# QCD Theory of the Hadrons and Filling the Yang–Mills Mass Gap

## Abstract

**:**

## 1. Introduction

## 2. A Brief Review of Maxwell’s Equations Using Duality and Differential Forms

## 3. Maxwell’s Yang–Mills Canonic Equations

^{V}, which are non-commuting, $\left[{G}_{\mu},{G}_{\nu}\right]\ne 0$. Specifically, for a simple group SU(N) with traceless N×N Hermitian generators ${\tau}_{i}={\tau}_{i}{}^{\u2020}$ with $i=1\dots {N}^{2}-1$ normalized to $\mathrm{tr}\left({\tau}_{i}{}^{2}\right)={\scriptscriptstyle \frac{1}{2}}$ for each ${\tau}_{i}$, and a commutator $\left[{\tau}_{i},{\tau}_{j}\right]=i{f}_{ijk}{\tau}_{k}$, these gauge fields are constructed via ${G}^{\mu}={\tau}_{i}{G}_{i}^{\mu}={G}^{\mu}{}^{\u2020}$ and so are likewise N×N Hermitian matrices. We may use the commutator to find that $\left[{G}_{\mu},{G}_{\nu}\right]=\left[{\tau}_{i},{\tau}_{j}\right]{G}_{i}{}_{\mu}{G}_{j}{}_{\nu}=i{f}_{ijk}{\tau}_{k}{G}_{i}{}_{\mu}{G}_{j}{}_{\nu}$. Likewise, while $\mathrm{tr}\left({G}^{\mu}\right)=\mathrm{tr}\left({\tau}_{i}{G}_{i}^{\mu}\right)=0$, it can be generally shown that $\mathrm{tr}\left(AB\right)=\mathrm{tr}\left({\tau}_{i}{\tau}_{j}{A}_{i}{B}_{j}\right)={\scriptscriptstyle \frac{1}{2}}{A}_{i}{B}_{i}$ for any $A={\tau}_{i}{A}_{i}$, $B={\tau}_{j}{B}_{j}$, which means that $\mathrm{tr}\left({G}_{\mu}{G}_{\nu}\right)=\mathrm{tr}\left({\tau}_{i}{\tau}_{j}{G}_{i}{}_{\mu}{G}_{j}{}_{\nu}\right)={\scriptscriptstyle \frac{1}{2}}{G}_{i}{}_{\mu}{G}_{i}{}_{\nu}$ for a product of two gauge fields. Although weak interactions use SU(2) and strong interactions SU(3), at the outset we shall not examine any particular gauge groups. Our immediate interest is to develop the counterparts to Maxwell’s equations generally, for any SU(N) Yang–Mills (YM) gauge theory.

## 4. Maxwell’s Yang–Mills Dynamic Equations

## 5. Populating Yang–Mills Magnetic Monopoles with Source Currents, by Inverting the Yang–Mills Electric Source Equation and Then Combining Both Maxwell Equations into One

## 6. Nonlinear Recursive Interactions Contained in the Inverse Yang–Mills Electric Equation

## 7. Introducing the Inverse Yang–Mills Electric Source Equation into the Yang–Mills Magnetic Monopoles, then Populating These Monopoles with Dirac Fermions

## 8. The Yang–Mills “Signal” Magnetic Monopole, without Perturbative “Noise”

^{μ}in (24) treated recursively in the manner of (22), what we have in (24) is clearly a “signal-plus-noise” monopole density. To explore the underlying QCD behaviors of these monopoles, we now shall study just the “signal” monopole density with all “noise” (which comprises about 99% of the observed rest energy of the proton and neutron and a large share of the rest energy of other baryons as well), removed.

## 9. Populating the Yang–Mills “Signal” Magnetic Monopole with Dirac Fermions: Two Alternatives, Each of Which Shows That the Signal Monopole Contains Exactly Three Fermions

## 10. Using the Gauge Group SU(3) to Establish Three Distinct Quantum States for the Three Fermions Populating a Yang–Mills Magnetic Monopole

_{QCD}is an exact symmetry because gluons in its adjoint representation are massless. However, for example, early theories of baryon flavor similarly placed (u, d, s) into the fundamental representation of SU(3) with an approximate flavor symmetry which is distinct from the exact color symmetry of SU(3)

_{QCD}. So, we must establish that this SU(3) group arising from the monopoles is truly synonymous with the exact SU(3) group of QCD, and not some independent SU(3).

## 11. The Yang–Mills Signal Magnetic Monopole Prior to Symmetry Breaking

_{QCD}requires massless gluons, whereas the denominators in (31) are all for massive vector bosons. So this looks like an approximate SU(3) flavor-type rather than an exact SU(3) color symmetry. Moreover, (31) has vanishing trace. Furthermore, knowing that all particle propagators have the general form $i{\Sigma}_{\mathrm{spins}}/\left({p}^{2}-{m}^{2}+i\epsilon \right)$ with spin sum ${\Sigma}_{\mathrm{spins}}$ being the completeness relation, we see that ${\psi}_{G}{\overline{\psi}}_{B}$ at the center of this numerator looks like a spin sum ${\Sigma}_{s}u\overline{u}=\overline{)p}+m$, but is not. This is because ${\psi}_{G}$ and ${\psi}_{B}$ in ${\psi}_{G}{\overline{\psi}}_{B}$ are not the same fermion but are two differently-colored fermions. This final point indicates the way forward, because if we can turn ${\psi}_{G}$ and ${\psi}_{B}$ in ${\psi}_{G}{\overline{\psi}}_{B}$ into the same fermion, we can use this as a fermion spin sum. Then, having a spin sum in the numerator and two massive boson propagators in the denominator, we can shuttle a degree of freedom from a boson to a fermion to simultaneously produce a fermion propagator and a massless boson propagator. This entails a form of spontaneous symmetry breaking which starts with (27) then breaks symmetry using (28), because the backbone of (28) does have the requisite same-fermion terms ${\psi}_{\left(\sigma \right)}{\overline{\psi}}_{\left(\sigma \right)}$, ${\psi}_{\left(\mu \right)}{\overline{\psi}}_{\left(\mu \right)}$ and ${\psi}_{\left(\nu \right)}{\overline{\psi}}_{\left(\nu \right)}$ needed to use the completeness relation.

## 12. Spontaneously Breaking Symmetry inside the Yang–Mills Signal Magnetic Monopole

_{QCD}, rather than the approximate symmetry of SU(3) flavor, bringing us closer to these R, G, B being true QCD states.

## 13. Incorporating the Massless Vector Boson Propagators into the Fermion Normalizations

## 14. Yang–Mills Magnetic Monopoles Have the Color-Neutral Singlet Wavefunction of Baryons, and Interact via Objects with the Color-Neutral Singlet Wavefunction of Mesons

## 15. Act of Confinement: Dynamical Hadronization from Maxwell’s Yang–Mills Equations

^{±}and Z from flowing across the monopole surface, because these are color-neutral.) Inside these signal monopoles, each of the three fermions has a net color charge in the fundamental representation of an SU(3) gauge group which is exact because its vector bosons are massless, and each of these massless vector bosons has a net bi-colored charge in the adjoint representation of SU(3). We therefore conclude that these fermions and massless gauge bosons are confined. Consequently, we further conclude that: the fermions in (39) are quarks; the now-massless gauge bosons are gluons; the signal monopole (39) is a baryon in a non-perturbative state with all “noise” filtered out; and the ${G}^{2}$ object (43) which net flows across the monopole-now-baryon surface is a meson.

_{QCD}as has been done ever since Gell-Mann [17] and Zweig [18] first discovered the quark model. The Yang–Mills magnetic monopoles dynamically make that postulate for us, all by themselves.

^{15}GeV, not far below the Planck scale ${E}_{P}=\sqrt{\u0127{c}^{5}/G}\approx 1.220\times {10}^{19}\mathrm{GeV}$. Specifically, the signal monopole obtained in (31) prior to symmetry breaking has $\mathrm{tr}\mathrm{tr}{p}^{\sigma \mu \nu}=0$ and is associated with energies above 10

^{15}GeV where quarks are free and can mingle with leptons, and where baryons with confined quarks and gluons are not yet formed. The signal monopole obtained in (39) after symmetry breaking has $\mathrm{tr}\mathrm{tr}{\Sigma}_{s}{p}^{\sigma \mu \nu}\ne 0$ and is associated with lower energies where free quarks and gluons no longer exist but are confined in color-neutral hadrons. So, the symmetry breaking to go from (27) to (28) in concert with (37) is now seen to take place at some energy ${E}_{X}$ a few orders of magnitude below the Planck energy. The pre-symmetry breaking (27) in which quarks are labelled with the spacetime indexes of the current densities which carry them into the monopole shows the pre-hadronization baryon above ${E}_{X}$, which at (31) has $\mathrm{tr}\mathrm{tr}{p}^{\sigma \mu \nu}=0$. We shall refer to these as the “plasma” labels. The post-break (28) in which the monopole is made indivisible with quarks now labelled independently from their current of origin shows the baryon below ${E}_{X}$ once hadronization is complete, which at (39) now has $\mathrm{tr}\mathrm{tr}{\Sigma}_{s}{p}^{\sigma \mu \nu}\ne 0$ with confined color. We shall refer to these as the “confinement” labels. The symmetry breaking from (27) to (28), which via (37) takes us from (31) to (39), then becomes synonymous with ${E}_{X}$ -scale hadronization which is dynamical, not ad hoc. This is what Close refers to as the “act of confinement.”

## 16. What Is a Baryon, and Who Ordered That?

## 17. Filling the Yang–Mills Mass Gap

## 18. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Calculation of the Inverse for the Yang–Mills Electric Source Equation

## Appendix B. The Yang–Mills Continuity Equation in Terms of Dirac Wavefunctions

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Yablon, J.R.
QCD Theory of the Hadrons and Filling the Yang–Mills Mass Gap. *Symmetry* **2020**, *12*, 1887.
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Yablon JR.
QCD Theory of the Hadrons and Filling the Yang–Mills Mass Gap. *Symmetry*. 2020; 12(11):1887.
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Yablon, Jay R.
2020. "QCD Theory of the Hadrons and Filling the Yang–Mills Mass Gap" *Symmetry* 12, no. 11: 1887.
https://doi.org/10.3390/sym12111887