Unusual high-energy phenomenology of Lorentz-invariant noncommutative field theories

It has been suggested that one may construct a Lorentz-invariant noncommutative field theory by extending the coordinate algebra to additional, fictitious coordinates that transform nontrivially under the Lorentz group. Integration over these coordinates in the action produces a four-dimensional effective theory with Lorentz invariance intact. Previous applications of this approach, in particular, to a specific construction of noncommutative QED, have been studied only in a low-momentum approximation. Here we discuss Lorentz-invariant field theories in which the relevant physics can be studied without requiring an expansion in the inverse scale of noncommutativity. Qualitatively, we find that tree-level scattering cross sections are dramatically suppressed as the center-of-mass energy exceeds the scale of noncommutativity, that cross sections that are isotropic in the commutative limit can develop a pronounced angular dependence, and that nonrelativistic potentials (for example, the Coloumb potential) become nonsingular at the origin. We consider a number of processes in noncommutative QED that may be studied at a future linear collider. We also give an example of scattering via a four-fermion operator in which the noncommutative modifications of the interaction can unitarize the tree-level amplitude, without requiring any other new physics in the ultraviolet.

Figure 1

Difermion production in the Yukawa theory, for the canonical parameter choice $a/2=b=c={\Lambda }^4$, and negligible fermion masses.

Figure 2

Differential cross section for two-into-two scattering in ${\varphi }^4$ theory, for the canonical parameter choice $a/2=b=c={\Lambda }^4$.

Figure 3

Differential cross section for diphoton production at a linear collider with $\sqrt{s}=1.5\mathrm{TeV}$. The curves shown correspond to the canonical parameter choice $a/2=b=c={\Lambda }^4$, with $\Lambda =0.1$, 1, 3, and 10 TeV.

Figure 4

Plot of the vertex integral function, for the canonical parameter choice $a/2=b=c={\Lambda }^4$.