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On a Singular Solution in Higgs (III) -Condensates and Representation of Certain f 0 ’ Kazuyoshi Kitazawa

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Kazuyoshi Kitazawa. On a Singular Solution in Higgs Field (III) -Condensates and Representation of Certain f 0 Mesons’ Masses. Journal of Physical and Application, 2013, 3 (2), pp.114-121. ￿hal-01468931￿

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On a Singular Solution in Higgs Field (III) - Condensates * and Representation of Certain f0 Mesons’ Masses

Kazuyoshi Kitazawa Mitsui Chemicals, Minato-Ku, Tokyo, Japan 105-7117

Abstract: We have recently discussed the and the basic structure of SM Higgs (H0) by obtaining asymptotic solution for the equation of motion of nonlinear Klein-Gordon type partial differential equation. In this paper, we will treat with above in 0 mind, - masses of (GB) of and of certain f0 mesons, ur- (ur-H ) which will consist of a number of

GBs and/or f0 above for respective fullerene structure, a representation of these f0 mesons’ masses by masses of π octet and GB, and transformation of ur-H0 into H0.

Key words: Higgs boson mass, glueball, , f0 , Bethe-Salpeter equation

ν : Nomenclature α, ξ : gauge parameter 2 2 1/2 G: gauge constant defined by (g + g’ ) ελ: infinitesimal Grassmsnn number

GF: Fermi constant εphoton ; energy of emitted I (p, p’, PB) : irreversible part of the process M: rest mass σ: string tension P: total momentum of T: operator of time ordered 1. Introduction Wμ: gauge field of W 0 Zμ: gauge field of Z In preceding paper [1] the mass and the basic structure of H a: constant were discussed by obtaining asymptotic solution for the c : velocity of light Euler-Lagrange equation of nonlinear Klein-Gordon type, in Higgs field with newly developed mass triangle method, and also e: phenomenological parameter of strength the basic structure of H0 by referring to a tightly bound virtual top g, g’ : gauge of SU(2), U(1), respectively. -pair (tt)*. Though we saw an intimate relation between H0 m: relativistic mass and (tt)*, in our calculation H0 had a smaller mass than the q: relative momentum predicted one by the dynamical strong coupling theory of top s: relativistically invariant distance from origin quark condensation [2], and also seemed to have a truncated- t; time Octahedron (tr-O) mass structure composed of heavy and light xi: coordinates of pseudo-scalar mesons’ masses of all 0. There we considered that the mass deference would come from different phase state of Greek letters each H0. Therefore, to understand this in detail we shall hereafter start with investigating the of H0 via a relativistic φ: isospinor scalar Higgs field energy equation with describing a phase transition diagram of H0, φ : scalar Higgs field after review of former result. Then we at first see that the ground φ (p, P ): BS amplitude Br B state mass of GB is calculated at 502.55 MeV/c2 which is Fa ,Fb : modified Feynman propagators expected as f0(500) meson's mass. The GBs will attract mutually with neighbors among original their components of in different colors, so that they could gradually form cluster. And

we show that our computed masses of f0(1370), f0(1500) and *Corresponding author: Kazuyoshi Kitazawa, f0(1710) are within each f0 meson's mass from experiment while Director of Mitsui Chemicals. they will construct respective fullerene structure for ur-H0 as well E-mail: [email protected] 0 2 as f0(500), provided that the mass of ur-H (120.611 GeV/c ) will

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consist of a number of masses of GB or f0 in which all (pure) 2 2 2 + (Z )2 2G 2m q q 2 2 2 G GB-fullerene may have an icosahedral (Ih) rotational .   q  cu cu  F i() cu   Finally we propose a representation by which f0 mesons masses 2 2 above are reproduced respectively with masses of several light  2m tt 2 2 2G (6) 0 0 ± 0 ± t i     F  pseudo-scalar mesons such as η, K , K _bar, K , π , π and GB, under the consideration of those junction networks. Where the It will be understood that first and second terms in right-hand mass of f0(1500) is described only by the mass of GB. And also side of Eq.(6) are both related to decay of to weak ur-H0 will transform into H0 under mass invariance through, for boson and to other quark. And we assume that the probability of instance, γf0 reaction to ηc as its component via radiative decay of mt-decay process obeys binominal distribution of being k-times

J/Ψ. Along with these discussions, a massive gluon propagator in n-trials (-) with r0 (0< r0 <1) as decay-mode parameter, for virtual top quark-pair decay is calculated by Bethe-Salpeter k n k nC k r00(1 r ) m t equation. k11 n  k k n  k  nkC r0(1  r 0 ) m Wbsdnk ( )  C r 0 (1  r 0 ) m Zcu ( ) 2. Review - formula of SM Higgs boson mass k11 n  k  nC k r0(1  r 0 ) (1  r 0 ) m W ( bsd )  r 0 m Z ( cu ) , 2.1. SM Higgs boson mass formula  mt  m W( bsd )/ r 0  m Z ( cu ) / (1  r0 ) (7) EOM of Higgs field [1] should have a solution at the point of Thus the stationary mass value of top quark is

2 vacuum expectation value (φ= ν), or φ = 0. When we choose an 11 2 2 2 24 2 224 M t 12 MMMMMMMW b  s  d  Z  c  u  asymptotic form for s → 0 as,  3 2 2 (s ) avs 1 exp s s , (1) 171.26(6)  GeV/c , with M b  4.68 GeV/c 1S Mass . (8)  0  s0   which is consistent with CDF/D0’s experimental result [4]. where s c22 t xi . i Since H0 is expected to be a composite ,

Then asymptotically. φ(0) ~ 0, φ’(0) ~ 0. And expanding near s 2 MMM  2  121.10(3) GeV/c . * H 0 t  → 0, we can take an asymptotic form near singular solution ()tt (9) 2 2  MMM *  0  0.49(2) GeV/c . (φ=0) as,  v, (   0) (2) ()tt H  So we have a Higgs mass formula from EOM as which is little smaller than masses of K± ,0 mesons, and is smaller 0 2 than mass of η meson. Therefore it is expected that H is to be a 22 22 m  2W W 2  g (Z  ) 2  G (3) scalar meson after emitting one photon from the virtual top     quark pair: (tt)*, 0 Finally we get the value of rest mass for H *0 (tt) H . (10) 22MMMWWZ 0.023 2 M0    120.611  GeVc (4) H 2 2 2 0.022  1 cos WWZMM 2.3. Basic structure of SM Higgs boson mass Let us consider that H0 mass is constructed basically by heavy 2.2. Top quark mass formula mesons’ masses of all spin 0, such as BBBB00  DD   (11) Extended EOM of Higgs [1] from Euler-  SSCCSS   Lagrange equation is It is expected that they will form a polyhedron composed of 2 3 2 2 211 2 2 2 planes of hexagon in space. Because effective number of the 3vm     g W W  G Z  0 2     planes turns out to be 4 (four), basic structure of H mass is t 24 represented as  1 2 1 M W W GM Z bb gWZ    mb  mc cc  mt tt M 0 ()3c,...(exper. values) 2 v i i i H 1  3 , 10 4 BBB00B  DD   m dd  m uu  m ss  0 (5)  c  SSCCSS    di u i s i   v Mi Then extended Higgs boson mass formula is 120.612 GeV/c2  . (12) 2 m 

2 2 2  2 2W W 2 g m q q 2 2 2G 3. Structure of ur-Higgs boson    q  bsd bsd F   i() bsd  

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substituting Eq.(14) into Eq.(13), we have a naïve (approximate) -Condensates and a representation of certain relativistic energy equation for the system as 2 (15) f0 mesons’ masses Mt 2. cphoton  eTotal / r  r

3.1 A naïve relativistic energy equation for the system So if we take r= 0.4 fm, then e Total =－48.0 GeVfm where

Before we later apply Bethe-Salpeter equation on (tt)*, we shall we adopt σ = 1.5 GeV/ fm [5]. Here we consider that there at first make a naïve relativistic energy equation for (tt)* with would be the latent heat between the molecular-like state: M considering Cornell potential V of in lattice QCD of =2Mt of the dynamical strong coupling theory of top quark

Wilson loop) [5] in Figs. 1 and 2. condensation and pure liquid state: M =Mt / √2 of our V( r )  e / r  r , (13) calculation. It is expected that the former would not be in tightly bound state. Then we could describe the diagram of where r : distance from the centre of the string phase transition into H0 as shown in Fig.3. We will later return to this diagram. It is interesting that the rate of outgoing energy from the system to the space (= +Q) by deficit of mass is fairly large during the condensation: 22 (16) Q 2 1 2  Mtt c2 M c   0.646

Fig.1 String Tension σ

Fig.3 Diagram of Phase Transition into SM Higgs boson.

3.2 Bethe-Salpeter equation with Goldstein approxima- tion Tightly bound -antifermion coupling which ex- changes with vector by Bethe-Salpeter equation (BS) [6] has long been investigated.[7,8,9] Firstly, Salpeter and Bethe Fig. 2 Quenched Wilson action SU(3) potential [5], constructed the relativistic equation for two interacted .

normalized to V (r0) = 0. Goldstein studied its solution by ladder approximation and discovered the continuous spectral solution with relevant . discrete ones.[6] There Goldstein argued the lack of physical Hence we shall write for a tightly bound energy as [1] interpretation for the continuous solution of highly singular behavior at the origin of coordinate space. Later, Kummer; V M c22 M2 c  (14) t. boundH0  t  photon Higashijima and Nishimura; Fukui and Seto; and others because we will treat the situation in next subsection that total discussed the continuous spectral solution in the fermion- mass of the system is zero, we here set its tightly bound (total) antifermion or in the spinor-spinor interactions. They excluded it energy to equalize to the mass of H0 which will be produced at from the reason of each difficulty of interpreting physical second stage from massless vector particle (gluon). Then by meaning, except that Higashijima and Nishimura considered it

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as a renormalized vertex function of the solution for the On the other hand, recently Iritani et al calculated gluon homogeneous BS.[7] Thus we shall hereafter apply BS for propagator’s functional forms [10] in the Landau gauge in SU(3), tightly bound fermion (top quark)-antifermion (anti-top quark) which fit their result of lattice QCD (LQCD), one of whose coupling which exchanges with vector particle, and reconsider candidates has similar form of Eq.(23). However they the physical meaning of continuous solution. The general form abandoned this form because of the deviation from LQCD for of BS is [9] small r . We consider in this case, it should rather be adopted that 1   , where 0<   1, KB  Br p , PBB  I B Br  p , P , (17) (24) 22 1   3 g 4     3   , where K   P  p  P  p , (18) B Faa BFb b B  according to the continuous solution of Goldstein for BS and 4 (19) also the behaviors of modified Bessel function of Kν(x). Since IB  d p I p,;. p PB   the value of Kν(x) is more gradually decreasing along with Then BS for fermion-antifermion bound state with total four decreasing of ν, the relevant value of ν will be certainly momentum is given explicitly [6] in the Bjorken-Drell metric, obtained by comparing to LQCD. We prefer that ν = 0, λ= 1. with as 4×4 matrix BS-amplitude of spinor having two x So this is just the case that Higashijima and Nishimura have legs; interpreted it as a renormalized vertex function [7]. Then we expect that the glueball would be produced successively by two 1111    SSq P x q, P  q P  gluons each of which has a length of and is made by the gluon 22     (20) from the vertex respectively, as shown in Fig.4. dq4    4 K;   q,; q P xqP, ,  2  

where (21) 4 11    qP,  d x 0 T x x P . x          22   

Hence we will have the Goldstein equation for abelian vector gluon model with the ladder approximation, putting P  0  Fig.4 Glueball producing process and x q,0   F q , [7]   5    Therefore the final stage of massive gluon propagator into  dq4  glueball (  r of Fig. 4) in the Landau gauge in SU(3) which is in 22  (22) mq  FFqq  22 . accord with LQCD is determined from Eq.(23) as 4 i  qq i 2 m1 2 After the Wick rotation and then the Fourier transform regarding 2 Dr() 2 K0  mrˆ  GeV/c , (25) Eq.(22), we will see that it has the continuous spectrum solution 4 mrˆ  1 for λ > 0, putting K as modified Bessel function of ν -th order, where  m GeV/c2 , rm  fm, ˆ  1 fm . -1 f(r) = (mr) Kν (mr). (23) Here the Compton of glueball in ground state is

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f (1710) will have respective fullerene structure according to 2 (26) 0 c  0.393 fm , provided m  502.55MeV/c mc their Ni –number: 90, 80, 70. The calculated f0 mesons’ masses by Eq.(27) and Eq.(28) are well consistent with (or at least Thus r   , and the value of m will be actually given in next c within) experimental values as shown in Table 1. subsection.

3.4 Fullerene structure of ur-SM Higgs Boson

3.3 Clustering force between 0 The fullerene structure of GB240 (≡ ur-H ) of Ih symmetry [11] Since the number of the kind for colored gluon is 8 (eight), the which consists of 110-hexagons and 12-pentagons, obeying color valence of glueball should be 4 (four) according to Fig. 5, Euler’s theorem [12], as shown in Fig. 6 with whose each which is same as carbon and is expected to be self-assembling hexagon of GB from two gluons on it. force between GBs.

Fig.5 Color valence of glueball (= 4)

In LQCD it is now believed that there might be several scalar Fig.6 Fullerene structure of ur-Higgs Boson mesons of f0(1370), f0(1500), f0(1710) all of which are supposed to have some contents of glueball of ground state. Then we can expect similar structure of the carbon fullerenes for these scalar mesons. After setting N1, N2, N3 as the fullerene number whose fullerene consists of f0 mesons above respectively, under the As far as carbon fullerenes, C20, C60, C80, C180, C240 have a consideration of similar structure to the carbon fullerenes of C90, common point group: Ih which is of the icosahedral symmetry. C80 and C70, we put 0 Thus we expect that GB80 (f0(1500)) and GB240 (ur-H ) also have

M  MNM0   , Ih. Therefore, inversely, we could expect that the f0 meson which ()fio ur-H iiGB has a fullerene structure of Ih symmetry, it may consist of pure 3 3 (27)   1 , N N . GB.  i GB  i i1 i1 3.4 A representation of certain f0 masses From Eq. 27 with MM 120.611 GeV/c2 , [1] ur-HH00 Over three decades ago, Rossi and Veneziano, also Igarashi et 2 as an element of GB -fullerene al [13] have described gauge invariant junction type baryonium MG  502.55MeV c, 240 ;

1230.292,   0.333,   0.375. (28)

2 Table 1. Comparison of f0 mesons’ mass values MeV/c

f0 meson Our calculation Experiment [4]

f0(500) 502.55 400-1200

f0(1370) 1340.1 1200-1500

f0(1500) 1507.6 1505±6

f0(1710) 1723.0 1720±6

Fig.7 Junction type of baryonium It should be noted that f0(1500) may be also a glueball for each element of GB80 since 0.333×3 =1. And f0(1370), f0(1500) and

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ji2 jm2 of S and M , where j, i and m denote respective 0 2 m   10  6f00500  2f 1500 . number of junction, junction-pair (equivalent to GB) and quark- 3) 4 BBB00 B DD   : pair (meson), as shown in Fig.7. Recently, Csörgó et al [14]  SSCCSS   showed gluon junction networks of truncated-polyhedrons. 00  BBBB 22f500 , Therefore we shall apply this junction type to be able to describe  SSCC 0   0 0 H with f0 mesons of the fullerene structure. For H with f0(1500)  24f0 500 , of pure GB’s we may construct it as an aggregation of 12 units  DDSS 8f0 500 , 10 12 0   of S0 and 30 units of S0 . For H with f0(1370) or with f0  4 B 00BBB D D   216f 500  72f 1500 . (1710) we must construct it as mixed one from a GB and several  SSCCSS   00 certain light pseudo-scalar mesons, because f0(1370) and f0 After all, we have the transformation under mass invariance that (1710) have both been interpreted that they are not consisted 0 H 240f00500 80f 1500 (31) only of GB’s. We propose their mass structure formulae:

3 70 0 0 240GB ur-H . GB 0 K   90  90  m  , f0 (1370)  2 70 75 40 0   K  4. Concluding remarks  3 90  90  90  mi So far, we have shown the condensation (molecular self- assemblage) of H0 into certain fullerene structure which is to be m f (1500)  3GB , (29) 0 mi constructed from a number of glueballs (f0(500)) of ground state

0 1  as well as of heavier f0 mesons, with phase transition diagram m f (1710) GB KK   4 . 0  3  and then Bethe-Salpeter equation. And we proposed a  mi representation by which certain f0 mesons masses are reproduced respectively with masses of several light pseudo- which give mass values of 1340.1 and 1723.0 MeV/c2 scalar mesons. The relation between recent experiments of LHC respectively, reproducing the calculated values in Table 1 which and our result for H0 mass will be discussed in next paper. [15] have been obtained from Eq.(27) and Eq.(28), as already we have seen. Here we should remind that H0 would be constructed References by 70×f0(1710)- or 90× f0(1370)-fullerene. So the factor (70/90) [1] K. Kitazawa, On a Singular Solution in Higgs Field, Theoretical 0  for K or K in Eq. (29-1) is considered. While the factor (1/3) and Applied Mechanics Japan 57, 2009, pp. 217-225; ditto (II), in Eq.(29-3) is expected from that 3× [fullerene number of H0 ibid. 58, 2010, pp. 61-70. 0 with f0(1500) of pure GB] = 240. Because, H with f0(1500) [2] Y. Nambu, BCS Mechanism, Quasi- and might have resonant mass of 3×(mass of GB for ground state) at Fermion Mass, Proceedings of XI Warsaw Symposium on each 80 vertexes. Elementary , 1988, pp.1-10. V.A. Miransky, M. Tanabashi, and K. Yamawaki, Is the t Quark 3.5 Transformation (decay) of ur-H0 into H0 Responsible for the Mass of W and Z ?, Mod. Phys. Lett..A4, 1989, pp.1043-1053; Dynamical Electroweak Reminding Eq.(10) and Eq.(12): with Large Anomalous Dimension and t

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