On a Singular Solution in Higgs Field () -A Representation of Certain f Mesons Masses. Kazuyoshi KITAZAWA Mitsui Chemicals (DPF Meeting 11, August 9-13, Brown Univ.)
Contents We have recently discussed the mass and the basic structure of SM Higgs boson (H ) by obtaining asymptotic solution for their equation of motion of nonlinear Klein-Gordon type PDE. In this paper, we ll treat with above in mind; Masses of glueball (GB) of ground state and of certain f mesons, Ur-SM Higgs boson (ur-h ) which will consist of a number of GBs and/or f above for respective fullerene structure, A representation of these f mesons masses by masses ofπoctet and GB, And transformation of ur-h into H. DPF Meeting 11: Aug. 9-13
== Introductory review -SM Higgs mass formula == EOM of Higgs field should have a solution at the point of vacuum expectation value v, or. When we choose an asymptotic form for () s near s as, ( s) avs 1 exp s s, i where s c t x x : relativistically invariant distance from origin i Then, asymptotically, And expanding near s ; 3 (), () 3 ( s ) avs 1 1 s s a vs s 1 s s v a s a as s 1 1 s, where,, < 1. DPF Meeting 11: Aug. 9-13 3
Higgs Mass Formula So, v, ( ) : Asymptotic singular solution Thus from EOM, Higgs field mass formula is ( ), m W W g (Z ) G, where, infinitesimal Grassmann number W c W W g m, W c (Z ) G m, where c, c :constant Z Z W Z m W W g (Z ) G mw mz 1 k mw mz, where cw cz cw cz k DPF Meeting 11: Aug. 9-13 4
H 1 M km W km Z (1) (1) H (1) (1) (1) (1) nwm nzm N M Generally, M ( M ) cos ( M )sin, W W Z Z nw where W, N Z n Z N n M ; n M W W Z Z H H H where W Z DPF Meeting 11: Aug. 9-13 5
1 w k, z k Thus, w z k 1 ( w z) 1 Thus, k 1 1 1 MW MZ 1 1 M ( ) H dm ( ) M H d M M M M W, then cos W 1.9934, MZ W W Z H 1cos W MW MZ DPF Meeting 11: Aug. 9-13 6
.3. M 1.611 GeV c H km ' km ' W 1 Z 1 tan tan cos W 41.4 deg DPF Meeting 11: Aug. 9-13 7
Extended EOM of Higgs scalar field ( ) from Euler-Lagrange equation 3 1 1 3v m g W W G Z t 4 1 gm WW W GM Z Z 1 1 v b i i i v m bb m c cc mt tt md dd mu uu ms ss i i i DPF Meeting 11: Aug. 9-13 8
Extended Higgs boson mass formula m W W g mq q q G bsd bsd F G mq q q G + (Z ) cu cu F m tt G i( cu), t i F i( bsd ) DPF Meeting 11: Aug. 9-13 9
Top quark mass formula We assume that the probability of m t -decay procs e s obeys a binominal distribution of beeing k-times in n-trials (-particles) with r as decay-mode parameter. C r (1 r ) m C r (1 r ) m C r (1 r ) m k nk k1 nk k nk1 n k t n k W ( bsd ) n k Z ( cu) C r (1 r ) (1 r ) m r m, k1 nk1 n k W ( bsd ) Z ( cu) m m / r m / (1 r ) t W ( bsd ) Z ( cu) r 1 DPF Meeting 11: Aug. 9-13 1
Stationary mass value of top quark r dm t dr, M M M M ( M M M M )( M M M ) W b s d W b s d Z c u M M M M M M M Z c u W b s d 1 1 4 4 M t M M M M M M M GeV/c, M 1 W b s d Z c u 171.6(6) with b 4.68 GeV/c 1S Mass. CDF / D s experimental result : 171.3 1.1 1. GeV/c C. Amsler et al., Physics LettersB667, 1 8 ; updated(1) 1 4 DPF Meeting 11: Aug. 9-13 11
H is expected to be a composite scalar meson M ( tt ) * M M 11.1(3) GeV/c H t M M M *.49() GeV/c ( tt ) H little smaller than masses of and is smaller than mass of (tt) * H, K mesons, meson. radiative decay of 1-photon emission DPF Meeting 11: Aug. 9-13 1
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Basic structure of SM Higgs boson mass (1) Constructed by heavy mesons masses of all spin, such as B B B B D D S S C C S S () Then expected that they will form a polyhedron composed of planes of hexagon, in space. And, effective number of planes of hexagon should be n eff M theory H. Int 4. M B B B B D D S S C C S S DPF Meeting 11: Aug. 9-13 14
Comparison of SM Higgs boson mass values M (3 ) 3 c, H c,...(exper. values) 1.61 GeV/c, M 1.611 where, GeV/c M i 1 4 H (theor. with M, M ) W Z B B B B D D S S C C S S DPF Meeting 11: Aug. 9-13 15
Outer shell from mass of SM Higgs boson D S B S B C B C D S B C B S D S B C B S D S B S B S B C B C D S D S B C B C D S D S B S B S B C B S D S B C B S D S B C D S D S B S B C DPF Meeting 11: Aug. 9-13 16
M Comparison of Central meson s mass values 3 c (theor.) M i M K K KK 3, 4 894. MeV/c, where, MeV/c 3 (exper.) c 3 98.31. 894.9 3.6 Inner shell from mass of 3η c K K K K K K K K K K DPF Meeting 11: Aug. 9-13 17
== Structure of ur-sm Higgs Boson == -A Representation of Certain f Mesons Masses. BETHE-SALPETER EQUATION WITH GOLDSTEIN APPROXIMATION The general form of B-S: K p, P I p, P, B Br B B Br where DPF Meeting 11: Aug. 9-13 Br Fa a KB Fa PB p Fb bpb p 4 IB d pi p, p; PB pp, : BS amplitude B B 1, : modified Feynman propagators Fb I p, p; P B : irreversible part of the process., 18
The B-S for fermion (t)-antifermion (t_bar) bound state with total four momentum 4 1 1 1 1 dq S q P x q, PS q P 4 K ; q, q ; P x, x qp, 4 where d x T x x P. 1 1 q P, 1 1 p q P, p q P where q : relative momentum, P : total momentum of bound state S : fermion propagator x :BS-amplitude of spinor T : operator of time ordered DPF Meeting 11: Aug. 9-13 19
The Goldstein equation for abelian vector gluon model*, with the ladder approximation after putting P μ = and x m q Fq q, F q 5 4 d q 4 i qq F i After the Wick rotation and then the Fourier transform, 1 r mr f K mr 1, where < 1. e 3 4 3, : gauge parameter, q. * K. Higashijima and A. Nishimura, Nucl. Phys., B113, 173 (1976) DPF Meeting 11: Aug. 9-13
The required form of (massive) gluon propagator: Dr () m 1 K mr ˆ GeV/c mr ˆ GeV/c r mˆ fm where m, fm, 1. Compton wavelength of glueball in ground state: c.393fm, provided mgb 5.55 m m c GB, 1 MeV/c. (Thus we may put also as r.) c DPF Meeting 11: Aug. 9-13 1
After all, virtual bound tt_bar decay is expected to have at last the glueball (1),(),,(N) producing process: DPF Meeting 11: Aug. 9-13
Color valence of glueball Color valence of glueball =4 DPF Meeting 11: Aug. 9-13 3
Clustering force between glueballs = Color valence In lattice QCD it is now believed that there might be several scalar mesons of f (137), f (15), f (171) all of which are supposed to have some contents of glueball (GB) of grand state. Then we can expect similar structure of the carbon fullerenes for these scalar mesons. DPF Meeting 11: Aug. 9-13 4
Glueball (GB) Cluster as ur-sm Higgs Boson By putting each fullerene number for these mesons as N 9, N 8, N 7 with M ur-h =1.611GeV/c, MGB 1 3 M 3 i1 f i ( ) ur-h o i GB i i M N M N 1, N. GB 3 i1, 5.55MeV c as an element of GB4-fullerene.9,.333,.375. 1 3, ; i The f (15) will have a glueball for each element of GB 8 since.333 3=1. DPF Meeting 11: Aug. 9-13 5
Comparison to Experimental Mass Values of Scalar Meson Our Calculation Experiment (PDG_1) f 6 GB 5.55 4 1 f 137 134.1 1 15 f 15 157.6 155 f 171 173. 17 6 6 6 6 DPF Meeting 11: Aug. 9-13 6
A Hexagon on ur-h (GB 4 ) DPF Meeting 11: Aug. 9-13 7
Fullerene structure of ur-sm Higgs Boson: GB 4 f(6) 4 I h symmetry 11-Hexagons 1-Pentagons by Euler s theorem for polyhedron DPF Meeting 11: Aug. 9-13 8
A representation of f masses with πoctet (light pseudoscalar meson) and GB mass m f (137) GB 3 7 K 9 9 7 75 4 K 3 9 9 9 m i, m f (15) 3GB, m i m f (171) 1 GB K K 3 4. m i DPF Meeting 11: Aug. 9-13 9
Transformation (decay) of ur-h into H Remind that ( tt )* H, where H condensates into GB fullerene (ur-h ), and M (3 H c c Mi 1) 3 c : 3 1S 3J 3 6f 6, B B B B D D S S C C S S ) 3, 1 4. 3 1S 18f 6 6f 15. ) 1 : c c 5 f 15 3f 6, 1 6f 6 f 15. DPF Meeting 11: Aug. 9-13 3
3) 4 B B B B D D S S C C S S : B B 6 B B 6 D D 6 f, 4f, S S C C S S 8f, B B B B D D 16f6 7 15 S S C C S S 4 f. After all, we have the transformation under mass invariance that H 4f 6 8f 15 ur-h. 4GB DPF Meeting 11: Aug. 9-13 31
Fullerenes of Icosahedral (I h ) Symmetry As far as carbon fullerenes, C, C 6, C 8, C 18, C 4 have a common point group: I h which is of the icosahedral symmetry. Thus we expect that GB 8 (f(15)) and GB 4 (ur-h ) also have I h. Therefore, inversely, we could expect that the f meson which has a fullerene structure of I h symmetry, it may consist of pure GB. DPF Meeting 11: Aug. 9-13 3