Noncommutative gauge theory with arbitrary U(1) charges

Abstract

It is well-known that the charge of fermion is 0 or $\pm1$ in the U(1) gauge theory on noncommutative spacetime. Since the deviation from the standard model in particle physics has not yet observed, and so there may be no room to incorporate the noncommutative U(1) gauge theory into the standard model because the quarks have fractional charges. However, it is shown in this article that there is the noncommutative gauge theory with arbitrary charges which symmetry is for example SU(3+1)$\ast$. This enveloping gauge group consists of elements $\exp i {\sum_{a=0}^8 T^a \alpha^a(x,\theta)\ast+ Q \beta(x,\theta)\ast} $ with $Q=\text{diag}(e,e,e,e') $ and the restriction $\lim_{\theta\to0}\alpha^0(x,\theta)=0.$ This type of gauge theory is emergent from the spontaneous breakdown of the noncommutative SU(N)$\ast$ or SO(N)$\ast$ gauge theory in which the gauge field contains the 0 component $A_\mu^0(x,\theta)$. $A_\mu^0(x,\theta)$ can be eliminated by gauge transformation. Thus, the noncommutative gauge theory with arbitrary U(1) charges can not exist alone, but it must coexist with noncommutative nonabelian gauge theory. This suggests that the spacetime noncommutativity requires the grand unified theory which spontaneously breaks down to the noncommutative standard model with fractionally charged quarks.