Unitarity of Compactified Five Dimensional Yang-Mills Theory

Compactified five dimensional Yang-Mills theory results in an effective four-dimensional theory with a Kaluza-Klein (KK) tower of massive vector bosons. We explicitly demonstrate that the scattering of the massive vector bosons is unitary at tree-level for low energies, and analyze the relationship between the unitarity violation scale in the KK theory and the nonrenormalizability scale in the five dimensional gauge theory. In the compactified theory, low-energy unitarity is ensured through an interlacing cancellation among contributions from the relevant KK levels. Such cancellations can be understood using a Kaluza-Klein equivalence theorem which results from the geometric “Higgs” mechanism of compactification. In these theories, the unitarity violation is delayed to energy scales higher than the customary limit through the introduction of additional vector bosons rather than Higgs scalars. ∗Electronic addresses: [email protected], [email protected], [email protected] The visible four-dimensional world may be only part of a higher dimensional space-time structure, with the extra spatial dimensions substantially larger than the traditional Planck length (10−33 cm), but too small to have been probed experimentally [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. If the gauge particles propagate in the higher-dimensional space, then from the four-dimensional viewpoint each gauge boson is associated with a Kaluza-Klein (KK) tower of massive vector bosons whose mass splittings are characterized by the (inverse) size of the extra dimensions. In this way compactification of higher dimensions leads to a “geometrical” mechanism for producing massive vector states. The high-energy behavior of the scattering of longitudinally-polarized massive vector bosons is potentially problematic in Yang-Mills theories, and can result in amplitudes growing with energy at tree-level [13, 14, 15, 16]. In the four-dimensional (4D) standard model these amplitudes are exactly canceled by the exchange of spin-0 Higgs particle [14, 17, 18, 19]. However, such Higgs scalar states do not exist in a compactified pure gauge theory. In this letter, we discuss the high-energy behavior of massive vector-boson scattering in the compactified five-dimensional (5D) Yang-Mills theory. We explicitly demonstrate that the scattering of the massive vector bosons is unitary at tree-level for low energies, and analyze the relationship between the scales of 4D unitarity violation and the nonrenormalizability of the 5D gauge theory. In the compactified theory we show that the low-energy unitarity is ensured through an interlacing cancellation among contributions from the relevant KK levels. We observe that this cancellation can be understood from a Kaluza-Klein equivalence theorem resulting from the geometric “Higgs” mechanism of compactification. As a consequence, the unitarity violation is delayed to energy scales higher than the customary limit of Dicus-Mathur and Lee-Quigg-Thacker [14, 17, 18, 19] through the introduction of additional vector bosons rather than Higgs scalars. The Lagrangian for five-dimensional Yang-Mills theory is given by L5 = − 1 2 Tr(F̂MN F̂ MN ) , F̂ a MN = ∂M Â a N − ∂N ÂM + g5CÂM ÂN , (1) where a is the gauge index, Cabc the structure constant, and g5 the 5D gauge coupling with dimension of (mass)−1/2. The five-dimensional coordinates are labeled by M,N ∈ (μ, 5) with μ ∈ (0, 1, 2, 3). For convenience, we may consider this 5D theory with a covariant gauge-fixing term [9], LGF = − 1 2ξ (∂ ÂM ) . (2) We expect this theory to have high-energy behavior similar to that of an effective 4D KK gauge theory. For instance, consider the elastic gauge-boson scattering, Âaj1Â b j2 → Âcj3Âdj4, where jk ∈ (1, 2, 3) denotes the polarization state of the massless 5D gauge field ÂM . For an SU(m) Yang-Mills theory, we may define the spin-0, gauge-singlet two-particle state, |Ψ0〉 = 1 √ 3 1 √ m2 − 1 3 ∑