We present a common solution to the puzzles of the light Higgs or quark masses and the need for a shift symmetry and large field values in high scale chaotic inflation. One way to protect, for example, the Higgs from a large supersymmetric mass term is if it is the Nambu-Goldstone boson (NGB) of a nonlinear sigma model. However, it is well known that nonlinear sigma models (NLSMs) with nontrivial Kähler transformations are problematic to couple to supergravity. An additional field is necessary to make the Kähler potential of the NLSM invariant in supergravity. This field must have a shift symmetry - making it a candidate for the inflaton (or axion). We give an explicit example of such a model for the coset space SU(3)/SU(2)×U(1), with the Higgs as the NGB, including breaking the inflaton's shift symmetry and producing a chaotic inflation potential. This construction can also be applied to other models, such as one based on E7/SO(10)×U(1)×U(1) which incorporates the first two generations of (light) quarks as the Nambu-Goldstone multiplets, and has an axion in addition to the inflaton. Along the way we clarify and connect previous work on understanding NLSMs in supergravity and the origin of the extra field (which is the inflaton here), including a connection to Witten-Bagger quantization. This framework has wide applications to model building; a light particle from a NLSM requires, in supergravity, exactly the structure for chaotic inflaton or an axion.
Hellerman, S., Kehayias, J., & Yanagida, T. T. (2015). Chaotic inflation from nonlinear sigma models in supergravity. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 742, 390–393. https://doi.org/10.1016/j.physletb.2015.02.019