On the CFT operator spectrum at large global charge

Hellerman, Simeon; Orlando, Domenico; Reffert, Susanne; Watanabe, Masataka (2015). On the CFT operator spectrum at large global charge. Journal of High Energy Physics, 2015(12), pp. 1-34. Springer 10.1007/JHEP12(2015)071

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We calculate the anomalous dimensions of operators with large global charge J in certain strongly coupled conformal field theories in three dimensions, such as the O(2) model and the supersymmetric fixed point with a single chiral superfield and a W = Φ3 superpotential. Working in a 1/J expansion, we find that the large-J sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge J is always a scalar operator whose dimension ΔJ satisfies the sum rule
J2ΔJ−(J22+J4+316)ΔJ−1−(J22+J4+316)ΔJ+1=0.04067
up to corrections that vanish at large J . The spectrum of low-lying excited states is also calculable explcitly: for example, the second-lowest primary operator has spin two and dimension ΔJ+3√. In the supersymmetric case, the dimensions of all half-integer-spin operators lie above the dimensions of the integer-spin operators by a gap of order J+12. The propagation speeds of the Goldstone waves and heavy fermions are 12√ and ±12 times the speed of light, respectively. These values, including the negative one, are necessary for the consistent realization of the superconformal symmetry at large J.

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