On the CFT operator spectrum at large global charge

Abstract

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 = Φ 3 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 ∆ J satisfies the sum rule 16 ∆ J+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 ∆ J + √ 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 + 1 2 . The propagation speeds of the Goldstone waves and heavy fermions are 1 √ 2 and ± 1 2 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.

Compensating strong coupling with large charge

“…is the original coupling, Q is the value of the global charge, and a, b are positive exponents. Finally in section five, we go back and generalize to any O(N ) model the computation of anomalous dimensions presented in [1], and provide support for the arguments and conclusions in that paper.…”

“…We find (as expected) that the low energy excitations of this sector are described by the general form of Goldstone's theorem in the non-relativistic regime. We also derive the unexpected result, first presented in [1], that the effective field theory describing such sector of fixed Q contains effective couplings λ eff ∼ λ b /Q a , where λ is the original coupling. Hence, large charge leads to weak coupling.…”

“…1 The aim of this paper is to show that when we consider quantum field theories with global symmetries, and we study the sector of the Hilbert space of states with large values of the global charge, we find not only that generically there are Goldstone excitations in the effective Lagrangian describing that sector of the theory, but furthermore, it seems that the effective couplings are related to the original couplings but suppressed by powers of the global charge. Hence the larger the charge, the weaker the coupling and thus the more reliable the results obtained in perturbation theory.…”

“…The slightly provocative title of this paper refers to the very intriguing results presented in [1] where the anomalous dimensions of operators with large global charge J in certain conformal field theories (cfts) in three-dimensions were obtained. In most cfts, the description in terms of local Lagrangians is not adequate, because the theory is frequently strongly coupled, and thus, the perturbative description is not valid.…”

Distinguishing a Higgs-like dilaton scenario with a complete bosonic effective field theory basis

“…Dilaton models are based on the fact that the SM Lagrangian is approximately scale invariant, once neglecting explicit scales associated to the EWSB mechanism and the dynamical conformal breaking due to QCD. In models where the scale invariance is broken spontaneously, a GB naturally arises [47,48] and can then be identified with the physical Higgs field: its mass is then protected by the GB shift symmetry and can only acquire (relatively small) values, close to the symmetry breaking scale.…”