On a Singular Solution
in Higgs Field (7)
- The stiffness of candidate for dark energy,
and the matter-antimatter consumption by
candidate for dark matter
Kazuyoshi KITAZAWA
RISP Japan
(PHENO 2014, University of Pittsburgh, 5 May)
Contents
We have recently discussed the degenerates into the
candidates for dark matter (DM) and for dark energy (DE)
from ur-Higgs boson, and also a long time behavior of DE.
In this talk, we’ll review and discuss with;
1. Short review
- The degenerates into DM and into DE from ur-Higgs boson
(ur-H0),
- A long time behavior of DE.
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2. The stiffness of DE
- A critical collision number,
- A formula for the disruption interval, the disruption timing,
etc.
3. A scenario of matter-antimatter (M-A) consumption by DM
- Some constituent parts of DM neighboring M-A,
- Washing-out into only certain matter.
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== Short review ==
- The degenerates into DM and into DE from ur-Higgs boson
K.K: PoS(EPS-HEP 2013)002
(ur-H0):
Ur-H0 (mass of 120.611 GeV/c2) can be transformed finally to the
excited Higgs boson (H0) with corrected mass of 125.28 GeV/c2,
through multi-photon resonances of its component by irradiated
γ-rays from neighboring  t t s onto it (ur-H0).

t
240
  GB
i 1
t
 fullerene : ur-H
i

ω
0

Bethe-Salpeter eq.
・・・
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Phase transition diagram -- from (tt_bar)* to Higgs boson
The rate of excited Higgs boson would be equivalent to the rate of
Atom itself. Then from the phase transition diagram,
1
1  2M t  M t 2 
R Atom  
 1   Q  
 1 
  0.03728,
Ne
Ne 
2M t

N e  m c 2 E  9.483,
m  125.28  120.611  4.669 GeV / c 2 , E  492.35 MeV.
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The rate of ‘total matter’ is estimated as
1

R total matter   1   Q  
.
2 2
Since the rate of fullerene of pure Glueballs (GBs) has been computed
as one third of the ‘total matter’ which was represented by the mixture
of fullerenes of pure GBs and the hybrid molecules (one-GB and several
certain light pseudo-scalar mesons) , the rate of remainder is
1

 0.2357  R DM 
1  1 3  R total matter  
3 2
that is, the rate of degenerate into DM from ur-H0.
And we re-calculate the rate of Atom by taking account of mass
contribution from QGP such as totally equivalent to top quark,
R Atom  R Atom   1 M t   M t  M b  M c  M s  n* ( gluon)  0.04609,
where we put n*  1  ,  : fine structure constant; and
( gluon) : mass of gluon  mass of GB 2.
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excited ur-H0 of multi-wall
developed fullerene: DE
ρ → π+π- /e+e- decays will be suppressed
by the outer-walls made of σ mesons
GB ⇒ σ(500) meson
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- A long time behavior of DE
K.K: DPF2013(Santa Cruz)_75
Gradual disruption of DE mesons by collisions of rather low-mass fullerene
into smaller ones consist of several σ mesons with some ω mesons:
σ40+ω×12
σ20×24 +12ω
σ60+ω×8+4ω
σ30×16+12ω
regular dodecahedron
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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.
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The P(t) behavior in the region around at P= -ρi /3 by Robertson-Walker eq.
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
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== The stiffness of DE ==
- A critical collision number
We think that the disruption of DE would be caused by weariness from
local micro deformation during several collisions. By considering the outerwall of DE consists of σ mesons as a membrane, we shall firstly express the
pressure produced by strong forces among σ mesons (attractive) and also
among ω mesons (repulsive) on local surface shown in figure below.
1200
force [N]
1000
800
600
400
200
r [fm]
0
0
1
2
3
4
5
-200
-400
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From Zoelly’s formula, classically, allowable buckling pressure is
2
 1  2 h  h 2 
2 EY
h
 
pa =




  , where h  R,
2 
2 
2
3 R 2R 
R 1   
3 1    R 
2 EY h
EY : Young modulus, ν: Poisson ratio, R: radius of DE.
Because we consider each damage at collision is independent, so that it is
cumulative. Then 1st disruption of DE will occur as soon as Miner’s law:
ni
i N  1
i
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is satisfied.
12
To see the critical collision number (NC) of DE, we need describe saturated
energy for deformation. The patterns of deformation are to be classified by
increasing N-value of S-N diagram (hence, the force decreases gradually):
1) Major part of energy is consumed
for deformation with fmax. In extreme
case, the repulsive movement will be
very small.
2) Completing moderate deformation
with f’max , then, DE leaves.
3) After the force reaches f’’max with
small deformation, DE leaves.
Apparently, f
''
'

f
max, (i )
max,
(i )
 fmax, (i ) ,
where i : number of disruption.
These (required) forces are thought to be larger as i is increasing because of
the increasing stiffness.
Then we formulate the energy balance for deformation and repulsion of DE.
ETotal  (i )  
( i )
0
l( i )
Fdisrupt (i )  dr   Fmovement (i )  dr
0
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

( i )
0
k(i )r  dr 
1
l( i )
l( i )
1
0
2


M  u r 
2
dr 
k( i )
2
 (i ) 2 
1
2
Mu
2
l  ,
(i )
 ( i ) : deformation along r , k( i ) : spring constant  stiffness of DE,
M : DE mass > 0, u  l  : leaving velocity along r at l( i ) .
(i)
Let K ( i ) j  cos   i  j , 0    i  j 

2
; then K ( i ) j will express the effectiveness of
jth collision in ith disruption. Where    j : colliding angle.
i
Moreover k(i) should be replaced by k(i+1) after ith disruption has occurred.
From Zoelly’s formula also,
2
2
k( i )  1 R( i )  c( i ) R( i ) .


And referring the property of jth colliding angle K(i)j above,
2



k


c( i ) 2


(i ) j
(i ) j
elastic
  K(i ) j  2   Enet  (i ) = 2  , where  (i )   (i ) R( i ) .
j 1 




N( i )
(i)
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Since c(i)j=c(i), then we have a formula for the saturated energy under local micro
deformation:
 (i ) 
NC ( i )

K ( i ) j  ( i ) j , or,  ( i ) 
2
j 1
NC ( i )



K ( i ) j k( i ) j k( i )  ( i ) j , where  ( i ) j   ( i ) j R( i ) j .
2
j 1
Regarding each critical collision, NC(i) is determined to satisfy this equation.
- A formula for the disruption interval, the disruption timing, etc.
Assuming that the mean free path (λ) is constant between ith and (i+1)th
disruption with random velocity in direction of DE with Maxwell distribution,
Interval (i i+1)
 DE ( i ) 

   allowable collision number : N C (i ) 
 u (li ) 
= NC (i )
=
M
2

2  DE (i ) DE (i ) u(li )

NC (i )
 DE ( i ) DE ( i ) ETotal ( i )   k( i ) 2   ( i )
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2
.
15
g

M  lim  M   2
 N 0
m

2
Where,

 3 kF 
N 1 
 0;
2 
10
M


2
(H. Kurasawa: Buturi,49(1994)628)
2
 


2
2
2
 3 
g
3 2 g
3 2  3 2 g


M  min  = lim 3 k F
 N  2  N 1  k F  k F
 N    0,
2
2
 N 0  5
m
m
10
m
5
 




k F : Fermi wave number,  N : density of nucleon.
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Behavior of Expansion of the Universe – Number of DE
n  t   n0 exp 1 a0 2  1 a 2  t  2 ,
where
n0 , a0 : at equilibrium of t  t0  1 .
Here we put a0=1.
(K.K: DPF2013)
The inflection point of a(t) is at n(t)/n0 = 1/e.
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Preliminary: Interval(i)→(i+1) – i th disruption
Where NC(i) ∝ exp(i-1).
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Mean mass of proton
and neutron is chosen
for the value of M.
Putting ΔM(min)→0 as an effective mass of DE at a limit, we obtain M=254.6
MeV/c2 at ρN=0.1331[1/fm3]. It is noteworthy that this critical nucleon mass
appears in reasonable agreement with ‘(GB mass)/2+ (total bare mass of
quarks which configure one proton and one neutron)/6’.
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Next we describe the momentum (p) of a DE at Nth and at (N+1)th collision
in which a disruption occurs. For the limit of ΔM→0 with wave number (k),
pN 1  pN   k( N 1)1  k( N 1)2   kN  Nk 0 .


Here we assumed that the given energy or momentum is equal in each
collision before the disruption. That is,
k 0  k1  k2   k N .  k( N 1)1  k( N 1) 2  ( N  1) k 0 .
Moreover if we put k( N 1)1  n1k and k( N 1) 2  n2 k ,
0
0
n1  n2  N  1. And we assume n1  n2  1.
Then if n1  n2 , always N is even; and if n1  n2 , always N is odd .
It will be usual that the disrupted two DEs have an equal momentum. So,
n1  n2 .
Thus in this case the disruption will occur after (odd)th collision.
In another case that disrupted two DEs have different (1:2) momentum,
the disruption will occur after (even)th collision.
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And, to think about the confluent behavior of original-DE and additional-DE (which
would be decayed from some DM in long time), we shall approximate the front
surface of DE as a three dimensional real spherical wave front (Ψ) in the universe:
  r, , , t  
A
r
cos  k  r  t    .
Then the front surface of original-DE from the center of universe is
A0
0 
r
cos  k 0  r  0 t  0  .
Also that of additional-DE is
 (i ) 
A( i )
R( i )

cos k ( i )  R ( i )  ( i ) t( i )   ( i )
.
By superimposing all of them,
   0
   (i )
i

A0
r
cos  k 0  r  0 t   0  
 A( i )

cos
k

R


t


.


R
(i )
(i )
(i ) (i )
(i ) 
i  (i )

 A0
A( i )
2

  Probability of presence         
 r
R( i )
i

2

 .

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== A scenario of matter-antimatter (M-A) consumption by DM ==
Hereafter we will propose a scenario of the matter-antimatter consumption
by a candidate for dark matter (DM) of 120.611 GeV/c2-mass, considering a
situation of that the DM has firstly encountered with M-A in the early
universe. Recently we showed that, (K.K: DPF 2013(Santa Cruz)_75)
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 
m f0 (1370)
   
     
 GB  3

90

 2  70
 3 90
70
0 

3GBm
m f0 (1710) 
 GB  K 0 

0
90
K 
m f0 (1500) 
K 
75

 
90
40
90



0

 mi
,
(possibly
be paired)
,
i
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
1
3

K  4

 .
 m
i
22
Then we shall focus on some parts of DM: π±, K±, K0. Since respective
meson consists of light quark pair of different flavor ;




   ud , du , K   us, su ; K 0  d s, K 0  s d ,
it is expected that constituent quarks of the meson are apt to have reunion
with one from neighboring M-A, for each own flavor as in same momentum:
-Remind the Cornell potential V of
quarkonium,
e
V (q q)    kr.
r
Therefore we consider that, in such
meson, the bond of constituent
quarks of different flavor tends to be
weakened when q or q_bar (M-A) of
same flavor in QGP state approaches.
Here the process of π－ shall be the
conjugate one of π＋; and for K±, dquarks are replaced by s-quarks.
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Also for K0, the constituent quarks would combine to that (M-A) of originally
same flavor respectively as well as K±.
Next we study on the behavior of remainder parts of DM：η, π0. These mesons
might have been resonant by the neighbor parts, i.e., fore-mentioned π± and K
mesons of virtual state, which would forced an increase of (η, π0) masses. Then
 ( q :q )* ( gg GB )
Re combination
0 0 *

i i
(  0 0 ) 

(



)


2
(
p

e
)

 2H ,
resonant
inverse reaction
Atomization
in nucleous
by which DM could have washed out both itself and M-A to only matter. Where
the consumed mass of M-A would be equal to the mass of produced ‘matter’
from the washing-out, assuming a quasi-static process. In such a static condition,
mass rate of the DM undergoes reaction- in total DM mass rate (  now: 0.268)
 mass rate of (M -A) in the early universe
 (now: 0.049 as only Matter).
It is interesting that this observed value of Planck 2013 is very near the
mean mass content of (η, π0) in DM90*f0(1370) and DM70*f0(1710) : 0.051, whose
value should be necessarily constant as far as these DMs are surviving.
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== NOTES ==
As seen we have calculated the content of Atom as 0.04609 based upon an
inverse reaction of γ ｆ0 from ur-Higgs boson into J/Ψ, and that decay to ηC (1S)
from ηC (2S) is much suppressed in observation. (K.K: 2013 Bormio Conference_71)
Here if we admit such a rare decay, the content will be extended a little; so be
put 0.049 after Planck 2013. Then the content of DM is also modified as
 0.049  0.04609 2 3 

 0.030,

0.04609
13

R DM   R DM  '   0.2357   
 R DM  mod  R DM  '  R DM   0.266,
R DE  mod  1  R Atom  mod  R DM  mod  1  0.049  0.266  0.685.
Where the factors of 2/3 and 1/3 are splitting rates from ur-Higgs boson
(GBs 240σ-mesons) into DMs and resonant state: (tr-O)* of Higgs bosons
(truncated-Octahedron) via fragmentation, respectively.
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Overview of degenerate of ur-Higgs
(fragmentation)
(ur-Higgs)*
(tr-O)*
decay
(γγ; …)
(fold)
decay
excite
(develop)
GeV/c2
ur-Higgs
125.28
excite
tr-O
Dark Matter
Dark Energy
120.611
(ΔM >0)
Meson (mass) level
Glueball level
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