This note is a further development of the recent paper [1] , in which the Higgs-like boson H(125) discovered in 2012 with mass 125.09 GeV [2] was tentatively assigned to a bound state of two charged gauge bosons W⁺W^- with a mass of E_B ≈ 110 GeV. The starting point is the action for gauge bosons ( [3] , 7.1.2),

where denotes the charged gauge bosons and the superscript 0 its time component. In ( [1] , 1), however, the flavour index l was limited to run from 1 to 3 to reflect the assumption that these gauge bosons W⁺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 ( [3] , §7.2.3).

Further development is the same as that in [1] with the change of SU(2) to SU(3). This change only affects the running coupling constant ( [1] , 19) where the coefficient of the logarithmic term is proportional to the eigenvalue of the quadratic Casimir operator C₂ which is 2 for SU(2) and 3 for SU(3) ( [4] , 18.106). The renormalized coupling constant ( [1] , 20) now reads

which corresponds to ( [4] , 18.133) for C₂ = 3 and is used here. The same computations that led to Table 1 in [1] will be repeated here using (2) and with much greater accuracy.

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_dd_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 £ 2¹⁰ = 1024. Since the backward integrations has to start from some large r value, taken to be ≈0.5 GeV^-1 in [1] , d_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_did_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 ( [1] , 17) is exactly satisfied. This requires that D_max = 0. Among the three parameters that fix ( [1] , 17), E_b and b₀ 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 [1] , only d_s = 2d_sm ≈ 0.001 GeV^-1 was used. As was mentioned near ( [1] , 18), a criterion for the existence of a bound state solution has been taken to be D_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 Table 1 of [1] and Table 1 below, only r_i ≈ 0.032 - 0.033 are of interest. In [1] , D_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 ( [1] , 18) with D_max = 2.69% and the 2 underlined entries in Table 1 of [1] with D_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.

The computations in [1] are repeated using the more accurate specifications above. The changed results are D_max = 2.69% → 2.67% in ( [1] , 18), and 2.71% → 2.52% and 2.23% → 1.44% in ( [1] , Table 1). But now the criterion becomes D_max < 1.5% and is not satisfied by the solution ( [1] , 18) with bare g and M_W values and such a bound state no longer exists. Using SU(2) representation, the L_f = 0.30 GeV^-1 case in ( [1] , Table 1) with D_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, b₀ and r_ci values mentioned above, the results are given in Table 1.

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 Table 1. A further reduction leads to D_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 Table 1 and also in Table 2.

The L_f = 0.35 case with D_max = 0.35% in Table 1 using SU(3) representation is far more close to 0 than does the 2.23% from Table 1 of [1] . It may be regarded as a solution to the bound state and is tentatively assigned to H(125) instead. The calculated mass E_b = 116.845 GeV is much closer to 125.09 GeV [2] than does 110.02 GeV in [1] . The wave functions for these cases are close to that given by the dotted curve in Figure 1 of [1] .

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 Table 1 appears to be compatible to some of the experi- mental conditions determining M_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.

The calculations leading to Table 1 has been repeated for L_f = 0.35 GeV^-1 using different C₂ in (2). The results are shown in Table 2.

This table shows that solutions according to the twice extrapolated criterion D_max < 0.375% exist only for 2.94 £ C₂ £ 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 [3] is far more successful in this region. Mass generation of W^± 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 ( [5] , p. 247). Such a condensate originated in the hypothesis (Nambu…) that vacuum contains a spin 0 background field that is spontaneously symmetry broken. It is reminiscent of the aether of the late 19^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 ( [3] , Appendix G, Sec. 6). They do not work.

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