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 2 glueball mass value calculated of grand state is around at 502.55 MeV/c 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 22 V  M0 c M2 c   , (2) In preceding paper 1) the author discussed the mass and the t. boundH  t  photon basic structure of SM Higgs boson with reference to bound top because we will treat the situation in next subsection that total quark-pair. Where we saw an intimate relation between them, mass of the system is zero, we here set its tightly bound (total) and we also derived a smaller mass of SM Higgs boson than the energy to equalize to the one with SM Higgs mass1) which will predicted one by the dynamical strong coupling theory of top be produced at second stage from massless vector particle. Then quark condensation. We consider that the mass deference comes by substituting eq.(2) into eq.(1), we have a relativistic energy from different state of each Higgs boson. Therefore we shall equation for the system as hereafter treat the phase transitions of SM Higgs boson to start 2 M2. c e / r  r (3) with studying a relativistic energy equation, and then applying  t  photon Total Bethe-Salpeter equation 2) for the states of the system. So if we take r  0.4 fm , then e  48.0 GeV fm where Total 3) ２．Formulation and the Result we adopt  1.5 GeV/fm . Here we assumed that bound 2.1 A Naïve Relativistic Energy Equation for the System energy between two gluons in a glueball is small. And, since we Before we later apply Bethe-Salpeter equation for a tightly consider that there would be some ‘latent heat’ between the * molecular-like state and solidification state of SM Higgs boson bound top quark-pair ()tt , we at first make a naïve relativistic (namely, we think, the former would not be in tightly bound * energy equation for ()tt with considering Cornell potential state), we will choose for a tightly bound energy with left-hand side of eq.(3). Then we could describe the diagram of phase Vr() of quarkonium in lattice QCD of Wilson loop 3): transition into SM Higgs boson as shown in Fig.2. It is very V( r )  e / r  r . (1) 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. 22 (4) Q 2 1 2  mtt c2 m c   0.646.

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 Fig. 2 Diagram of Phase Transition into SM Higgs boson

1

2.2 Bethe-Salpeter Equation with Goldstein Approximation 2.3 Crystallization as SM Higgs Boson Tightly bound fermion-antifermion coupling which exchanges In lattice QCD it is now believed that there might be several 2) with vector particle by Bethe-Salpeter equation has long been scalar mesons of f0(1370), f0(1500), f0(1710) all of which are investigated.4) Firstly, Salpeter and Bethe constructed the relativ- supposed to have some contents of glueball (GB) of grand state. istic equation for two interacted nucleons. Goldstein studied its So we shall write down them, relating to the glueball mass and solution by ladder approximation and discovered the continuous SM Higgs boson mass as spectral solution with relevant discrete ones.4) There Goldstein MMM     argued the lack of physical interpretation for the continuous GGG NNN   M 0 , (13) solution of highly singular behavior at the origin of coordinate 1  2   3  H 1    2    3 space. Later, Kummer; Higashijima and Nishimura; Fukui and

Seto; and others discussed the continuous spectral solution in the 1  2   3  1. (14) 5) fermion- antifermion or in the spinor-spinor interactions. They As the number of the kind for colored gluon is 8, the glueball’s excluded it from the reason of each difficulty of interpreting color valence should be 4 (cf. Fig.3), which is same as carbon. physical meaning, except that Higashijima and Nishimura After setting N1, N2 N3 as the fullerene number of f0(1370), considered it as a renormalized vertex function of the solution f0(1500), f0(1710), under the consideration of similar structure to for the homogeneous BS. Thus we shall hereafter apply BS for the carbon fullerenes of C90, C80 and C70 respectively, we put tightly bound fermion (top quark)-antifermion (anti-top quark) NNN90,  80,  70. (15) coupling which exchanges with vector particle, and reconsider 1 2 3 0 2 the physical meaning of continuous solution. The general form From eqs.(13), (14), (15) with MH =120.611GeV/c , we have 6) of BS is 2 MG  502.55 MeV c, as an element of GB 240-fullerene; K p , P p , P (5) (16) B Br BB  I B Br  , 0.292,   0.333,   0.375. 1 1 2 3 where K  P  p  P  p , (6) B Fa a B Fb b B  4 I  d p I p,;. p P (7) B   B  pP, : BS amplitude Br  B     Fa ,  Fb : modified Feynman propagators  I p,; p PB  : irreversible part of the process Then BS for fermion-antifermion bound state with total four 5) momentum P is given explicitly in the Bjorken-Drell metric,

1111    Fig. 3 Color valence of glueball SSq P x q, P  q P  22     (8)

4 References dq 1) K.-Y. Kitazawa: On a Singular Solution in Higgs Field,    4 K;   q,; q P xqP, ,  2  Theoretical and Applied Mechanics Japan, 57, pp.217-225, 2009; ditto (II), ibid, 58, pp.61-70, 2010. 4 11    (9) 2) E.E. Salpeter and H.A. Bethe: A Relativistic equation for where qP,   d x 0 T x x P . x        bound state problems, Phys. Rev., 84, pp.1232-1242, 1951. 22    3) E. Eichten et al.: The Spectrum of Charmonium, Phys. Rev. Hence we will have the Goldstein equation for abelian vector 4), 5) Lett., 34, pp.369-372, 1975; Erratum,ibid,36, p.1276,1976. gluon model with the ladder approximation after putting Gunnar S. Bali, QCD forces and heavy quark bound states, P  0 and  q,0   F q , Phys. Rep., 343, pp.1-136, 2001.  x  5    4) J.S. Goldstein: Properties of the Salpeter-Bethe Two Nucle- 4  22  dq on Equation, Phys. Rev., 91, pp.1516-1524, 1953. mq FFqq   . (10) 5) W. Kummer: Exact Solution of the Bethe-Salpeter Equation   2  2 4 i qq  i for Fermions, Nuovo Cimento, XXXI, pp.219-246,1964. After the Wick rotation and then the Fourier transform regarding K. Higashijima and A. Nishimura: A Solution to the eq.(10), we will see that it has the continuous spectrum solution Goldstein Problem and the Possibility of Dynamical Chiral for   0 , putting K as modified Bessel function of -th order Symmetry Breaking, Nucl. Phys., B113, pp.173-188, 1976.  A. Nishimura and K. Higashijima: Exact Solution of the 1 fr   mr K  mr . (11) Spinor-Spinor Bethe-Salpeter Equation and their Gauge  Dependence, Prog. Theor. Phys., 58, pp.908-918, 1976. On the other hand, recently Iritani et al calculated gluon propa- 7) I. Fukui and N. Seto: On the Structure of the Continuous gator’s functional forms in the Landau gauge in SU(3), which Spectra in the Spinor-Spinor Bethe-Salpeter Equation, Prog. fit the result of lattice QCD, one of whose candidates has similar Theor. Phys., 65, pp.1026-1040, 1981. form of eq.(11). However they abandoned this form because of 6) N. Nakanishi: General Survey of the Theory of the the deviation from the data of lattice QCD near r  0. We Bethe-Salpeter Equation, Suppl. Prog. Theor. Phys., 43, consider in this case, it should rather be adopted that pp.1-81, 1969. 7) T. Iritani et al: Gluon-propagator functional form in the   1, where 0< 1, (12) Landau gauge in SU(3) lattice QCD: Yukawa-type gluon according to the continuous solution of Goldstein for BS. We propagator and anomalous gluon spectral function, Phys. prefer from the result of lattice QCD that   0,  1. Rev., D80, 114505, 2009. 2