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Group Theory and Its Application to Particle Physics

Global Gauge Symmetry

The global gauge symmetry SU(2) X U(1) can be used to explain the empirical relation of Gell-Mann and Nishijima:

Q = t3 + (Y/2) ---------- (19)

where Q is the electric charge, t3 is the third component of the isospin, and Y is the "hypercharge". The hypercharge was invented to explain the state of the electron and neutrino prior to spontaneous symmetry breaking, which endows mass and charge to these particles. Before the occurence of such event these two particles differ only by the weak charge t3. The hypercharge was introduced to relate the difference in electric charge to the difference in t3. As shown in Table 01, Y = -1 or 1/3 for the SU(2) isodoublet, and Y = -2 or -2/3 for the U(1) isosinglet. These values of Y reflect the charge state of these particles before symmetry breaking. To satisfy this formula for electro-weak interaction with SU(2) X U(1) symmetry where the SU(2) group splices together with the U(1) group (no mixing), the isodoublet L in Eq.(17) has to be an eigenvector of SU(2) with t = 1/2 and the isosinglet R an eigenvector of U(1) with t = 0. With the known values of Q and t3, Y can be computed from Eq.(19). Similar assignment can be given to the u (up), d (down), and s (strange) quarks as listed in Table 01 below:

Particle Q t t3 Y
0 1/2 1/2 -1
eL -1 1/2 -1/2 -1
eR -1 0 0 -2
u 2/3 1/2 1/2 1/3
d -1/3 1/2 -1/2 1/3
s -1/3 0 0 -2/3

Table 01 Gell-Mann-Nishijima Relation

In strong interaction with SU(3) symmetry, the hypercharge for the quarks is generated by the 8/(3)1/2 generator (see Table 01 and diagrams at the top of Figure 03). The hypercharge for the hadrons, which is the composite particle with quarks, is just the sum of the individual quarks (see Table 02 and 03). The vaule of Y is related to the other quantum number in strong interaction by the formula: Y = S + B, where S is the strangeness number (0 for u and d, -1 for s), and B is the baryon number (+1/3 for quark, -1/3 for anti-quark, +1 for baryon, -1 for anti-baryon, and 0 for other particles). It was used by Gell-Mann to arrange the hadrons into geometrical patterns known as the eight-fold way (also see Figure 03). It is a plot with Y against t3. The isosinglet with one member for t = 0 is located at the center. Two isodoublets for t = 1/2 are arranged in the top and bottom row with Y = +1 and -1 respectively. The three members of the SU(3) isotriplet for t = 1 occupy the middle row, the t3 = 0 member shares the center with the isosinglet member. The particles with same isospin (same multiplet) are supposed to be similar except for the value of t3. The mass splitting among the isospin members is caused by the electromagnetic interactions as shown in Figure 04.

Eightfold Way Mass Splitting

Figure 03 Eightfold Way [view large image]

Figure 04 Mass Splitting [view large image]

Another application of global gauge symmetry is to build mesons and baryons from quarks in strong interaction, where the wave functions for the quarks u, d, and s constitute the fundamental representation of SU(3):
---------- (20)
Note that the three quarks in Eq.(20) is called "flavor" quark associated with global SU(3). The other three kinds of quark related to the local SU(3) gauge transformation is called "colour" quark participating in strong interaction.

Beginning with the fundamental representation in Eq.(20), all representations of SU(N) can be generated by taking the multiple (tensor) products. For example, the bound states of mesons can be constructed according to the following scheme:
---------- (21)
where the bar "-" denotes anti-particle.

Table 02 lists the relation between the meson wave functions and the quark pairs. The value within the brackets is the rest mass in Mev.

In these tables J is the total angular momentum quantum number. The mesons in the middle column belongs to J = 0, while those in the right column belongs to J = 1. In terms of the meson wave functions the octet in Eq.(21) can be re-written in the form of Eq.(22). Similarly, the baryon wave functions are related to the quark triplet as shown in Table 03 and Eq.(23).

This is clearly related to the Eightfold Way (Figure 03) originally proposed by Gell-Mann and Ne'eman in 1961.

Note: These elementary particles were referred to as resonances in the 1960s. Now they are considered as the excited states of hadrons with some of their constituent quarks boosted into higher energy levels. Most of them have a very short lifetime about 10-23 sec.

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