On a Singular Solution in Higgs Field (III) - The Phase Transition and Crystallization of SM Higgs Boson

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On a Singular Solution in Higgs Field (Ⅲ) - The Phase Transition and Crystallization of SM Higgs Boson.

Kazuyoshi KITAZAWA, Mitsui Chemicals, Inc.

FAX: 03-6253-4244, E-mail: [email protected]

The phase transition and crystallization of SM Higgs boson are discussed by considering a naïve relativistic energy equation with Cornell potential and then Bethe-Salpeter equation (BS) with Goldstein approximation for tightly bound fermion (t) -antifermion (t_bar) coupling which exchanges with vector particle. We consider that the well known continuous spectrum solution of BS should not be excluded but be interpreted as it closely relates to propagator of the exchanged gluon. Then SM Higgs boson will be constructed from a number of mesons each of which is composed of two gluons (glueball), after emitting a virtual photon. A glueball mass value calculated of grand state is around at 502.55 MeV/c² which is expected as f₀(600) (σ meson) mass. Moreover the mass values of its resonant states are consistent with the known result of lattice simulation and within upper-f₀ mesons’ masses from experiment. And we predict that they finally crystallize into certain fullerene structures, as SM Higgs boson, through their color ‘valence’ of components. So the phase transition of SM Higgs boson is completed to transit to a state of truncated-Octahedron (tr-O) structure by receiving ‘melting heat' from another SM Higgs boson before crystallization near itself.

１．Introduction

In preceding paper¹⁾the author discussed the mass and the basic structure of SM Higgs boson with reference to bound top quark-pair. Where we saw an intimate relation between them, and we also derived a smaller mass of SM Higgs boson than the predicted one by the dynamical strong coupling theory of top quark condensation. We consider that the mass deference comes from different state of each Higgs boson. Therefore we shall hereafter treat the phase transitions of SM Higgs boson to start with studying a relativistic energy equation, and then applying Bethe-Salpeter equation ²⁾ for the states of the system.

２．Formulation and the Result

2.1 A Naïve Relativistic Energy Equation for the System Before we later apply Bethe-Salpeter equation for a tightly bound top quark-pair(t t)^*, we at first make a naïve relativistic energy equation for(t t)^*with considering Cornell potential

( )

V r of quarkonium in lattice QCD of Wilson loop³⁾:

( ) / .

V r  e rr (1)

r q○---●q

Fig. 1 Quenched Wilson action SU(3) potential ³⁾, normalised to V (r0) = 0.

Hence we may write for a tightly bound energy as

 

0

2 2

. H 2 ,

t bound t photon

V M c  M c  (2)

because we will treat the situation in next subsection that total mass of the system is zero, we here set its tightly bound (total) energy to equalize to the one with SM Higgs mass¹⁾ which will be produced at second stage from massless vector particle. Then by substituting eq.(2) into eq.(1), we have a relativistic energy equation for the system as



^M^t ²



^c²^^^photon ^^^e^Total^/^r^^^r^. ⁽³⁾

So if we taker 0.4 fm, then e_Total 48.0 GeV fmwhere we adopt  1.5 GeV/fm³⁾. Here we assumed that bound energy between two gluons in a glueball is small. And, since we consider that there would be some ‘latent heat’ between the molecular-like state and solidification state of SM Higgs boson (namely, we think, the former would not be in tightly bound state), we will choose for a tightly bound energy with left-hand side of eq.(3). Then we could describe the diagram of phase transition into SM Higgs boson as shown in Fig.2. It is very interesting that the rate of outgoing energy from the system to the space (= +Q) by deficit of mass is fairly large at this point of time, as eq.(4). We will later return to the diagram of Fig.2.



^{2 1} ²



² ² ² ^0.646.

Q m ct m ct

_    (4)

Fig. 2 Diagram of Phase Transition into SM Higgs boson

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2 2.2 Bethe-Salpeter Equationwith Goldstein Approximation Tightly bound fermion-antifermion coupling which exchanges with vector particle by Bethe-Salpeter equation²⁾ has long been investigated.⁴⁾ Firstly, Salpeter and Bethe constructed the relativistic equation for two interacted nucleons. Goldstein studied its solution by ladder approximation and discovered the continuous spectral solution with relevant discrete ones.⁴⁾ There Goldstein argued the lack of physical interpretation for the continuous solution of highly singular behavior at the origin of coordinate space. Later, Kummer; Higashijima and Nishimura; Fukui and Seto; and others discussed the continuous spectral solution in the fermion- antifermion or in the spinor-spinor interactions.⁵⁾ They excluded it from the reason of each difficulty of interpreting physical meaning, except that Higashijima and Nishimura considered it as a renormalized vertex function of the solution for the homogeneous BS. Thus we shall hereafter apply BS for tightly bound fermion (top quark)-antifermion (anti-top quark) coupling which exchanges with vector particle, and reconsider the physical meaning of continuous solution. The general form of BS is⁶⁾

 ^, _B  ^, _B,

B Br B Br

K  p P I  p P^ (5)

   

 ¹

where K_B  _Fa^ _aP_Bp ^_Fb _bP_Bp ^ , (6)

 

I_B d p I p p P4 ^ , ^; _B . ⁽⁷⁾ _Brp P, _B: BS amplitude

_Fa,_Fb: modified Feynman propagators I p p P , ; _B: irreversible part of the process

Then BS for fermion-antifermion bound statewith total four momentum P_is given explicitly⁵⁾in the Bjorken-Drell metric,

 

     

1 1

4

4 ; ,

1 1

,

2 2

, ; 2

,

q P

q P q P q P

q q P

S S

d q K

x

x 

  





 

    

 



    

   

    

 

 

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 

4 1 1

,

2 2

where x q P d x 0T x xP .⁽⁹⁾ Hence we will have the Goldstein equationfor abelian vector gluon model^{4), 5)} with the ladder approximation after putting

0

P_  and x_^q^{, 0} ₅ _ ^F ^q ,

 ² ² ^{ } 2 _ _2 ^{ }

4

. 4

d q

q q

i q q

m q

i

F ^ F

 

 

  

 



 ⁽¹⁰⁾

After the Wick rotation and then the Fourier transform regarding eq.(10), we will see that it has the continuous spectrum solution for 0, puttingK_as modified Bessel function of-th order

   ¹  .

f r  mr ^ K_ mr (11)

On the other hand, recently Iritani et al calculated gluon propagator’s functional forms⁷⁾ in the Landau gauge in SU(3), which fit the result of lattice QCD, one of whose candidates has similar form of eq.(11). However they abandoned this form because of the deviation from the data of lattice QCD near r 0. We consider in this case, it should rather be adopted that

where 1 , 0< 1,

    (12)

according to the continuous solution of Goldstein for BS. We prefer from the result of lattice QCD that  0, 1.

2.3 Crystallization as SM Higgs Boson

In lattice QCD it is now believed that there might be several scalar mesons of f0(1370), f0(1500), f0(1710) all of which are supposed to have some contents of glueball (GB) of grand state.

So we shall write down them, relating to the glueball mass and SM Higgs boson mass as

H0

1 2 3

1 ^G 2 ^G 3 ^G ,

N M N M N M

      M

   

 

   

      ⁽¹³⁾

1 2 3 1.

     (14) As the number of the kind for colored gluon is 8, the glueball’s color valence should be 4 (cf. Fig.3), which is same as carbon.

After setting N1, N2 N3 as the fullerene number of f0(1370), f₀(1500), f₀(1710), under the consideration of similar structure to the carbon fullerenes of C90, C80 and C70 respectively, we put

1 90, 2 80, 3 70.

N  N  N  (15)

From eqs.(13), (14), (15) with M_H⁰=120.611GeV/c², we have

2

1 2 3

502.55 MeV c as an element of 240-fullerene 0.292, 0.333, 0.375.

, ;

MG GB

  



   ⁽¹⁶⁾

Fig. 3 Color valence of glueball

References

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