Quantum Field Theory

The SO(10) Group and Unification (Revised 2021)

The 16 fermions in the standard model are summarized in Table 06, where the superscript denotes electric charge, and the subscript denotes color charge. All the quarks in the SU(3) gauge group participate in strong interaction, all left-handed fermions undertake weak interaction, and those fermions carrying electric charge are eligible for electromagnetic interaction.

Gauge-Group / Handedness	SU(3)	SU(2)	U(1)
Left-handed	u_r^2/3, u_g^2/3, u_b^2/3, d_r^-1/3, d_g^-1/3, d_b^-1/3	⁰, e^-1
Right-handed	u_r^2/3, u_g^2/3, u_b^2/3, d_r^-1/3, d_g^-1/3, d_b^-1/3		e^-1	⁰

Table 06 Fermions in Standard Model

An alternate arrangement to unify all these fermions into the 10 dimensional SO(10) group with the introduction of two color charges for weak interaction is proposed :

There would be 5 generalized color charges including the red (r), green (g), blue (b) in strong interaction, the yellow (y) and purple (p) for 3rd components of isospin t₃ in weak interaction (Figure 05b).

The Table in Figure 05b lists all the fermions of the standard model in the last column. The generalized color charges are in the middle. The value of hypercharge Y for each fermion is calculated from the colored charges in solid circles (using the formula as shown). The hollow circles would yield a value of Y for its opposite partner, either anti-fermion or fermion (anti-fermion has a "bar" sign on top).

The color charges in strong interaction and those charges for the weak interaction are now lumped together to form the generalized hypercharge Y = - ( r + g + b ) / 3 + ( y + p ) / 2. This is not exactly the electric charge, for example, the left-handed neutrino and electron carries 0 and -1 electric charge respectively yielding the average value -1/2 for Y. The hypercharge was invented to explain the state of the electron and neutrino prior to Spontaneous Symmetry Breaking (SSB), which endows mass and charge Q to these particles. After SSB, Q = Y + t₃. For example, t₃ = (1/2, -1/2) for (

, e)^L and (u, d)^L while t₃ = 0 for the right handed sector.

Figure 05b SO(10) Group
[view large image]

The right-handed neutrino (

)^R is the odd-man/woman out in the standard model. It is there to provide a possible mechanism for the mass of the neutrinos. It does not involve in any of the interactions - that's why it has not been observed (smelling like dark matter ?).

Figure 05b,a depicts a summary of all the charges and their fermion carriers using r (red), g (green), b (blue) for the color charges associated with the u and d quarks in strong interaction. The 3rd component isospin in weak interaction is represented by y (yellow) and p (purple) respectively (double dots denotes both strong and weak interactions). The electric charge in electromagnetic interaction is given by the formula Q = Y + t₃. There are altogether 16 fermions including the non-existing right-handed neutrino. The "L" and "R" in the diagram stand for the "Left-handed" and "Right-handed" sectors of the SU(2)U(1) group; while Y is denoted in the subscript.

The SO(10) group breaks into many schemes. One of them is the SU(3)SU(2)U(1) carrying charges of (r, g, b), (y, p), and Q respectively. It is this particular grouping that fit all the elementary fermions snugly together without much room to maneuver. The breaking down to various subgroups occurred long time ago soon after the Big Bang (see "Quantum Field History").

To complete the list of all particles in the standard model, table 07 below displays all the force mediating bosons, where the single quote (') denotes anti-charge similar to the + charge of the positron in the electromagnetic interaction:

Charge	Gauge Field(s)	Gauge Particle(s)
r	G_rr', G_rg', G_rb', gluon fields	Gluons from red color charge
g	G_gg', G_gr', G_gb', gluon fields	Gluons from green color charge
b	G_bb', G_bg', G_br', gluon fields	Gluons from blue color charge
w	G_ww', G_wz', vector meson fields	W vector mesons
z	G_zz', G_zw', vector meson fields	Z⁰ vector meson
e	A, electromagnetic field	Photon

Table 07 Gauge Bosons

There is actually only 8 independent bosons for SU(3) and 3 independent bosons for SU(2) since the trace (trace of a matrix A :
TrA = a_nn) of their matrix representations is equal to zero - a condition to reduce the number of independent bosons by one. This is a requirement for the unitarity of the gauge group.

Since the force mediating gluon itself carries color charges, the corresponding Feynman diagram is interpreted in slightly different way than those in QED. Figure 05c shows two quarks with blue and red colors moving toward each other initially. They interact by gluon

exchange such that the blue quark emits a blue-antired gluon thereby transforming itself into a red quark, while the red quark absorbs this gluon to become a blue quark. In other word, the gluon manages the creation and annihilation of the red-antired color charge pair, while fetches and donates the blue color charge from one quark to another resulting in the exchange of color charge on the quarks during the course of the interaction.

Figure 05c Strong Interaction [view large image]

Since all the color charges are SO(10) group members that can be turned into each others, it is expected that at high probing energy, which enables penetration through the shielding by the virtual particles, the coupling strength of the various kinds of charges would merge into one unifying point. As shown by the diagram on the left of Figure 05d, it almost works, but not quite (the experimental errors is indicated by the width of the lines). The expectation comes true by introducing supersymmetry (SUSY), it even brings the outcast gravity to close proximity of the merging point (see diagram on the right of

Figure 05d Unification by SUSY [view large image]

Figure 05d). This additional symmetry requires a new partner for all the particles. They have the same charges (of all kinds) as their known partner, but different masses (heavier) and spins (integer

1/2 integer). Supersymmetry in effect doubles the number of particles in the standard model.

See "Unification of Coupling Strength" in "Unification of Force and Substance" by Frank Wilczek, 2016.