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Higgs-Like Boson and Bound State of Gauge Bosons *W ^{+}W^{-}* II ()

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^{}

*W*with a mass of

^{+}W^{-}*E*≈ 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.

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*W*II.

^{+}W^{-}*Journal of Modern Physics*,

**7**, 1304-1307. doi: 10.4236/jmp.2016.711115.

Received 30 May 2016; accepted 8 July 2016; published 11 July 2016

1. Introduction

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),

(1)

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.

2. SU(2) vs SU(3)

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 C_{2} which is 2 for SU(2) and 3 for SU(3) ( [4] , 18.106). The renormalized coupling constant ( [1] , 20) now reads

(2)

which corresponds to ( [4] , 18.133) for C_{2} = 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.

3. Detailed Computation

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 [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_{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 ( [1] , 17) is exactly satisfied. This requires that D_{max} = 0. Among the three parameters that fix ( [1] , 17), E_{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
_{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.

Table 1. This table is the same as Table 1 of [1] with ( [1] , 20) replaced by (2) here to reflect the change of the eigenvalue of the Casimir operator C_{2} from 2 to 3. Only the underlined two cases satisfy the extrapolated criterion D_{max} < D_{err} ≈ 0.75% below and can be solutions. k_{di} is the number of integration steps needed to reach r_{ci}, the printout r_{c} value nearest to the joint distance r_{i} in ( [1] , 17), for three different integration step lengths, 4d_{sm}, 2d_{sm} and d_{sm}. * denotes that k_{di} = 60 was excluded due the above k_{d} £ 2^{10} = 1024 limitation in the backward integration.

4. New Results

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}, b_{0} 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.

Table 2. Deviations D_{max} for different Casimir operator eigenvalues C_{2} for L_{f} = 0.35. The underlined values satisfy the above twice extrapolated criterion D_{max} < 0.375%. C_{2} = 3 for SU(3) is preferred by the bound state.

5. Variation of C_{2}

The calculations leading to Table 1 has been repeated for L_{f} = 0.35 GeV^{-}^{1} using different C_{2} 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 £ 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.

6. Consequences

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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Hoh, F.C. (2016) Journal of Modern Physics, 7, 36-42. http://dx.doi.org/10.4236/jmp.2016.71004 |

[2] |
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[3] | Hoh, F.C. (2011) Scalar Strong Interaction Hadron Theory. Nova Science Publishers, Hauppauge. |

[4] | Lee, T.D. (1981) Particle Physics and an Introduction to Field Theory. Harwood Academic Publisher, New York. |

[5] |
Wilczek, F. (2005) Nature, 433, 239-247. http://dx.doi.org/10.1038/nature03281 |

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