Inevitable emergence of composite gauge bosons

A simple theorem is proved: When a gauge-invariant local field theory is written in terms of matter fields alone, a composite gauge boson or bosons must be formed dynamically. The theorem results from the fact that the Noether current vanishes in such theories. The proof is carried out by use of the charge-field algebra at equal time in the Heisenberg picture together with the well-established analyticity of the form factor of the current. While there is no need of diagram calculation for the proof, we demonstrate in the leading $1/N$ expansion of the existing models what the theorem means in diagrams and how the composite gauge boson emerges.

Figure 1

Analyticity of $F(t,\lambda )$ and $1/F(t,\lambda )$ in the complex $t$ plane. The cross in the left-side figure indicates the pole due to a bound state of $J^{PC}=1^{--}$ for $F(t,\lambda )$. The crosses in the right-side figure are due to possible poles of $1/F(t,\lambda )$, that is, zeros of $F(t,\lambda )$.

Figure 2

The form factor $F(t,\lambda )$ in the leading $1/N$ order ($t=q^2$). Each bubble in the left-side figure gives the function $K(t)$ in Eq. (27) and its iteration generates a vector-bound state in the right-side figure.