
In terms of dimensionless fields, the low energy effective
action of string theory is typically 𝜑 can be any field that becomes
S=
𝑚𝑠8
𝑔𝑠2
+
=
=
𝑑10 𝑥 𝑔 𝐴 𝜑 𝑅
𝑚𝑠8
𝑔𝑠2
𝑑10 𝑥 𝑔𝐵(𝜑) 𝜕𝜑
𝑚𝑠8
𝑉6
2
𝑔𝑠
+
scalar in 4D.
We assume that the tree-level
potential is zero.
+ 𝐶𝑙𝑜𝑜𝑝 𝑚𝑆10 𝑑10 𝑥 𝑔𝑉(𝜑) + ⋯
4
𝐶𝑙𝑜𝑜𝑝 =
𝑑 𝑥 𝑔𝐴(𝜑)𝑅
𝑚𝑠8
𝑉6
2
𝑔𝑠
𝑀𝑃2
2
𝑑 4 𝑥 𝑔𝐵(𝜑) 𝜕𝜑
∼ 10−7 in 10D
4
+ 𝐶𝑙𝑜𝑜𝑝 𝑚10
𝑉
𝑑
𝑥 𝑔𝑉(𝜑) + ⋯
𝑠 6
2
𝑚
𝑠
2
𝑀𝑃 = 2 (𝑚𝑠6 𝑉6 )
𝑔𝑠
4
𝑑 𝑥 𝑔𝐴(𝜑)𝑅
+𝑀𝑃2 𝑑 4 𝑥 𝑔𝐵(𝜑) 𝜕𝜑
2
𝑆9
2 2𝜋 10
2
+ 𝐶𝑙𝑜𝑜𝑝 𝑔𝑠2 𝑀𝑃2 𝑚𝑠2 𝑑 4 𝑥 𝑔𝑉(𝜑) + ⋯
In the Einstein frame we have
S= 𝑀𝑃2 𝑑 4 𝑥 𝑔 𝑅
+𝑀𝑃2 𝑑 4 𝑥 𝑔𝐶(𝜑) 𝜕𝜑
2
+ 𝐶𝑙𝑜𝑜𝑝 𝑔𝑠2 𝑀𝑃2 𝑚𝑠2 𝑑 4 𝑥 𝑔𝑈(𝜑) + ⋯
In terms of the canonical-like field this becomes
S=
2
𝑀𝑃
𝑑4 𝑥 𝑔 𝑅
+𝑀𝑃2 𝑑 4 𝑥 𝑔 𝜕𝜒
2
+ 𝐶𝑙𝑜𝑜𝑝 𝑔𝑠2 𝑀𝑃2 𝑚𝑠2 𝑑 4 𝑥 𝑔 𝑊(𝜒) + ⋯
If we do a fine tuning we have a saddle point for 𝑾 𝝌 :
W
𝜒
~𝑂(1)
S= 𝑀𝑃2 𝑑4 𝑥 𝑔 𝑅
+𝑀𝑃2
4
𝑑 𝑥 𝑔 𝜕𝜒
2
+
𝐶𝑙𝑜𝑜𝑝 𝑔𝑠2 𝑀𝑃2 𝑚𝑠2
4
𝑑 𝑥 𝑔 𝑊(𝜒) + ⋯
𝐶𝑙𝑜𝑜𝑝 ∼ 10−7
Higher derivatives in the effective action can be consistently neglected
because
𝑯~ 𝑪𝒍𝒐𝒐𝒑 𝒈𝒔 𝒎𝒔 ≪ 𝒎𝒔 .
If we assume the value of the potential matches with that of the
SM, we have
𝑪𝒍𝒐𝒐𝒑 𝒈𝟐𝒔 𝑴𝟐𝑷 𝒎𝟐𝒔 ∼ 𝟏𝟎𝟔𝟒 𝐆𝐞𝐕 𝟒 .
2
𝑚
𝑠
2
𝑀𝑃 = 2 (𝑚𝑠6 𝑉6 )
𝑔𝑠
Then we obtain
𝒈𝒔 𝒎𝒔 ∼ 𝟏𝟎𝟏𝟔.𝟓 GeV ,
𝒎𝟔𝒔 𝑽𝒔 =
𝒈𝟒𝒔 𝑴𝟐𝑷
~
𝟐
𝟐
𝒈𝑺 𝒎𝑺
𝒈𝟒𝒔 ⋅ 𝟏𝟎𝟑 ~𝟏.
not bad
Inflation at higher criticality
If one does not dislike the fine tunings, we can tune such that
𝑊 = 𝑊0
𝜒
1− 1−
𝜒𝑐
𝑛+1
.
In terms of the canonical field 𝝋 = 𝑴𝑷 𝝌 , the potential becomes
𝜑
2 2
2
𝑉 𝜑 = 𝐶𝑙𝑜𝑜𝑝 𝑔𝑠 𝑀𝑃 𝑚𝑆 𝑊
.
𝑀𝑃
From this we obtain
𝜖=
1
2
𝑛!
1 𝜒𝑐𝑛+1
𝑛−1 𝑛 𝑁𝑛 𝑛+1 !
2
𝑛−1
∼
⇒ 𝑛𝑆 = 1 − 6𝜖 + 2𝜂 ∼ 1 −
𝑛
𝑛𝑆
2
3
0.933 0.95
1
𝑁
2𝑛
𝑛−1
2𝑛
𝑛−1 𝑁
,
𝜂=−
𝑛
𝑛−1 𝑁
.
.
4
5
6
⇒𝑛≥4
0.955 0.958 0.96
favored.
The potential is smoothly connected to the SM potential:
𝑉𝑆𝑀 ∼ 10−6 𝜑4 .
Example: 𝒏 = 𝟓, 𝝌𝒄 = 𝟏
−7
𝜖 = 2.3 × 10 ,
𝑉𝑐
4
𝑀𝑃
𝑉𝑐
𝑀𝑃4
= 1.2 × 10−13 .
Predictions on the Higgs portal DM
MPP imposes constraints on the Higgs
portal DM.
Hamada, Oda, HK: arXiv: 1404.6141
Higgs portal dark matter
We consider 𝒁𝟐 invariant Higgs portal DM:
𝟏
𝟏 𝟐 𝟐 𝝆 𝟒 𝜿 𝟐 †
𝟐
ℒ = ℒ𝑺𝑴 + 𝝏𝝁 𝑺 − 𝒎𝑺 𝑺 − 𝑺 − 𝑺 𝑯 𝑯
𝟐
𝟐
𝟒!
𝟐
mHiggs =125GeV
mtop =172.892 GeV
κ
ρ
The effect of 𝑺 on 𝝀 is opposite to that of top quark.
Top Yukawa lowers 𝝀 for smaller values of 𝝁,
while 𝜿 increases 𝝀 almost uniformly in 𝝁.
𝝀
Suppose that at some scale 𝚲 we have:
𝝀 𝚲 = 𝟎, 𝜷 𝝀 𝚲 = 𝟎.
If top mass is increased, 𝝀 becomes negative
at some scale. Then we can recover that
𝝀 𝚲′ = 𝟎, 𝜷 𝝀 𝚲′ = 𝟎 by increasing 𝜿.
MPP gives a relation between 𝒚𝒕 and 𝜿.
The scale of the tangent point decrease.
increasing Top Yukawa
increasing 𝜿
𝝁
From the natural abundance, the DM mass is
related to 𝜿:
𝜿
𝒎𝑫𝑴 ∼ 𝟑𝟑𝟎𝐆𝐞𝐕 ×
𝟎. 𝟏
Naturalness and Big Fix
Even if CC problem is solved, we still have to
explain the magnitude of the weak scale.
In terms of the ordinary field theory it involves
a fine tuning, but it may become natural if
one consider modification or correction to the
field theory.
There are several attempts to slightly extend
the framework of the local field theory in order
to explain such fine tunings.
• multiple point criticality principle
Froggatt, Nielsen, Takanishi
• baby universe and maximal entropy principle
Coleman
Okada, Hamada, Kawana, HK
• classical conformality
Meissner, Nicolai,
Foot, Kobakhidze, McDonald, Volkas
Iso, Okada, Orikasa
• asymptotic safety
Shaposhnikov, Wetterich
Summary
• It is possible that SM is directly connected to the Planck scale.
We may have only a minor modification of SM below the string
scale.
• We still need some mechanisms to explain the smallness of the
CC and Higgs mass.
For example, we can consider MPP or MEP.
• We can justify this picture by examining e.g. the inflation, DM
and precise measurements of the top and Higgs masses.
• We may start thinking about Planck scale physics seriously.
Our universe might close to one of the perturbative vacuum.
Appendix
A mechanism to have MEP
Consider Euclidean path integral which
involves the summation over topologies,
Coleman (‘88)
not consistent
  dg  exp   S  .
topology
Then there should be a wormhole-like configuration in
which a thin tube connects two points on the universe.
Here, the two points may belong to either the same
universe or the different universes.
If we see such configuration from the side of the large
universe(s), it looks like two small punctures.
But the effect of a small puncture is equivalent to an
insertion of a local operator.
Therefore, a wormhole contribute to the path
integral as
 dg   c  d
ij
4
x
x d y g ( x ) g ( y ) O ( x ) O ( y ) exp   S  .
4
i
j
i, j
Summing over the number of wormholes, we have

1
4
4
i
j
  ci j  d x d y g ( x ) g ( y ) O ( x ) O ( y ) 

N 0 n !  i , j



 exp   ci j  d 4 x d 4 y g ( x ) g ( y ) O i ( x ) O j ( y )
 i, j
n

.

Thus wormholes contribute to the path integral as


4
4
i
j
ci j  d x d y g ( x ) g ( y ) O ( x ) O ( y )  .
 dg  exp  S  
i, j

bifurcated wormholes
⇒ cubic terms, quartic terms, …
y
The effective action becomes a factorized multilocal form:
Seff 
c
i
Si   ci j Si S j   ci j k Si S j Sk  ,
i
Si 
d
ij
D
i jk
x g ( x )Oi ( x ) .
By introducing the Laplace transform


exp   Seff  S1 , S2 ,     d  w  1 , 2 ,  exp    i Si  ,
 i

we can express the path integral as


Z    d   exp   Seff    d  w    d   exp    i Si .
 i

 
Coupling constants are not merely constant but to
be integrated.
A solution to the cosmological constant problem
Z 
 d w     dg  exp   

g R  
g .
S
r
S
4

r
1


2
4
d

w

dr
exp


r


r
 




 exp 1/   ,   0
 d  w    no solution,   0
𝚲~𝟎 dominates irrespectively of w    .
Euclidean action is not bounded from below.
Difficulty Action is not bounded from below.
Problem of the Wick rotation
WDW eq.
H total   0
H total  H universe  H graviton-matter
H universe
 1 2
 
pa 
 2a

 ←wrong sign

a : size of the universe
“Ground state” does not exists.
Wick rotation is not well defined.
H universe
H graviton-matter is bounded from below.
H universe
is bounded from above.
H matter ,
This is not a merely technical problem,
but it is the reason why the universe can evolve from
t
Problem: Consider the Minkovski quantum
theory given by the following factorized multilocal action:
Seff 
c
i
i
Si 
Si   ci j Si S j   ci j k Si S j Sk  ,
ij
i jk
D
d
x
g
(
x
)
O
(
x
)
.
i

T. Okada, HK: arXiv:1110.2303, 1104.1764
HK: Int. J. of Mod. Phys. A vol. 28, nos. 3 & 4 (2013)
1340001
The path integral is given by


Z    d   exp  i Seff    d  w    d   exp  i  i Si .
 i

Coupling constants are not merely constants but to
be integrated.
 
superposition of
various states
corresponding to
various↓couplings
Coupling constants that
give the maximum
entropy dominate.