_{1}

The Higgs-like boson discovered at CERN in 2012 is tentatively assigned to a newly found bound state of two charged gauge bosons
W^{+}W^{-} with a mass of
E_{B} ≈ 117 GeV, much closer to the measured 125 GeV than 110 GeV predicted in a paper with the same title earlier this year. The improvement is due to a shift from the earlier SU(2) representation assignment for the gauge bosons to the more realistic SU(3) one and that the computations are carried out with much greater accuracy.

This note is a further development of the recent paper [^{+}W^{-} with a mass of E_{B} ≈ 110 GeV. The starting point is the action for gauge bosons ( [

where ^{+}W^{-} belong to a SU(2) representation, the lowest ranked one of a SU group. In this representation, the other gauge bosons, the massive neutral Z and the massless A are absent.

Now, H(125) was generated in high energy proton-proton collisions in which all these 4 gauge bosons appear and a SU(3) representation is more appropriate. The 4 extra gauge bosons of the 8 gauge bosons in this representation degenerate to the 4 observed ones ( [

Further development is the same as that in [_{2} which is 2 for SU(2) and 3 for SU(3) ( [

which corresponds to ( [_{2} = 3 and is used here. The same computations that led to

In the Fortran 77 “dverk” integration subroutine, denote the integration step length by d_{s}. The inter gauge boson distance r that enter the computations is r_{c} = k_{d}d_{s}, where k_{d} is the number of steps needed to reach r_{c}. Only at these discrete r_{c} values can the solutions be printed out. This subroutine only allows k_{d} £ 2^{10} = 1024. Since the backward integrations has to start from some large r value, taken to be ≈0.5 GeV^{-}^{1} in [_{s} has a minimum d_{sm} = 0.5/1024 ≈ 0.000488 GeV^{-}^{1}. Let r_{ci }be the r_{c }closest to r_{i} and r_{ci} = k_{di}d_{s}. Three step lengths, d_{s} = d_{sm}, 2d_{sm} and 4d_{sm}, corresponding to k_{di}, k_{di}/2 and k_{di}/4, respectively, for a given r_{ci} will be used.

A bound state solution exists when the three conditions of ( [_{max} = 0. Among the three parameters that fix ( [_{b} and b_{0} are continuous and can be specified to any degree of accuracy. But the third parameter r_{i} is according to the last paragraph limited to the discrete r_{ci} which can differ from r_{i} by sm ¹ 0. Therefore, D _{max} ¹ 0 and minima of D _{max} are sought. For such minima encountered here, it is sufficient to specify D _{max} up to 0.01%. This error margin leads to that E _{B} needs be accurate up to 0.001 GeV and b _{0} up to 0.0001. _{}

In [_{s} = 2d_{sm} ≈ 0.001 GeV^{-}^{1} was used. As was mentioned near ( [_{max} < D_{err}, an error due to the finite integration step length; D_{err} = d_{s}/r_{ci} = 2d_{sm}/r_{ci}. As is seen in _{i} ≈ 0.032 - 0.033 are of interest. In [_{err} = 2d_{sm}/r_{i} ≈ 3% and the criterion D_{max} < 3% was used. This criterion is not absolute or derivable but is regarded as a plausible first approximation. It is satisfied by the solution ( [_{max} = 2.69% and the 2 underlined entries in _{max} = 2.71% and 2.23%.

Here, D_{err} = d_{sm}/r_{i} ≈ 1.5% and D_{err} = 4d_{sm}/r_{i} ≈ 6% are also considered. The corresponding criteria are D_{max} < D_{err} ≈ 1.5% and D_{max} < D_{err} ≈ 6%, respectively.

L_{f} GeV^{-}^{1} | 0.20 | 0.20 | 0.30 | 0.30 | 0.34 | 0.35 | 0.36 | 0.40 | 0.50 |
---|---|---|---|---|---|---|---|---|---|

k_{di} | 16 | 33, 66 | 17 | 33, 66 | 17, 34, 68 | 17, 34, 68 | 17, 34, 68 | 17, 34, 68 | 15, 30* |

D_{max} % | 33.57 | 30.18 | 6.20 | 1.71 | 0.50 | 0.35 | 1.04 | 2.42 | 32.73 |

b_{0} | 1.2753 | 1.3466 | 1.5277 | 1.4108 | 1.6792 | 1.7143 | 1.7486 | 1.8760 | 1.9161 |

E_{b} GeV | 104.635 | 105.337 | 112.708 | 113.179 | 115.951 | 116.845 | 117.773 | 121.825 | 131.227 |

The computations in [_{max} = 2.69% → 2.67% in ( [_{max} < 1.5% and is not satisfied by the solution ( [_{W} values and such a bound state no longer exists. Using SU(2) representation, the L_{f} = 0.30 GeV^{-}^{1} case in ( [_{max} = 2.71% → 2.52% is also no longer a solution. The L_{f} = 0.35 GeV^{-}^{1} case with D_{max} = 2.23% → 1.44% is barely < 1.5% but will not survive the extrapolated criterion D_{err} → 0.75% below.

Now, employ (2) with SU(3) and the more accurate E_{B}, b_{0} and r_{ci} values mentioned above, the results are given in

For L_{f} = 0.20 and 0.30, k_{di} = 33 and 66 refer to the same r_{ci}. Since 33/2 = 16.5 is not an integer, k_{di} = 17 and 16 refer to this r_{ci} + and - 2d_{sm} respectively. It is seen that D_{max} is lower for the smallest step length d_{sm} accompanying k_{di} = 66, as expected. The four cases with k_{di} = 17, 34 and 68 are accompanied by the step lengths d_{s} = 4d_{sm}, 2d_{sm} and d_{sm}, respectively, correspond to the same r_{ci} and yield the same integration results. This shows that the computer accuracy is independent of these step lengths; only printouts do. Extrapolating these cases by reducing d_{s} one more step down to d_{sm}/2, D_{err} → d_{sm}/2r_{i}. The so-extrapolated criterion becomes D_{max} < 0.75% which is satisfied only by the two underlined cases in _{max} < 0.375% which is only satisfied by the L_{f} = 0.35 case. If the step length is reduced by half once more, D_{max} < 0.1875% and there is no solution for any case in

The L_{f} = 0.35 case with D_{max} = 0.35% in _{b} = 116.845 GeV is much closer to 125.09 GeV [^{ }cases are close to that given by the dotted curve in

Equation (2) shows that L_{f} is an infrared cutoff and represents the size of the normalization box for a W^{±} boson. Its mass M_{W} = 80.385 GeV is interpreted to have been determined when this boson is separated from other interacting particles by >L_{f}. L_{f} = 0.35 GeV^{-}^{1} in _{W}. Also, E_{b} gets closer to the measured 125 GeV with increasing L_{f}. But why D_{max} is small enough for the present bound state to exist only when L_{f} ≈ 0.35 GeV^{-}^{1} is not understood.

C_{2} | E_{B} GeV | b_{0} | D_{max} %_{ } | k_{di} |
---|---|---|---|---|

2.0 | 108.101 | 1.4113 | 21.30 | 16 |

2.0 | 109.833 | 1.4119 | 1.44 | 33, 66 |

2.0 | 110.083 | 1.4348 | 22.35 | 17 |

2.4 | 112.351 | 1.5818 | 5.57 | 17, 34, 68 |

2.8 | 115.372 | 1.6645 | 1.83 | 17, 34, 68 |

2.9 | 116.098 | 1.6899 | 0.62 | 17, 34, 68 |

2.94 | 116.396 | 1.6996 | 0.23 | 17, 34, 68 |

2.97 | 116.594 | 1.7094 | 0.35 | 17, 34, 68 |

2.99 | 116.771 | 1.7119 | 0.26 | 17, 34, 68 |

3.0 | 116.845 | 1.7143 | 0.35 | 17, 34, 68 |

3.02 | 116.995 | 1.7193 | 0.54 | 17, 34, 68 |

3.1 | 117.579 | 1.7407 | 1.43 | 17, 34, 68 |

3.2 | 118.345 | 1.7651 | 2.18 | 17, 34, 68 |

The calculations leading to _{f} = 0.35 GeV^{-}^{1} using different C_{2} in (2). The results are shown in

This table shows that solutions according to the twice extrapolated criterion D_{max} < 0.375% exist only for 2.94 £ C_{2} £ 3.0. The bound state bosons W^{+}W^{-} prefer SU(3) and reject SU(2).

To obtain more precise prediction on the existence of bound state solutions, the Fortran 77’s “dverk” subroutine may be replaced by more modern routines including finite element method.

If the 2012 H(125) is indeed a bound state W^{+}W, it can no longer be the SM Higgs and at least the low energy end of SM is without foundation and has to be abandoned. No appreciable predictive power is lost; SM has not been able to account for basic hadron spectra and decays. SSI [^{±} comes from pseudoscalar mesons when the relative time between the both quarks in the meson is taken into account.

Further, the presence of a Higgs condensate, disregarding the requirement that its isospin must be >0, will lead to a cosmological impasse ( [^{th} century that permeates the vacuum as a medium carrying light waves. Such attempts to put physics into the vacuum are against the historical examples that new physics come from some basic principles and mathematics ( [

F. C. Hoh, (2016) Higgs-Like Boson and Bound State of Gauge Bosons W^{+}W^{-} II. Journal of Modern Physics,07,1304-1307. doi: 10.4236/jmp.2016.711115