On a Singular Solution
in Higgs Field (6)
- A long time behavior of the candidate for
dark energy
Kazuyoshi KITAZAWA
RISP Japan
(DPF 2013 Santa Cruz, 15 Aug)
Contents
We have recently discussed the calculated mass correction
for SM Higgs boson (H0), and also the degenerates into a
candidate of dark matter and dark energy from ur-Higgs
boson (ur-H0) and excited ur-H0 respectively
In this talk, we’ll review and discuss with;
1. Short review
- Corrected Higgs mass (125.28 GeV/c2) formula for the
excited states, and its multiwall-fullerene representation
of glueballs, having ρ mesons inside, as an excited state
of ur-H0
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
2
- Almost of ur-H0 degenerate into the hybrid molecules of a
glueball and pseudo-scalar mesons as a candidate of dark
matter; and the excited ur-H0 degenerate into the quasicrystals of fullerene consists of σ mesons and some ω
mesons, as the candidate of dark energy
2. A long time behavior of the candidate for dark energy
- Gradual disruption of dark energy of almost non-mass
fullerene into smaller ones consist of several σ mesons
by collisions
- The working force to expand the volume of universe is
thought to be the result of these fullerenes’ mutual
repulsive strong force between respective ω mesons
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
3
- The observed accelerating expansion of the universe
would be explained with the scale factor (a) in RobertsonWalker metric by describing a long time disruption-behavior
- The accelerating expansion will continue until number of ω
meson in the disrupted fullerene becomes around one or
zero; then the expansion slows down and at last the
contraction of universe begins after equilibrium by central
gravity
- A certain decay route into dark energy from dark matter
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
4
== Short review ==
- corrected Higgs mass formula with the respective mesons’
(K.K: BORMIO 2013)
masses of excited states:

M (H0 )*   3 c0 (1p), 3 (1990) 3 (1990)  4 B S  B S 

Mi


BC B C

D

S
DS


 125.28 GeV c 2 .
- Its multiwall-fullerene representation of glueballs, having
ρ mesons inside, as an excited state of the ur-H0
M (H0 )*    40  f 6  3100    6   GB40    (770)
Mi
Mi
  6   GB40   3 (770)  (770) .
Mi
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
5
- Higgs mass originally derived from an asymptotic solution in Higgs
field and its mass formula by mesons’ masses of truncated-octahedron
Higgs Mass Formula
(K.K: TAMJ 57(2009) 217; 58(2010) 61)
  2v, (   0) : Asymptotic singular solution
Thus from EOM, Higgs field mass formula is

 (  )  0,



2
2
2



m  2  W W
 2  g    (Z )2  2  G  ,


where     , infinitesimal Grassmann number
2
M H0 
2M W
1  cos 2 W

2M W M Z
MW 2  M Z 2
 120.611 GeV c
2
truncated-Octahedron
(tr-O)




M H0 (3c ,...(exper. values) )   3c , 10    4 B S 0 B S 0 B C  B C  D S  D S  


Mi
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
6
Phase transition diagram -- from (tt_bar)* to Higgs boson
K.K: JPSA 3 (2013) 114
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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- Degenerate into the candidate of dark matter
Only (1/Ne) of these ur-Higgs bosons can be transformed finally to the
H0 with corrected mass (125.28 GeV/c2) through multi-photon
resonances of its component by irradiated γ-rays from neighbors.

t
240
0
GB

fullerene
:
ur-H
 
i 1
t
i
ω

Ne  m c2 E  9.483
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013

・・・
8
A hexagon and a pentagon on GB240 (= ur-Higgs)
Each glueball feels the color of neighbor glueball, and is supposed to have color
valence of 4 (four).
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
9
- Degenerate into the candidate of dark energy
(K.K: EPS HEP 2013)
excited ur-H0 of multi-wall
developed fullerene
ρ → π+π- /e+e- decays will be suppressed
by the outer-walls made of σ mesons
GB ⇒ σ meson
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
10
Overview of degenerate ur-Higgs
(fragmentation)
(ur-Higgs)*
(tr-O)*
decay
(γγ; γ, …)
(fold)
decay
excite
(develop)
GeV/c2
ur-Higgs
125.28
excite
tr-O
Dark Matter
Dark Energy
Glueball level
120.611
(ΔM >0)
Meson (mass) level
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
11
The content ratios
Although all rates of the candidates are obtained now,
more over, we should refine these values as follows.
In order to take account of mass contribution from QGP, such
as totally equivalent top quark, we re-calculate the rate of
*
Atom;
M t  M b  M c  M s  n ( gluon )
R Atom   R Atom  
Mt
 0.03728 
R DM 
171.266  4.65  1.275  0.095  137.036   0.50255 2 
171.266
 0.04609, where we put n*  1  .
 R DM   0.2357,




 R dark energy   1  R Atom   R DM   0.7182
all of which are completely in accordance with the result of
WMAP (9 Years).
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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Comparison to observations
- WMAP: content of matter, dark matter and dark energy
WMAP (9 Years)
This calculation
Atom
0.0463 ± 0.0024
Dark Matter
0.233 ± 0.023
0.04609
0.2357
Dark Energy
0.721 ± 0.025
0.7182
- Fermi-LAT: mass of dark matter
Fermi-LAT
129.8  2.4713
This calculation
 120.611
GeV c2
And … - CDMS-II WIMP mass: 8.6 GeV/c2
14   WIMP mass   This calc. (120.611; possible ancestor of the WIMP mass)
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
13
2. A long time behavior of the candidate for dark energy
Finite nucleus by mean-field theory of σ-ω model
    r  ,  =  ,0(r)
By inserting into Dirac equation.

0
i



m

U
r


  V  r    x   0,
 
where U  r   g   r  , V  r   g   r  .
We shall regard the developed fullerene as a finite nucleus of the
limit of N(p, n)→0, namely, m(p, n)→0.
Because the masses of degenerate fullerenes are thus to be some ultimately
small effective mass which has been reduced by σ-potential, the removing
speed by the repulsive force between their ω mesons would not be so small,
overcoming the attractive force by σ mesons, after approaching very near
each other.
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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- Gradual disruption of dark energy of almost non-mass fullerene
into smaller ones consist of several σ mesons with some ω mesons
by collisions
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
15
σ40+ω×12
σ20×24 +12ω
regular dodecahedron
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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A disruption figure like ‘soap bubble’ of the candidate for
dark energy
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
17
The rate of the candidate of dark energy
The rate of dark energy per unit volume of the universe is to be expressed as
R    ,
where   N : the macroscopic cross section,
 : the flux of the dark energy,
N : the density of the target dark energy,
 : the microscopic cross section.
So it is needed at least to keep the expansion of universe that NΦ>0.
Moreover for accelerating expansion, dR/dt=d(NσΦ)/dt >0 is needed.
The respective acceleration expression by force balance
M *   f  f   f G ,
where
 f  f    V ,  r 
r
 m r
    2 e  m r
e
    g
 g 2
r  4 
r
r
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
 

 
18

 1
2  m r
2  m r
2  m r
2  m r 
g
e

g
e

m
g
e

m
g



  
,

 
  e
4r  r

M * : effective mass for respective dark energies,
m , : mass of ω-, σ-meson,
 : acceleration of respective dark energies,
f , f , fG : forces from  mesons,  meson and central gravity,
V , : potentials from ω-, σ-meson,
g , 2 4 :ω-, σ-meson coupling constant of fundamental interaction,
r : distance between respective mesons.
Although at the early time the outgoing
acceleration would have been apt to
cancel between dark energies, the
expansion might have gradually begun
at the point far from the center of the
universe.
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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Where the effective mass of a candidate for dark energy of
fullerene could be approximately estimated by taking a limit
for the nucleon density in a known formula of nuclear matter:
M  M Us  M  4
*
g 2
m 2
kF
0
3
d k
 2 3
M
M
*
*2
k
2


g 2
3 k F 2  
 lim  M  2  N 1 
2 
 N 0
m
10
M

 


 M  0.
M : mass of nucleon
U s : potential energy of σ meson
k F : Fermi wave number
 N : density of nucleon
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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The force between the young dark energies at short distance
(computed by σ-ω model)
1200
force [N]
1000
800
600
400
200
r [fm]
0
0
1
2
3
4
5
-200
-400
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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- The working force to expand the volume of universe is thought
to be the result of these fullerenes’ mutual repulsive strong force
between respective ω mesons.
The ω mesons are mutually repulsive not only in their fullerenes
but also with intercalated ω mesons of another fullerenes.
- The observed accelerating expansion of the universe would be
explained with the scale factor (a) in Robertson-Walker metric by
describing a long time disruption-behavior:
a
4 G

   3P  ,
a
3
So when (ρ+3P) < 0, i.e., P < -ρ/3, the universe is in the
accelerating expansion, whose minus pressure is produced by
a lot of newly outgoing fullerenes under repulsive force.
And,
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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- The accelerating expansion will continue until number of ω
meson in the disrupted fullerene becomes around one or zero;
then the expansion slows down and at last the contraction of
universe begins after equilibrium by central gravity which may
feel effective mass (M*) from σ potential, with some quasi-static
clustering of the disrupted fullerenes under attractive force
between their σ mesons.
Let dn(t) as the infinitesimal number of active (outgoing or
repulsive) dark energies against da(t) of the universe, then
dn  t   n  t  a 3  t  da  t   n  t   n0 exp 1 a0 2  1 a 2  t  2  ,
where n0 , a0 : at equilibrium of t  t0  1
On the other hand,
P t   n t  a2 t  ;
 P  t   n0 exp  1 a0 2  1 a 2  t  2  a 2  t  .
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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We can then compute the formula of P(t) with an inflection
point of a(ti) as
ai 2
P  t    2 i exp  1 2  1 ai 2  1 a 2  t  .
3a  t 
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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Since the acceleration of a(t) is then expressed as

ai 2
4 G
a
a t    t   2
i exp 1 2  1 ai 2  1 a 2  t  
3
a t 




,

the inflection point should satisfy


ai 2
  t   2 i exp 1 2  1 ai 2  1 a 2  t    0.
a t 
Actually it is noted that a(ti)= ai, and ρ(ti)= ρi surely satisfy above
condition. Then we may write:
i) t  ti(1)  decelerating expansion 
a  t   ai(1) ,
ii) ti (1)  t  ti(2)  accelerating expansion 
ai (1)  a  t   ai(2) ,
iii) ti(2)  t  t0  decelerating expansion* 
ai(2)  a  t   a0 .
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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*: The number of active fullerene is insufficient to produce the breedingcollision or disruption when ti(2) < t < t0. At t= ti(2), each fullerene would
no longer have only one or zero ω meson inside. Each interval is to be
determined by the stiffness against the disruption of the fullerene.
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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The P(t) behavior in the region around at P= -ρi /3
dP

dt
2 ai
2
3
3 a (t )


i  1 
1   da
1 1


exp


 2  a 2 a 2 (t )    dt  0.
2
2 a (t ) 
  i

1
2
2
1   
3
2   da  
1  d a
1 1

i  exp   2  2     
 3
     1  2  2   0,
2
3
dt
3 a (t )
2
a
a
(
t
)
a
(
t
)
a
(
t
)
  dt   2a (t )  dt 
  i
  
2
2 ai
d P
2

2 3a (t )  2
2

 da  .


 
2
2
dt
a (t )  2 a (t )  1  dt 
2
d a
2
Where we assigned
d 2a
And,
dt 2 a (t )a
da
dt
 0.
a ( t )  ai (1)
0 ;
i (1)
then a(t ) P(MINIMUM)  1
2,
ai (2)  2 3
with a0  1.
And, by numerical calculation,
ai (1)  0.62013
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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After all, we can describe that the breeding collision and disruption
of the outgoing developed fullerene would be a factor of the
accelerating expansion of the universe, i.e. the candidate of dark
energy. And the repulsive potential energies between ω mesons of
the fullerenes are thus to be consumed at respective disruptions to
expand the universe.
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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- A certain decay route into dark energy from dark matter
As seen, the candidate for dark matter to which major part of urHiggs boson may degenerate, is expected to decay into the
candidate for dark energy. We propose a structural model of
nucleus of the candidate for dark matter from the stand point of a
former state of the dark energy.
We’ll reconstruct a model so as the
nucleus consists of many neutrons,
protons and their antiparticles with
several σ mesons and some ω mesons
(i.e., σ-ω model), which could also be
regarded as a descendant of the
degenerate from ur-Higgs boson.
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
29
Because we have considered that
GB  f 0  500    meson; ur-Higgs  80*f 0 1500 
DM90f0 1370  90 * f 0 1370 
*: molecular state
DM70f0 1710  70 * f 0 1710 
These f0 mesons’ masses were able to be represented by masses of π-octet
(pseudo-scalar mesons) and mass of GB which was computed at 502.55 MeV/c2:
m f0 (1370)
   
     
 GB  3

90

 2  70
 3 90
m f0 (1500) 
m f0 (1710)
3GBm
70
0 

K 
0
90
K 
75

 
90
40
90



0
 
 mi
(possibly be paired)
,
i

0
 GB  K 

,

1
3

K  4

 .
 m
i
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
30
By interpreting as an effective mass as well for mass of these f0
mesons because the light quarks in 3π+π- of them are equivalent
to that of (nn_bar +pp_bar); and so on, we approximately write:
DM 90 f0 (1370) : N1  M n, p  1  90 M   120.611 GeV/c 2 ,
DM 70 f0 (1710) : N 2  M n, p   2  70 M   120.611 GeV/c 2 ,
N1  N 2  4i,
N1, N 2  4 j , 4 j ,
1   2 ,
where N1 , N 2 : total number of included neutrons, protons and their antiparticles
1 ,  2 : mass effective factor of  meson,
M n , p : mean mass of neutron and proton,
M  : mass of  meson (502.55MeV/c 2 ),
i; j , j  : integer.
And by the condition that α1 ,α2< 1, we have
N1  164, N2  156,
1  0.7377,  2  0.7351.
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
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A scheme of nucleon-decay cycle in the nucleus might be
n  p  e   e , (merely happens in an almost -stable nucleus)
 n  p  e   e (thus merely happens in an almost -stable nucleus)
while the p  e   
 lifetime: 300, 000 years  ;
then these e  , e  captures by current p, p will be happened in long time:
p  e   n  e , p  e   n  e .
As a result, one n, n pair would be dripped out for the sake of
keeping stability of nucleus, with (2e  e )   .
The number of neutron is supposed to be maximum at the early time.
And this number, i.e. the effective mass of dark matter, very gradually
decreases with dripping out the neutron pair and so on above in long
time. Hence, finally such a dark matter will reach the dark energy
which could have only few neutron.
K. Kitazawa: DPF 2013 Santa Cruz,
15 Aug. 2013
32