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Fields and strings in the era of
post LHC
GCOE Symposium
Feb. 12. 2013
Kyoto University
Hikaru Kawai
LHC gave beautiful results
In some sense, they indicate
“the worst scenario”.
Higgs particle was discovered,
but nothing else.
Especially, no sign of the SUSY.
We need to reconsider the meaning of SUSY.
Why did some people like SUSY?
The only scientific reason is that it was
thought to solve the naturalness problem.
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 expected
as follows:
naturalness problem (cont.’d)
gS 2
.
2
mS
dimension -2 (Newton constant)
GN
dimension 0
(gauge and Higgs couplings)
g1 , g2 , g3
H
dimension 2 (Higgs mass)
mH 2
unnatural ! → mH
2
100GeV 
2
2
g S mS
2
gS ,
gS 2 .
g S 2mS 2 .
10
18
GeV 
2
dimension 4
(vacuum energy or cosmological constant)
 mS 4 .
unnatural ! !→ 
 2 ~ 3meV 
4
2
g S mS
4
10
18
GeV 
4
SUSY is the symmetry between
bosons and fermions
Bosons and fermions cancel the UV
divergences:
2
m
H  0.
⇒
+
bosons
fermions
+
2
⇒ mH
⇒   0.
⇒ 
M SUSY 2 .
M SUSY 4 .
However, SUSY must be spontaneously
broken at some momentum scale MSUSY ,
because such symmetry does not exists at
the low energy scale. Below the momentum
scale, cancellation does not work.
Possibility of SUSY as the solution to the
naturalness problem
Therefore, if MSUSY is close to mH , the Higgs
mass is naturally understood, although the
cosmological constant is still a big problem.
History
Actually when the Z- and W- bosons were
discovered in 1983, there was a strong
motivation to expect that MSUSY is close to
the weak scale, and the Higgs mass is
protected by SUSY.
But it turned out not the case. No signal of
SUSY was observed around the weak scale.
Present status of SUSY
After 30 years the Higgs particle was finally
discovered at mH ~ 125 GeV.
Furthermore, no signal of new physics is observed
below 1 TeV, which indicates that
MSUSY >1 TeV ,
if it ever exists.
2
2
1

suggests that
100
The inequality mH / M SUSY
we need to fine tune the Higgs mass parameter at
least by factor 100. In other words,
the probability that the Higgs mass is naturally
explained by SUSY is less than 1%.
Possible solutions to the naturalness problem
It seems that we would better peruse the solutions
other than SUSY.
1. We do not have to mind. We should simply take
them as they are.
2. Anthropic principle.
The parameters should be such that we can exist.
a) The wave function of the universe is a
superposition of various worlds having
different low energy effective Lagrangians:
  1  2  3 
We are in one world described by one of them,
whose parameters must be such that we exist.
Anthropic principle. (cont.’d)
b) The universe has different parameters place
by place. We are sitting at one place, where
the parameters are such that we can exist.
3. The parameters are fixed by some nonperturbative effect of quantum gravity/string
theory such as Coleman’s baby universe
mechanism.
They are not totally nonsense, but it is difficult
to judge what is correct.
Simply accept the observed parameters and
analyze to what energy SM is valid
Most particle physicists believe that the SM is the
low energy effective theory of string theory.
Then the question is whether SM or its minor
modification is valid to the string scale mS, or
completely new physics appears before reaching it.
If it is the former case, we have a “desert” in the
sense that we have nothing other than the SM
physics below mS.
In order to examine it, we consider the SM
Lagrangian with cutoff momentum Λ, and evaluate
its bare parameters.
If no inconsistency appears, it means that SM can
be valid to Λ.
[with Y. Hamada and K. Oda]
Bare parameters of the cutoff theory (1)
mHiggs =126GeV
mtop =172 GeV
No inconsistency arises
below the string scale.
U(1)
SU(2)
Ytop SU(3)
Higgs self coupling
log10 Λ[GeV}
Higgs mass 2
Bare parameters of the cutoff theory (2)
mHiggs =126GeV
mtop =190 GeV
Higgs field becomes unstable.
Ytop
SU(3)
Higgs mass 2
SU(2)
U(1)
log10 Λ[GeV}
Higgs self coupling
Bare parameters of the cutoff theory (3)
mHiggs =126GeV
mtop =150 GeV
Higgs self coupling
No inconsistency arises,
but the Higgs self
coupling tends to diverge.
U(1)
SU(2)
SU(3)
Ytop
log10 Λ[GeV}
Higgs mass 2
Bare parameters of the cutoff theory (4)
mHiggs =100GeV
mtop =172 GeV
Higgs field becomes unstable.
Ytop
SU(3)
SU(2)
U(1)
Higgs mass 2
log10 Λ[GeV}
Higgs self coupling
Bare parameters of the cutoff theory (5)
mHiggs =150GeV
mtop =172 GeV
Higgs self coupling
Ytop SU(3)
No inconsistency arises,
but the Higgs self
coupling tends to diverge.
SU(2)
U(1)
log10 Λ[GeV}
Higgs mass 2
Summary of the bare parameters
• It is possible that the SM is valid to the string scale. In
other words, desert seems probable.
• The experimental value of the Higgs mass seems to be just
on the stability bound. Nature seems to like the marginal
value.
• The bare Higgs mass becomes close to zero at the string
scale. It implies that SUSY is restored at the string scale.
Actually there are many string vacua in which SUSY is
spontaneously broken at the string scale.
• The Higgs self coupling also becomes close to zero at the
string scale. It indicates that the Higgs potential becomes
almost flat around the string scale, which opens the
possibility that the Higgs field plays the roll of inflaton.
Higgs field as inflaton (1)
If we allow a fine tuning of the parameters of the
SM, the Higgs field can play the role of inflaton.
[with Y. Hamada and K. Oda]
The effective potential of the Higgs field is given by
6 6 8 8
4
Veff      c    2   4   .
mP
mP
Here the first term on the RHS is determined by the
low energy renormalizable theory, that is the
standard model, and the other terms are so called
Plank suppressed terms that depend on the
underlying microscopic theory.
c is a constant of order the coupling constants:
c ~ 0.1 .
Higgs field as inflaton (2)
In the case of string theory, we have
2
1
g
S
6 , 8 ,
gS 2 ,
,
2
2
mP
mS
and typically 6 , 8 ,
0.1 .
As we have seen, in some
parameter region the Higgs
coupling λ(Λ) takes
minimum near the Plank
scale, whose value is slightly
negative.
mHiggs =126GeV
mtop =172 GeV
log10 Λ[GeV}
Higgs field as inflaton (3)
Then the first term of Veff looks as follows.
  c   4
log10  c 
Therefore if we add the second term, we can
obtain a saddle point by tuning one parameter.
  c   4 
6
mP
6

2
log10  c 
Higgs field as inflaton (4)
Because the logarithmical change of λ in the first
term is slow compared to the power behavior of
the second term, the saddle point arises near the
left zero of λ(Λ) , and we can approximate it as
 c 
.
 0 
  c  b log 
  c 
0
log10  c 
Furthermore, at the saddle point Φ is smaller than
the Planck scale by about factor 100 so that the
higher order Planck suppressed terms are negligible.
Higgs field as inflaton (5)
Therefore it suffices to consider
 c  4 6 6
Veff    b log 
  2  .
mP
 0 
Then it is easy to see that this function has a saddle
2
 0 
point if
1/ 2
2
3e 6 

bc
,

 mP 
e1/ 4  0
at
sp 
,
c
and the value there is given by
4
Veff
6  sp 
b2
 
.
 
2 2
mP 
4  mP 
366
There is a difficulty.
From the density perturbation of the early universe,
it is desirable to have V
eff
2 2
P
m 
which indicates b 10
5
1010 ,
and
0
mP
103.5.
The former can be satisfied by tuning the
parameters in such a way that the Higgs self
coupling λ(Λ) is almost tangent to the Λ axis.
But the latter becomes hard to satisfy then.
There is a difficulty. (cont’d)
mHiggs =125GeV
mtop =171.316 GeV
b 10
0
mP
5
101.5  103.5
0
b can be set to the desired value by
doing one-parameter tuning, but
Λ0 is too large then.
The value of Λ0 does not change
much even if the other couplings
are changed.
Introduce an extra field
In order to change the shape of the function λ(Λ) ,
we introduce an extra scalar field that can be
identified with dark matter:
2
1
 4  2 †
L             SM
2
4!
mHiggs =125GeV
mtop =172.895 GeV
κ
2
Then we can
change the
position of Λ0
as we want.
ρ
0
Fields in the era of post LHC
• We might have nothing other than the SM particles below
the string scale.
• There is a small room for the other field, and it seems that
SM with a little modification is the right theory.
• It is better not to insist on SUSY but to think about what is
really needed.
• Some parameters such as the cosmological constant, Higgs
mass, and strong CP phase are unnatural, but it might not
be correct to try to solve them within the framework of
field theory.
• Dark matter is really needed and should be explained in
terms of field theory.
• It is not clear whether inflation should be explained in the
field theory context or not, but it is worth trying.
Gravity is different.
Gravity is different from the other interactions in the sense
that it is not renormalizable.
If the theory is renormalzable, the effects of short distance
quantum fluctuations can be absorbed to the redefinition
of the parameters of the particles such as mass and
coupling constants.
It means the picture of point particles is valid even after
quantization.
~
mass renormalization
~
coupling constant renormalization
Gravity is different. (cont’d)
On the other hand, if the theory is not renormalizable,
short distance quantum fluctuations are too large to
absorb to the parameters of the particles.
It means that the notion of point particle is no longer valid
if we consider the quantum effects of gravity.
In other words, if we want to include quantum gravity,
we need to think about extended objects instead of point
particles.
mass renormalization
Not possible
coupling constant renormalization
Not possible
String theory
In string theory, we consider one-dimensional objects
like a rubber band on which some internal degrees of
freedom are sitting.
The mass spectra and interactions are completely
determined by those freedom.
spin
graviton
2
Gauge fields
quarks and leptons
Higgs
1
0
heavy particles
standard model
,
m2
..
, .
String as unification
• Gravity is automatically contained.
• Effects of short distance quantum fluctuations are
so small that there is no UV divergence.
• Gauge and matter fields also appear naturally.
• But their structures depend on the choice of the
internal degrees of freedom.
• There are infinitely many theories that have
various space-time dimensions, gauge group,
number of generations, …etc. , depending on the
choice.
It is believed that they are different vacuum
(ground state) of the same theory.
Non-perturbative effects in string theory
As we have seen, there are many theories corresponding
to the choice of the internal degrees of freedom.
However, if infinitely many strings are condensed
properly to the vacuum of one theory, it is converted to
that of another theory.
vacuum of
one theory
⇒
infinitely
many strings
＝
vacuum of
another theory
It is expected that all the theories are different vacua of
one theory.
( non-perturbative effects = effects caused by infinitely many objects)
String is one.
9D theory 9D theory
#2
#1
4D theory
#1
nonperturbative
string
theory
10D Heterotic E8× E8
10D Type II A
11D M-theory
tunneling
string condensate
Each vacuum is perturbatively stable.
Non-pertirbative effects cause transitions.
The most important question is whether a unique vacuum
exists or not after taking the non-perturbative effects into
account.
If it is the case, it should be our SM universe, and we should
be able to explain all the parameters of SM from one theory.
Matrix model
We need to find a formulation of string theory which
can describe the non-perturbative effects.
In other words, we need to find a good definition of
string theory.
One possibility is the IIB matrix model:
1
1   2

S  Tr  [ A , A ]     [ A ,  ] 
2
4

A    1, ,10  ,
 10D Majorana-Weyl  : N  N hermitian
O   dAd O  A,  exp   S  A,   .
Evidences for the IIB Matrix model
(1) World sheet regularization
Green-Schwartz action in the Schild Gauge
1   2 1
S   d  ( {X , X }     {X  ,  } )
4
2
2
Regularization by matrix
{ , }→[ , ]

→
Tr
1
1   2 1
S   2 Tr ( [ A , A ]     [ A ,  ] )
g
4
2
(2) Loop equation and string field
Wilson loop = string field
w(k  ())  Tr ( P exp( i  d k  ( ) A   fermion) )
⇔ creation annihilation operator of | k  () 
loop equation → light-cone string field
This can be shown with some
assumptions .
x   x 0  x 9  const .
(3) effective Lagrangian and gravity
x
(1) 
1 a
(1) 

x ( 2 ) 1 a ( 2 )

Integrate out
this part.
The loop integral gives the
exchange of graviton and dilaton.
Seff  
1
(1)
(1)
( 2)
( 2)
{
const

tr
(
f
f
)
tr
(
f
f






)
(1)
( 2) 8
(x  x )
 const  tr ( f (1)  f (1)  ) tr ( f ( 2)   f ( 2)   )    }
Emergence of Space-time
A remarkable feature of the matrix model is that the
space-time itself emerges dynamically from the matrix
degrees of freedom Aμ .
There are several mechanisms for the emergence of
space-time.
(1) Aμ as the space-time coordinates
mutually commuting Aμ ⇒ space-time
(2) Aμ as non-commutative space-time
non-commutative Aμ ⇒ NC space-time
(3) Aμ as derivatives
as in the naive large-N reduction
Einstein equation follows.
The classical EOM of the IIB matrix model is
 Aa  Aa , Ab   0.
If we impose the Ansatz
( a  1.. D )

  a 
Aa  

0 ( a  D  1..10).
it becomes
0  ( a ) , ( a ) , ( b )    0  a , a , b   (a Rab cd )Ocd  Rab cac
 a Rab cd  0 , Rab  0  Rab  0 .
The Einstein equation follows from the EOM of the
IIB matrix model
multiverse in the matrix model
Multiverse appears naturally if we consider block diagonal
configurations.
b,
C( a )   b
0
0
b,
C( a )
b
・
・
matrix model
Each
block
represents
a universe.
quantum gravity
Factorized action from IIB matrix model
Y. Asano, A Tsuchiya, HK
The low energy effective action around the multiverse is
not a simple local action but has the multi-local form:
Seff 
c
i
Si   ci j Si S j   ci j k Si S j Sk  ,
i
Si 
ij
i jk
D
d
 x g ( x )Oi ( x ) .
y
x
y
z
x
Coleman (‘88)
Consider the path integral which involves the
summation over topologies,
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
universe.
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.
Summing up wormholes, we obtain the multi-local action.
Solution to the naturalness broblem?
The effective action of quantum gravity/string is given by
Seff 
c
i
Si   ci j Si S j   ci j k Si S j Sk  ,
i
Si 
ij
i jk
D
d
 x g ( x )Oi ( x ) .
The path integral is given by


Z    d  exp  i Seff    d  w      d  exp  i  i Si .
 i

Coupling constants are not merely constant but to be
integrated.


If   d  exp  i  i Si  is peaked strongly around some values
 i

of λ , it means that the coupling constants are dynamically
fixed to those values.
If not, we have a kind of parallel world consisting of world
with different couplings.
A good point of matrix model
The matrix model is well defined, and in principle it is
possible to determine which case is true.
There are some attempts to perform Monte Carlo
simulations of the IIB matrix model, and they observed
an expanding four dimensional universe.
[Nishimura et.al.]
Although it is in a very primitive level at present,
numerical analyses seem to work to examine the vacuum
of the matrix model.
Fields and Strins in the era of post LHC
• There might be a desert to the Plank scale.
• It is not disappointing, but it means that there is
no obstruction for connecting the SM physics to
the Plank scale one.
• We can start serious comparison between string
theory and the SM model.
• To do so, it is important to know the precise SM
data such as neutrino mass and mixing, and what
is really needed besides the SM.