Pseudoscalar Top-Bottom Quark-Antiquark Composite as the Resonance with 28 GeV at the LHC: Hadron Masses and Higgs Boson Masses Based on the Periodic Table of Elementary Particles ()

Ding-Yu Chung^{}

Utica, Michigan, USA.

**DOI: **10.4236/jmp.2018.914164
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Utica, Michigan, USA.

This paper posits that the observed resonance with 28 GeV at the LHC is the pseudoscalar top-bottom quark-antiquark composite which has the calculated mass of 27.9 GeV derived from the periodic table of elementary particles. The calculated mass is for the mass of . In the periodic table of elementary particles, t quark (13.2 GeV) in the pseudoscalar top-bottom quark-antiquark composite is only a part of full t quark (175.4 GeV), so pseudoscalar (26.4 GeV) cannot exist independently, and can exist only in the top-bottom quark-antiquark composite. As shown in the observation at the LHC, the resonance with 28 GeV weakens significantly at the higher energy collision (13 TeV), because at the higher collision energy, low-mass pseudoscalar in the composite likely becomes independent full high-mass vector moving out of the composite. The periodic table of elementary particles is based on the seven mass dimensional orbitals derived from the seven extra dimensions of 11 spacetime dimensional membrane. The calculated masses of hadrons are in excellent agreement with the observed masses of hadrons by using only five known constants. For examples, the calculated masses of proton, neutron, pion (π^{±}), and pion (±^{0}) are 938.261, 939.425, 139.540, and 134.982 MeV in excellent agreement with the observed 938.272, 939.565, 139.570, and 134.977MeV, respectively with 0.0006%, 0.01%, 0.02%, and 0.004%, respectively for the difference between the calculated and observed mass. The calculated masses of the Higgs bosons as the intermediate vector boson composites are in excellent agreements with the observed masses. In conclusion, the calculated masses of the top-bottom quark-antiquark composite (27.9 GeV), hadrons, and the Higgs bosons by the periodic table of elementary particles are in excellent agreement with the observed masses of resonance with 28 GeV at the LHC, hadrons, and the Higgs bosons, respectively.

Keywords

LHC, CMS, Resonance, b Quark Jet, Periodic Table of Elementary Particles, Top Quark, Bottom Quark, Hadron Masses, Mass Calculation, Higgs Boson

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Chung, D. (2018) Pseudoscalar Top-Bottom Quark-Antiquark Composite as the Resonance with 28 GeV at the LHC: Hadron Masses and Higgs Boson Masses Based on the Periodic Table of Elementary Particles. *Journal of Modern Physics*, **9**, 2638-2656. doi: 10.4236/jmp.2018.914164.

1. Introduction

In the search for resonances in the mass range 12 - 70 GeV produced in association with a b quark jet and a second jet, and decaying to a muon pair, the CMS Collaboration at the LHC recently reported an excess of events above the background near a dimuon mass of 28 GeV [1] . The search is carried out in two mutually exclusive event categories from proton-proton collisions at center-of-mass energies of 8 and 13 TeV. The first category involves a b quark jet in the central region and at least one jet in the forward region, while the second category involves two jets in the central region, at least one of which is identified as a b quark jet, no jets in the forward region. At the 8 TeV collision, the first category has 4.2 standard deviations, while the second category has 2.9 standard deviations. At the 13 TeV collision, the first category has 2.0 standard deviations, while the second category results in a 1.4 standard deviation deficit.

This potential new particle at 28 GeV does not match the properties of any of particles in the standard model. It is also puzzling that the resonance at 28 GeV weakens, disappears, or gets inverted at 13 TeV. This paper posits that the resonance with 28 GeV observed recently at the LHC is the pseudoscalar top-bottom quark-antiquark composite which has the calculated mass of 27.9 GeV derived from the periodic table of elementary particles in good agreement with the observed 28 GeV. The calculated mass is the mass of three pseudoscalar b quarks and one pseudoscalar t quark which represent the composite of b_{p}b̅_{p} + (b_{p}b̅_{p} + t_{p}t̅_{p})/2 where p = pseudoscalar. (The quark in pseudoscalar meson is denoted as “pseudoscalar quark”, while the quark in vector mesons is denoted as “vector quarks” which has higher mass than pseudoscalar quark.) As described in the periodic table of elementary particles, pseudoscalar t quark (13.2 GeV) is only a part of full t quark (175.4 GeV), so pseudoscalar t_{p}t̅_{p} (26.4 GeV) cannot exist independently, and can exist only in the top-bottom quark-antiquark composite. As shown in the observation at the LHC, the resonance with 28 GeV weakens significantly at the higher energy collision (13 TeV), because at the higher collision energy, low-mass pseudoscalar tt̅ in the composite likely becomes independent full high-mass vector tt̅ moving out of the composite. To account for the observed two jets, the composite has two jets consisting of a bb̅ jet and a b + t jet, where bb̅ jet for (b_{p}b̅_{p} + t_{p}t̅_{p})/2 is more stable than b + t jet which decays faster into the jet in the forward region to constitute the first category of the search by the CMS Collaboration at the LHC.

The periodic table of elementary particles is based on the seven mass dimensional orbitals derived from the seven extra dimensions of 11 spacetime dimensional membrane particles [2] [3] [4] . The seven mass dimensional orbitals include the seven principal mass dimensional orbitals for stable baryonic matter leptons (electron and neutrinos), gauge bosons, gravity, and dark matter and the seven auxiliary mass dimensional orbitals for unstable leptons (muon and tau) and quarks, and calculate accurately the masses of all elementary particles and the cosmic rays by using only five known constants [5] [6] . Hadron masses can be calculated in excellent agreement with the observed masses of hadrons. For examples, the calculated masses of proton, neutron, pion (π^{±}), and pion (π^{0}) are 938.261, 939.425, 139.540, and 134.982 MeV in excellent agreement with the observed 938.272, 939.565, 139.570, and 134.977 MeV, respectively with 0.0006%, 0.01%, 0.02%, and 0.004%, respectively for the difference between the calculated and observed mass.

Section 2 describes the periodic table of elementary particles and the mass formulas. Section 3 deals with quarks and hadrons. Section 4 explains the top-bottom quark-anti-quark composite as the resonance at 28 GeV. Section 5 describes the Higgs boson doublet as the intermediate vector boson composites.

2. The Periodic Table of Elementary Particles and the Mass Formulas

The periodic table of elementary particles [2] [3] [4] is based on the seven mass dimensional orbitals derived from the seven extra dimensions of 11 spacetime dimensional membrane. The seven mass dimensional orbitals include the seven principal mass dimensional orbitals for stable baryonic matter leptons (electron and neutrinos), gauge bosons (all forces), gravity, and dark matter (five sterile dark matter neutrinos) and the seven auxiliary mass dimensional orbitals for unstable leptons (muon and tau) and quarks (d, u, s, c, b, and t) as in Figure 1 and Table 1.

Table 1. The periodic table of elementary particles for baryonic matter and dark matter.

d = principal mass dimensional orbital number, a = auxiliary mass dimensional orbital number.

Figure 1. The seven principal mass dimensional orbitals (solid lines) denoted by the principal mass dimensional orbital number d and the seven auxiliary mass dimensional orbitals (dash-dotted lines) denoted by the auxiliary mass dimensional orbital number a.

The periodic table of elementary particles calculates accurately the particle masses of all leptons, quarks, gauge bosons, the Higgs boson, and the cosmic rays by using only five known constants: the number (seven) of the extra spatial dimensions in the observed four-dimensional spacetime from the eleven-dimensional membrane, the mass of electron, the masses of Z and W bosons, and the fine structure constant [5] [6] . The calculated masses are in excellent agreements with the observed masses. For examples, the calculated masses of muon, top quark, pion, neutron, and the standard model Higgs boson are 105.55 MeV, 175.4 GeV, 139.54 MeV, 939.42 MeV, and 126 GeV, respectively, in excellent agreements with the observed 105.65 MeV, 172.4 GeV, 139.57 MeV, 939.27 MeV, and 126 GeV, respectively.

The seven mass dimensional orbitals are arranged as F_{5} B_{5} F_{6} B_{6} F_{7} B_{7} F_{8} B_{8} F_{9} B_{9} F_{10} B_{10} F_{11} B_{11}, where F_{d} and B_{d} are mass dimensional fermion and mass dimensional boson, respectively. As described in the previous papers [2] [3] , the mass of mass dimensional fermion and the mass of mass dimensional boson are related to each other with three simple formulas as the follows.

${M}_{\text{d},B}={M}_{\text{d,}F}/{\alpha}_{\text{d}}$ (1)

${M}_{\text{d}+\text{1,F}}={M}_{\text{d,}B}/{\alpha}_{\text{d}+\text{1}}$ (2)

${M}_{\text{d}+\text{1,B}}={M}_{\text{d},B}/{\alpha}_{\text{d}+\text{1}}^{2}$ (3)

where d is the mass dimensional orbital number, F is fermion, and B is boson. Each dimension has its own α_{d}, and all α_{d}’s except α_{7} (α_{w}) of the seventh dimension (weak interaction) are equal to α, the fine structure constant of electromagnetism. The given observed masses are the mass of electron for F_{6} and the mass of Z boson for B_{7}. From Equations (1) and (3), α_{w} = α_{7} = α of week interaction = (M_{B}_{6}/M_{B}_{7})^{1/2} = (M_{F}_{6}/α/M_{B}_{7})^{1/2} = (M_{e}/α/M_{Z})^{1/2} = 0.02771. Therefore, the masses of gauge bosons are as in Table 2.

The lowest energy gauge boson (B_{5}) at d = 5 is the Coulomb field for electromagnetism. The second gauge lowest boson (B_{6}) at d = 6 is basic gluon (g* = 70 MeV ≈ one half of pion) is the strong force as the nuclear force in the pion theory [7] where pions mediate the strong interaction at long enough distances (longer than the nucleon radius) or low enough energies. At short enough distances (shorter than the nucleon radius) or high enough energies, gluons emerge

Table 2. The masses of the principal mass dimensional orbitals (gauge bosons).

to confine fractional charge quarks. B_{6} is denoted as basic gluon, g*. The third lowest boson (B_{7}) at d = 7 is Z^{0} for the weak interaction.

F_{11} (8.275 × 10^{16} GeV) relates to spin 3/2 gravitino, while B_{11} (1.134 × 10^{19} GeV) relates to spin 2 graviton. In supersymmetry, gravitino and graviton mediate the supersymmetry between fermion and boson in space dimension and gravitation. There are 10 space dimensions in the 11 spacetime dimensional membrane. As a result, the supersymmetry involves 10F_{11} + B_{11}, which is equal to 1.217 × 10^{19} GeV in excellent agreement with the Planck mass (1.221 × 10^{19} GeV) derived from observed gravity as (ћc/G)^{1/2} where c is the speed of light, G is the gravitational constant, and ħ is the reduced Planck constant.

The lepton mass formula and the quark mass formula are derived from the incorporation of basic gluon (g* = B_{6} = 70 MeV) to electron. The incorporation of basic gluon as flux quanta follows the the composite fermion theory for the FQHE (fractional quantum Hall effect) [8] [9] . In the composite fermion model for FQHE, the formation of composite fermion is through the attachment of an even number of magnetic flux quanta to electron, while the formation of composite boson is through the attachment of an odd number of magnetic flux quanta to electron. In the same way, the formation of composite fermion is through the attachment of an even number of basic gluons to electron, while the formation of composite boson is through the attachment of an odd number of basic gluons to electron. The formation of composite boson is equal to the formation of composite di-leptons, so the formation of composite lepton is through the attachment of one half of an odd number of basic gluons to electron. As a result, the muon (µ) mass formula is as follows.

$\begin{array}{c}{M}_{{\mu}_{7}}=Me+3{M}_{\text{g*}}/2\\ =Me+3{M}_{\text{e}}/2\alpha \\ =105.5488\text{\hspace{0.17em}}\text{MeV}\end{array}$ (4)

which is in excellent agreement with the observed 105.6584 MeV [10] for the mass of muon. The mass of t follows the Barut lepton mass formula [11] as follows.

${M}_{\text{lepton}}={M}_{e}+\frac{3{M}_{e}}{2\alpha}{\displaystyle \underset{a=0}{\overset{n}{\sum}}{a}^{4}}$ (5)

where a = 0, 1, and 2 are for e, µ_{7}, and τ_{7}, respectively. The calculated mass of τ_{7} is 1786.2 MeV in good agreement with the observed mass as 1776.82 MeV.

According to Barut, the second term, $\underset{a=0}{\overset{n}{\sum}}{a}^{4}$ of the mass formula is for the

Bohr-Sommerfeld quantization for a charge-dipole interaction in a circular orbit. The more precise calculated mass of tfor the tau lepton mass formula is as follows.

$\begin{array}{c}{M}_{\tau}=Me+\left(\frac{3{M}_{e}}{2\alpha}-{M}_{e}\right){\displaystyle \sum {2}^{4}}\\ =Me+\left(17\frac{3{M}_{e}}{2\alpha}-17{M}_{e}\right)\text{}\\ =1777.47\text{\hspace{0.17em}}\text{MeV}\end{array}$ (6)

which is in excellent agreement with observed 1776.82 MeV, and means that during this dipole-interaction in a circular orbit for τ, an electron with total mass of 17M_{e} is lost. 17M_{e} is shown as the observed 17 MeV for 34M_{e} in the light boson (17 ee̅) [12] [13] .

Quark has fractional charge (±1/3 or ±2/3), 3-color gluons (red, green, and blue) for 3g*, and both the principal mass dimensional orbitals and axillary mass dimensional orbitals, so similar to Equation (4), d and u in the principal mass dimensional orbital involves e/3 or 2e/3 and 3g* as follows.

$\begin{array}{l}\text{principal mass dimensional orbital at}\text{\hspace{0.17em}}\text{d}=6\\ {M}_{\text{principal}\text{\hspace{0.17em}}q}=\frac{1\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}2Me}{3}+\frac{3\left(3{M}_{\text{g*}}\right)}{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}2Me}{3}+\frac{3\left(3{M}_{\text{B6}}\right)}{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}2Me}{3}+\frac{9Me}{2\alpha}\end{array}$ (7)

For quarks in the auxiliary mass dimensional orbitals, 3-color basic gluons (3g*) become 3-color auxiliary basic gluons ( $3{\text{g}}_{a7}^{\ast}$ ) at d = 7. Based on Equation (2), auxiliary basic gluon is derived from muon as follows.

${M}_{3{\text{g}}_{a7}^{\ast}}={M}_{{\mu}_{7}}{\alpha}_{w}$ (8)

Similar to Equation (5), the masses of quarks in the auxiliary mass dimensional orbital are as follows.

$\begin{array}{l}\text{auxiliary mass dimensional orbitalat}\text{\hspace{0.17em}}\text{d}=7\\ {M}_{\text{auxiliary}\text{\hspace{0.17em}}q7}=\frac{3\left(3{M}_{{g}_{a7}^{\ast}}\right)}{2}{\displaystyle \underset{a=1}{\overset{n}{\sum}}{a}^{4}}=\frac{9{M}_{{\mu}_{7}}{\alpha}_{w}}{2}{\displaystyle \underset{a=1}{\overset{n}{\sum}}{a}^{4}}\end{array}$ (9)

The quark mass formula at d = 7 is the combination of Equations (7) and (9) as follows.

${M}_{q}{}_{7}=\frac{1\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}2Me}{3}+\frac{9Me}{2\alpha}+\frac{9{M}_{{\mu}_{7}}{\alpha}_{w}}{2}{\displaystyle \underset{a=1}{\overset{n}{\sum}}{a}^{4}}$ (10)

where a = 1, 2, 3, 4, and 5 for u_{7}/d_{7}, s_{7}, c_{7}, b_{7}, and t_{7}, respectively.

The quark mass at a = 5 for the auxiliary mass dimensional orbital at d = 7 is the maximum mass below the mass of B_{7}, so the next auxiliary mass dimensional orbital has to start from B_{7}.There are b and t at d = 8, so it is necessary to have µ_{8}for the masses of b and t. Like µ_{7} in Equation (4), the mass of µ_{8} is as follows.

$\begin{array}{c}{M}_{{\mu}_{8}^{0}}=2Me+3{M}_{{\text{g}}_{7}^{\ast}}/2\\ =2Me+3{M}_{{\text{B}}_{\text{7}}}/2\\ =2Me+3{M}_{{\text{Z}}^{\text{0}}}/2\\ =136.78\text{\hspace{0.17em}}\text{GeV}\end{array}$ (11)

Since at d = 7, there are 3-color basic gluons, at d = 8, 3-color basic gluons are not needed, and only one basic gluon ( ${\text{g}}_{7}^{\ast}$ ) at d = 7 is used. Similar to Equations (7) and (9). The quark mass formulas for the principal and auxiliary mass dimensional orbitals are as follows.

$\begin{array}{l}\text{principal mass dimensional orbital at}\text{\hspace{0.17em}}\text{d}=7\\ {M}_{\text{principal quark}}=3{M}_{{g}_{7}^{\ast}}/2=3{M}_{{B}_{7}}/2=3{M}_{Z}/2\end{array}$ (12)

$\begin{array}{l}\text{auxiliary mass dimensional orbital at}\text{\hspace{0.17em}}\text{d}=8\\ {M}_{\text{auxiliary quark}}=\frac{3\left({M}_{{g}_{a8}^{\ast}}\right)}{2}{\displaystyle \underset{{a}^{\prime}=1}{\overset{{n}^{\prime}}{\sum}}{{a}^{\prime}}^{4}}=\frac{3{\mu}_{8}^{0}\alpha}{2}{\displaystyle \underset{{a}^{\prime}=1}{\overset{{n}^{\prime}}{\sum}}{{a}^{\prime}}^{4}}\end{array}$ (13)

The quark mass formula at d = 8 is the combination of Equations (12) and (13) as follows.

${M}_{q8}=\frac{3{M}_{Z}}{2}+\frac{3{M}_{{\mu}_{8}^{0}}\alpha}{2}\text{}{\displaystyle \underset{{a}^{\prime}=1}{\overset{{n}^{\prime}}{\sum}}{{a}^{\prime}}^{4}}$ (14)

where a' = 1 and 2 for b_{8} and t_{8}, respectively.

Combining Equations (10) and (14), the quark mass formula is as follows.

${M}_{\text{quark}}=\frac{1\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}2Me}{3}+\frac{9Me}{2\alpha}+\frac{9{M}_{{\mu}_{7}}{\alpha}_{w}}{2}{\displaystyle \underset{a=1}{\overset{n}{\sum}}{a}^{4}}+\frac{3{M}_{Z}}{2}+\frac{3{M}_{{\mu}_{8}^{0}}\alpha}{2}{\displaystyle \underset{{a}^{\prime}=1}{\overset{{n}^{\prime}}{\sum}}{{a}^{\prime}}^{4}}$ (15)

where a =1, 2, 3, 4, and 5 for d/u. s, c, b, and t, respectively, and a' = 1 and 2 for b and t respectively. The calculated masses for d, u, s, c, b, and t are 328.4 MeV, 328.6 MeV, 539 MeV, 1605.3 MeV, 4974.6 MeV, and 175.4 GeV, respectively. In the standard model, there are three generations of leptons. Extra-muon µ_{8} is outside of the three generations of leptons in the standard model, so µ_{8} is absent as shown in Table 2. As shown in Table 2, to be symmetrical to the absent µ_{8}, b_{8} quark is also absent. The calculated mass of top quark is 175.4 GeV in good agreement with the observed 172.4 GeV. The calculated masses are comparable to the quark masses proposed by De Rujula, Georgi, and Glashow [14] , Griffiths [15] , and El Naschie [16] .

3. Quarks and Hadrons

For baryons, the quarks in the periodic table of elementary particles are baryonic quarks. Mesons have vector mesons with parallel spins and pseudoscalar mesons with antiparallel spins. Since parallel spins have higher energy than antiparallel spins, vector mesons have higher masses than pseudoscalar mesons. For higher-mass vector mesons (parallel spins), vector t, b, c, and s are baryonic t, b, c, s, and vector d and u are baryonic d and u plus basic gluon (g* = 70 MeV). For lower-mass pseudoscalar mesons (antiparallel spins), pseudoscalar t as t_{7} is a part of vector t (= t_{7} + t_{8}), pseudoscalar b and c are vector quarks minus g*, and pseudoscalar d, u, and s are derived from g* as shown in Table 3.

The calculated masses and the observed masses [10] of baryons are listed in Table 4. The binding energy for each d or u quark involves the auxiliary mass dimensional orbital at d = 7 from Equation (9). The primary binding energy E_{Q}_{1} for d or u quark from Equation (9) is as follows.

${E}_{Q1}=9{M}_{{\mu}_{7}}{\alpha}_{w}/2=13.162\text{\hspace{0.17em}}\text{MeV}$ (16)

Table 3. The masses of quarks.

d = principal mass dimensional mass orbital number, a = auxiliary mass dimensional number, generation = generation of lepton-quark in the standard model.

Table 4. The masses of baryons.

The secondary binding energy E_{Q}_{2} for d or u quark is as follows.

${E}_{Q2}=9{E}_{Q1}{\alpha}_{w}/2=1.641\text{\hspace{0.17em}}\text{MeV}$ (17)

The tertiary binding energy E_{Q3} for d or u quark bond is as follows.

${E}_{Q3\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{quark}}=9{E}_{Q2}{\alpha}_{w}/2=0.205\text{\hspace{0.17em}}\text{MeV}$ (18)

The binding energy E_{QQ} for each dd, uu, and du bond is 2E_{Q}.

${E}_{QQ}=2{E}_{Q}$ (19)

The mass of neutron (ddu) involves the mass of 2d and u subtracting the binding energy of E_{QQ}_{1} and E_{QQ}_{2} for two quark bonds (2du’s) as follows.

${M}_{N}=Mu+2Md-2{E}_{Q{Q}_{d/u}1}+2{E}_{Q{Q}_{d/u}2}=939.425\text{\hspace{0.17em}}\text{MeV}$ (20)

The calculated mass of neutron is in excellent agreement with the observed value 939.565 MeV with the % mass difference between the calculated and the observed masses = −0.01%.

Proton (duu) is more stable than neutron, so it involves the additional binding energy from the tertiary binding energy E_{QQ3}. For the mass of proton, the baryon number conservation involves the loss of the mass of positron to prevent the decay into positron. Proton becomes permanently stable. The proton mass formula is as follows.

${M}_{P}=2Mu+Md-2{E}_{Q{Q}_{d/u}1}+2{E}_{Q{Q}_{d/u}2}-2{E}_{Q{Q}_{d/u}3}-Me=938.261\text{\hspace{0.17em}}\text{MeV}$ (21)

The calculated mass of proton is in excellent agreement with the observed value 938.272 MeV with the % mass difference between the calculated and the observed masses = −0.0006%.

Being less stable than du bond, the primary binding energy for us bond is one-third of the primary binding energy for du as follows.

${E}_{Q1}=3{M}_{{\mu}_{7}}{\alpha}_{w}/2=4.387\text{\hspace{0.17em}}\text{MeV}$ (22)

The secondary binding energy E_{Q}_{2} for u and s is as follows.

${E}_{Q2}=3{E}_{Q1}{\alpha}_{w}/2=0.182\text{\hspace{0.17em}}\text{MeV}$ (23)

Only one bond (with binding energy) or less per baryon is allowed for the baryons with s, c, and b. The mass of Sigma (Σ^{+}) as uus is as follows.

${M}_{N}=2Mu+2Ms-{E}_{Q{Q}_{us}1}+{E}_{Q{Q}_{us}2}=1187.9\text{\hspace{0.17em}}\text{MeV}$ (24)

which is in excellent agreement with the observed value 1189.4 MeV. The binding energy of ds is zero. For example, Sigma (Σ−) with dds has the mass of d + s + d which is 1195.9 MeV in excellent agreement with the observed 1197.4 MeV.

In the two baryons with the same quark composition, the difference in the masses between the two baryons is equal to the multiple of g*, and one baryon has morebond (with binding energy) than the other baryon, so a bond is added or subtracted in one of the two baryons. For example, the two baryons, Lambda (Λ^{0}) and Sigma (Σ^{0}), are uds. Lambda (Λ^{0}) has the mass of u + d + s - g* − E_{QQus1} + E_{QQus2} which is 1117.7 MeV in excellent agreement with the observed 1115.7 MeV. One bond is subtracted in Sigma (Σ^{0}) which has the mass of u + d + s which is 1196.1 MeV in excellent agreement with 1192.6 MeV.

The binding energies of dd, uu, du, uc, and ub are the same. The binding energies of ds, dc, and db are zero. For example, bottom Lambda (
${\Lambda}_{\text{b}}^{+}$ ) with udb has the mass of u + d + b − E_{QQub1} + E_{QQub2} which is 5608.7 MeV in excellent agreement with the observed 5619.4 MeV. Only one bond or less is allowed for the baryons with s, c, and b except in the two baryons with the same quark composition where one bond is added in one of the two baryons. For example, the two baryons, charmed Sigma (
${\Sigma}_{\text{c}}^{+}$ ) and charmed Lambda (
${\Lambda}_{\text{c}}^{+}$ ), are udc, Charmed Sigma (
${\Sigma}_{\text{c}}^{+}$ ) has the mass of u + d + c + 3g* − E_{QQuc1} + E_{QQuc2} which is 2449.4 MeV in excellent agreement with the observed 2452.9 MeV. One bond is added in charmed Lambda (
${\Lambda}_{\text{c}}^{+}$ ) which has the mass of u + d + c + g* − E_{QQuc1} + E_{QQuc2} − E_{QQud1} + E_{QQud2} which is 2286.3 MeV in excellent agreement with the observed 2286.5 MeV.

Without d/u, the baryons have no binding energy. For example, charmed Omega ( ${\Omega}_{\text{c}}^{0}$ ) with ssc has the mass of s + s + c which is 2683.4 MeV which is in excellent agreement with the observed 2695.2 MeV.

The calculated masses and the observed masses [10] of mesons are in Table 5. Since parallel spins have higher energy than antiparallel spins, vector mesons with parallel spins have higher masses than pseudoscalar mesons with antiparallel spins. For higher-mass vector mesons (parallel spins), vector t, b, c, and s are baryonic t, b, c, s, and vector d and u are baryonic d and u plus basic gluon (g*). For lower-mass pseudoscalar mesons (antiparallel spins), pseudoscalar t is a part of vector t, pseudoscalar b and c are vector quarks minus g*, and pseudoscalar d, u, and s are derived from g*.

The mass of π^{±} is the mass of 2g* minus the mass of e^{±} as proposed by Peter Cameron [17] . The calculated mass of π^{±} is 139.5395 MeV which is in excellent agreement with the observed 139.5702 MeV. π^{±} has much longer mean lifetime than other mesons to indicate that the composite of π^{±} is not normal composite of u and d quarks. Another pseudoscalar meson with long mean lifetime is K^{+} (us̅) which has the composition of 7g* + 7e with the calculated mass of 493.754 MeV in excellent agreement with the observed 493.677 MeV.

The mass of π^{0} involves the composite of pseudoscalar u and pseudoscalar d quarks as (u_{p}u̅_{p} + d_{p}d̅_{p})/2. The binding energy for pseudoscalar meson involves the auxiliary mass dimensional orbital at d = 7 similar to the binding energy in u and d quarks for baryons as Equation (16).The binding energy for pseudoscalar u and d does not involve 3 colors as in 3-color gluons, so similar to Equation (16), the primary binding energy for pseudoscalar u and d quarks at d = 7 with α_{w} as follows.

${E}_{Q1}=3{M}_{g*}{\alpha}_{w}/2=2.911\text{\hspace{0.17em}}\text{MeV}$ (25)

The secondary binding energy is as follows.

${E}_{Q2}=3{E}_{Q1}{\alpha}_{w}/2=0.121\text{\hspace{0.17em}}\text{MeV}$ (26)

The binding energy E_{QQ} for each dd, uu, and du bond is 2E_{Q}. π^{0} is (u_{p}u̅_{p} + d_{p}d̅_{p})/2, so similar to Equation (20), the mass of π^{0} is as follows.

${M}_{{\pi}^{0}}=M{u}_{p}+M{d}_{p}-{E}_{QQ1}+{E}_{QQ2}=134.982\text{\hspace{0.17em}}\text{MeV}$ (27)

which is in excellent agreement with the observed value 134.9766 MeV.

The binding energy for the ds bond and the us bond is three times of the d/u quark bond to form the composite boson with three flux quanta as follows.

${E}_{Q1}=3\times 3{M}_{g*}{\alpha}_{w}/2=8.732\text{\hspace{0.17em}}\text{MeV}$ (28)

The secondary binding energy is as follows.

Table 5. The masses of mesons and the Higgs bosons.

${E}_{Q2}=9{E}_{Q1}{\alpha}_{w}/2=1.089\text{\hspace{0.17em}}\text{MeV}$ (29)

The mass of η as (u_{p}u̅_{p} + d_{p}d̅_{p} + s_{p}s̅_{p})/2 is as follows.

${M}_{\eta}=M{u}_{p}+M{d}_{p}+M{s}_{p}-{E}_{QQ1}+{E}_{QQ2}=548.663\text{\hspace{0.17em}}\text{MeV}$ (30)

which is in excellent agreement with the observed value 547.862 MeV.

The mass of η’ as (u_{p}u̅_{p} + d_{p}d̅_{p})/2 + s_{p}s̅_{p} is as follows.

${M}_{{\eta}^{\prime}}=M{u}_{p}+M{d}_{p}+2M{s}_{p}-2{E}_{QQ1}+2{E}_{QQ2}=956.764\text{\hspace{0.17em}}\text{MeV}$ (31)

which is in excellent agreement with the observed value 957.78 MeV.

The mesons with c, b, and t have no binding energy. For d_{p} and u_{p} in the pseudoscalar mesons with c and b, d_{p} = d_{b} = d_{v} - g*, and u_{p} = u_{b} = u_{v} - g*. For example, the mass of D meson (D^{0}) with c_{p} u̅_{p} is c_{p} + u_{p} = c_{p} + u_{v} - g* which is 1863.9 MeV in excellent agreement with the observed 1864.8 MeV.

The binding energy for vector dd, uu, and du bonds involves the same binding energy as baryonic d/u quark bond as Equations (16) and (17), so vector ρ^{+} as u_{v}d̅_{v} with the binding energy derived from Equations (16) and (17) is as follows.

${M}_{\rho}=M{u}_{v}+M{d}_{v}-{E}_{QQ1}+{E}_{QQ2}=774.07\text{\hspace{0.17em}}\text{MeV}$ (32)

which is in excellent agreement with the observed value 775.11 MeV. As in the baryons with the same quark composition, charged rho meson (ρ^{+}) and omegameson (ω) have the same composition as u_{v}d̅_{v}, so 1/2 bond is subtracted in omegameson (ω) which has the mass as follows.

${M}_{\omega}=M{u}_{v}+M{d}_{v}-{E}_{Q1}+{E}_{Q2}=785.59\text{\hspace{0.17em}}\text{MeV}$ (33)

which is in excellent agreement with the observed value 782.65 MeV.

The binding energy for vector ds bond and us bond is twice of the binding energy for d/u quarks. The mass for kaon (K^{*+}) with u_{v}s̅_{v} is as follows.

$\begin{array}{l}{M}_{{K}^{*+}}=M{u}_{v}+M{s}_{v}-2{E}_{QQ1}+2{E}_{QQ2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=M{u}_{v}+M{s}_{v}-52.648\text{\hspace{0.17em}}\text{MeV}+6.565\text{\hspace{0.17em}}\text{MeV}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=891.60\text{\hspace{0.17em}}\text{MeV}\end{array}$ (34)

which is in excellent agreement with the observed value 891.66 MeV. The binding energy for vector ss bond has the opposite sign for E_{QQ2}, so phi meson (ϕ) as s_{v}s̅_{v} has the mass of s_{v} + s_{v}―52.648 MeV - 6.565 MeV which is 1018.9 MeV in excellent agreement with the observed 1019.5 MeV. The mesons with c, b, and t have no binding energy. For example, D meson (D*^{0}) with c_{v}u̅_{v} has the mass of c_{v} + u_{v} which is 2004.0 MeV which is in excellent agreement with the observed 2007.0 MeV.

The masses of the mesons of c/b without d/u follow the meson mass formula by Malcolm H. MacGregor [18] to match the masses of mesons derived from the quark mass formula as Equation (15). The MacGregor’s meson mass formula derived from the muon mass formula as Equation (4) is as follows.

${M}_{meson}=Me\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}2Me+2n\left(3{M}_{\text{g*}}/2\right)$ (35)

where one e is for charge meson and 2e for neutral meson, and n (integer) is determined by the masses of mesons calculated from the quark mass formula as Equation (15). For example, the calculated mass of vector strange D meson (
${\text{D}}_{\text{s}}^{\ast +}$ ) as c_{v}s̅_{v} is c_{v} + s_{v} which is 2144.4 MeV. To match 2144.4 MeV, the MacGregor’s meson mass formula generates e + 20(3g*/2) which is 2101.3 MeV in excellent agreement with the observed 2112.1 MeV. The mass different between pseudoscalar strange D meson (
${\text{D}}_{\text{s}}^{+}$ ) as c_{p}s̅_{p} and vector strange D meson (
${\text{D}}_{\text{s}}^{\ast +}$ ) as c_{v}s̅_{v} is 2g*. Strange D meson (
${\text{D}}_{\text{s}}^{+}$ ) has the mass of e + 20(3g*/2) - 2g* which is 1961.2 MeV in excellent agreement with the observed 1968.3 MeV.

The MacGregor’s meson mass formula in Equation (35) for the mesons of c/b without d/u/s, n = the multiple of 3to simulate baryonic quark which uses 3µ as in Equation (9). For example, vector upsilon meson (γ) with b_{v}b̅_{v} has the mass of 2e + 90(3g*/2) which is equal to 9454.4 MeV in excellent agreement with the observed 9460.3 MeV. The mass difference between vector bb̅ and pseudoscalar bb̅ is the mass of g*. Pseudoscalar bottom eta meson (η_{b}) with b_{p}b̅_{p} has the mass of 2e + 90(3g*/2) - g* which is equal to 9384.4 MeV in excellent agreement with the observed 9398.0 MeV.

Pseudoscalar and partial t quark is t_{7}, while vector and full t quark is t_{7} + t_{8}. Vector and full t quark with enormous mass is extremely short-lived, so top quark-antiquark does not have time before they decay to form hadrons, resulting in “bare” t quark and antiquark. The calculated mass of t is 175.4 GeV in good agreement with the observed 172.4 GeV.

The summary of binding energies in hadrons is in Table 6. The binding energies are derived from the auxiliary mass dimensional orbital at d = 7 as in Equations (16), (17), (18). (19), (22), (23), (25), (26), (28), (29), (32), and (34). In general, the relatively stable hadrons with d, u, and s quarks have binding energies, while relatively unstable hadrons with c, b, and t quarks and without u and d do not have binding energies. The baryons with the u and s/c/b bonds have binding energies, and the baryons with the d and s/c/b bonds do not have binding

Table 6. The binding energies in hadrons.

energies. The mesons with c, b, and t quarks do not have binding energies. Pseudoscalar mesons have lower binding energies than baryons and vector mesons. In the two hadrons with the same quark composition, one hadron has more bond (with binding energy) than the other, so a bond is added or subtracted in one of the two hadrons.

4. The Top-Bottom Quark-Antiquark Composite

In the search for resonances produced in association with a b quark jet and a second jet, and decaying to a muon pair, the CMS Collaboration at the LHC recently reported an excess of events above the background near a dimuon mass of 28 GeV. The search is carried out in two categories from proton-proton collisions at center-of-mass energies of 8 and 13 TeV. The first category involves a b quark jet in the central region and at least one jet in the forward region, while the second category involves two jets in the central region, at least one of which is identified as a b quark jet, no jets in the forward region. At the 8 TeV collision, the first category has 4.2 standard deviation, while the second category has 2.9 standard deviations. At the 13 TeV collision, the first category has 2.0 standard deviations, while the second category results in a 1.4 standard deviation deficit.

As shown in Figure 1, Table 1, Table 3, and Table 5, pseudoscalar t quark is t_{7} (13.2 GeV), while vector and full t quark (175.4 GeV) is t_{7} + t_{8}. This paper posits that the resonance with 28 GeV observed recently at the LHC is the pseudoscalar top-bottom quark-antiquark composite which has the calculated mass of 27.9 GeV derived from the periodic table of elementary particles in good agreement with the observed 28 GeV as shown in Table 5. The calculated mass is the mass of three pseudoscalar b quarks and one pseudoscalar t quark which represent the composite of b_{p}b̅_{p} + (b_{p}b̅_{p} + t_{p}t̅_{p})/2. As described in the periodic table of elementary particles, pseudoscalar t quark is only a part of full t quark, so pseudoscalar t_{p}t̅_{p} (26.4 GeV) cannot exist independently, and can locate within a composite, such as the top-bottom quark-antiquark composite. As shown in the observation at the LHC, the resonance with 28 GeV weakens significantly at the higher energy collision (13 TeV), because at the higher collision energy, low-mass pseudoscalar tt̅ in the composite likely becomes independent full high-mass vector tt̅ moving out of the composite. The presence of the top-bottom quark-antiquark composite weakens, disappears, or gets inverted at 13 TeV as shown at the LHC. Normally, the presence of resonance gets stronger at the higher collision energy.

To account for the observed two jets, the composite has two jets consisting of a bb̅ jet and a b + t jet for (b_{p}b̅_{p} + t_{p}t̅_{p})/2, where bb̅ jet is more stable than b + t jet which decays faster into the jet in the forward region to constitute the first category of the search by the CMS Collaboration at the LHC. Since t_{p}t̅_{p} is less stable than b_{p}b̅_{p}, so the decay of the b + t jet is faster to allow the greater standard deviations for the first category than for the second category. The sum of the standard deviations from both categories is greater than 5.

5. The Higgs Boson Doublet

One important open theoretical issue about the Higgs boson is the triviality problem [19] . Within the perturbation theory, the Higgs boson mass squared is proportional to the self-coupling. However, the scalar self-coupling for the scalar Higgs boson leads to triviality or non-interaction which is inconsistent to the interactive Higgs boson. To deal with the triviality problem, Cea and Cosmai [20] [21] established the non-perturbation non-trivial rescaling of the Higgs condensate to avoid the vanishing self-coupling, resulting in the generation of the heavy Higgs boson with 754 GeV. According to Cea, the theoretical expectations of the predicted heavy Higgs boson (754 GeV) are in fairly good agreement with the observations at the LHC Run 2with an estimated statistical significance of more than five standard deviations [22] .

This paper proposes that derived from the non-trivial rescaling of the Higgs condensate, the Higgs boson doublet consists of the high Higgs boson from the upward rescaling of the Higgs condensate and the low Higgs boson from the downward rescaling. The perturbative Higgs boson became the non-perturbative Higgs boson doublet irreversibly during the spontaneous symmetry breaking. The observed mass of the high Higgs boson is 750 GeV [22] [23] [24] [25] , and the observed mass of the low Higgs boson is 125 GeV [26] [27] .

This paper also proposes that the Higgs bosons are the intermediate vector boson composites whose condensate provides the masses directly to the intermediate vector bosons during the spontaneous symmetry breaking. The Higgs bosons consist of the perturbative Higgs boson (W^{+}W^{−}Z^{0} = 252 GeV = M_{W+} + M_{W−} + M_{Z0}), the low Higgs boson (W^{+}W^{−}Z^{0}/2 = 126 GeV) from the downward rescaling, and the high Higgs boson (3W^{+}W^{−}Z^{0} = 756 GeV) from the upward rescaling. The perturbative Higgs boson is absent, and there is no Higgs boson pair [28] . The low Higgs boson as W^{+}W^{−}Z^{0}/2 with respect to W^{+}W^{−}Z^{0} is like one quark in a di-quark meson, while the high Higgs boson as 3W^{+}W^{−}Z^{0} with respect to W^{+}W^{−}Z^{0} is like one baryon consisting of three quarks. In this way, the Higgs boson doublet from the rescaling provides the structures of the quark compositions for mesons and baryons which are not like leptons without composite structures. The calculated masses (126 GeV and 756 GeV) of the Higgs boson doublet are in excellent agreements with the observed masses (125 GeV and 750 GeV) [22] - [27] . For the periodic table of elementary particles, the Higgs mechanism assigns the mass of B_{7} as the mass of Z^{0}. The mass of B_{7} produces α_{w} which determines the masses of quarks as in the quark mass formula in Equation (15).

6. Summary

This paper posits that the observed resonance with 28 GeV at the LHC is the pseudoscalar top-bottom quark-antiquark composite which has the calculated mass of 27.9 GeV derived from the periodic table of elementary particles. The calculated mass is the mass of three pseudoscalar b quarks and one pseudoscalar t quark to represent the composite of b_{p}b̅_{p} + (b_{p}b̅_{p} + t_{p}t̅_{p})/2 where p = pseudoscalar. (The quark in pseudoscalar meson is denoted as “pseudoscalar quark”, while the quark in vector mesons is denoted as “vector quark” which has higher mass than pseudoscalar quark.) In the periodic table of elementary particles, pseudoscalar t quark (13.2 GeV) is only a part of full t quark (175.4 GeV), so pseudoscalar t_{p}t̅_{p} (26.4 GeV) cannot exist independently, and can exist only in the top-bottom quark-antiquark composite. As shown in the observation at the LHC, the resonance with 28 GeV weakens significantly at the higher energy collision (13 TeV), because at the higher collision energy, low-mass pseudoscalar t_{p}t̅_{p} in the composite likely becomes independent full high-mass vector t_{v}t̅_{v} moving out of the composite. To account for the observed two jets, the composite has two jets consisting of a bb̅ jet and a b + t jet, where bb̅ jet for (b_{p}b̅_{p} + t_{p}t̅_{p})/2 is more stable than b + t jet which decays faster into the jet in the forward region to constitute the first category of the search by the CMS Collaboration at the LHC.

The periodic table of elementary particles is based on the seven mass dimensional orbitals derived from the seven extra dimensions of 11 spacetime dimensional membrane particles. The seven mass dimensional orbitals include the seven principal mass dimensional orbitals for stable baryonic matter leptons (electron and neutrinos), gauge bosons, gravity, and dark matter and the seven auxiliary mass dimensional orbitals for unstable leptons (muon and tau) and quarks, and calculate accurately the masses of all elementary particles and the cosmic rays by using only five known constants. For baryons, the quarks in the periodic table of elementary particles are baryonic quarks. For high-mass vector mesons (parallel spins), vector t, b, c, and s are baryonic t, b, c, s, and vector d and u are baryonic d and u plus basic gluon (g* = 70 MeV) which has the mass of electron/α where α is the fine structure constant of electromagnetism. For low-mass pseudoscalar mesons (antiparallel spins), pseudoscalar t is a part of vector t, pseudoscalar b and c are vector quarks minus g*, and pseudoscalar d, u, and s are derived from g*. The binding energies among quarks are derived from the auxiliary mass dimensional orbital. With these masses and binding energies of quarks, the masses of hadrons can be calculated in excellent agreement with the observed masses of hadrons by using only five known constants in the periodic table of elementary particles. For examples, the calculated masses of proton, neutron, pion (π^{±}), and pion (π^{0}) are 938.261, 939.425, 139.540, and 134.982 MeV in excellent agreement with the observed 938.272, 939.565, 139.570, and 134.977 MeV, respectively with 0.0006%, 0.01%, 0.02%, and 0.004%, respectively for the difference between the calculated and observed mass. The calculated masses of the Higgs bosons as the intermediate vector boson composites are in excellent agreements with the observed masses. In conclusion, the calculated masses of the top-bottom quark-antiquark composite (27.9 GeV), hadrons, and the Higgs bosons by the periodic table of elementary particles are in excellent agreement with the observed masses of resonance with 28 GeV at the LHC, hadrons, and the Higgs bosons, respectively.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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