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Constraints from precision electroweak measurements reveal no evidence for new physics up to 5 -7 TeV, whereas naturalness requires new particles at around 1 TeV to address the stability of the electroweak scale. We show that this "little hierarchy problem" can be cured by introducing a symmetry for new particles at the TeV scale. As an example, we construct a little Higgs model with this new symmetry, dubbed T -parity, which naturally solves the little hierarchy problem and, at the same time, stabilize the electroweak scale up to 10 TeV. The model has many important phenomenological consequences, including consistency with the precision data without any finetuning, a stable weakly-interacting particle as the dark matter candidate, as well as collider signals completely different from existing little Higgs models, but rather similar to the supersymmetric theories with conserved R-parity.

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Goldstone's theorem states that there is a massless mode for each broken symmetry generator. It has been known for a long time that the naive generalization of this counting fails to give the correct number of massless modes for spontaneously broken spacetime symmetries. We explain how to get the right count of massless modes in the general case, and discuss examples involving spontaneously broken Poincaré and conformal invariance.

Little Higgs theories are an attempt to address the "little hierarchy problem," i.e., the tension between the naturalness of the electroweak scale and the precision electroweak measurements showing no evidence for new physics up to 5 -10 TeV. In little Higgs theories, the Higgs mass-squareds are protected at one-loop order from the quadratic divergences. This allows the cutoff of the theory to be raised up to ∼ 10 TeV, beyond the scales probed by the current precision data. However, strong constraints can still arise from the contributions of the new TeV scale particles which cancel the one-loop quadratic divergences from the standard model fields, and hence re-introduces the fine-tuning problem. In this paper we show that a new symmetry, denoted as T -parity, under which all heavy gauge bosons and scalar triplets are odd, can remove all the tree-level contributions to the electroweak observables and therefore makes the little Higgs theories completely natural. The T -parity can be manifestly implemented in a majority of little Higgs models by following the most general construction of the low energy effective theoryà la Callan, Coleman, Wess and Zumino. In particular, we discuss in detail how to implement the T -parity in the littlest Higgs model based on SU (5)/SO(5). The symmetry breaking scale f can be even lower than 500 GeV if the contributions from the higher dimensional operators due to the unknown UV physics at the cutoff are somewhat small. The existence of T -parity has drastic impacts on the phenomenology of the little Higgs theories. The T -odd particles need to be pair-produced and will cascade down to the lightest T -odd particle (LTP) which is stable. A neutral LTP gives rise to missing energy signals at the colliders which can mimic supersymmetry. It can also serve as a good dark matter candidate.

We construct an SU (6)/Sp(6) non-linear sigma model in which the Higgses arise as pseudoGoldstone bosons. There are two Higgs doublets whose masses have no one-loop quadratic sensitivity to the cutoff of the effective theory, which can be at around 10 TeV. The Higgs potential is generated by gauge and Yukawa interactions, and is distinctly different from that of the minimal supersymmetric standard model. At the TeV scale, the new bosonic degrees of freedom are a single neutral complex scalar and a second copy of SU (2) × U (1) gauge bosons. Additional vector-like pairs of colored fermions are also present.

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