Exponential suppression of the cosmological constant in nonsupersymmetric string vacua at two loops and beyond

Two independent criteria are presented that together guarantee exponential suppression of the two-loop cosmological constant in nonsupersymmetric heterotic strings. They are derived by performing calculations in both the full string theory and in its effective field theory, and come respectively from contributions that involve only physical untwisted states, and contributions that include orbifold twisted states. The criteria depend purely on the spectrum and charges, so a model that satisfies them will do so with no fine-tuning. An additional consistency condition (emerging from the so-called separating degeneration limit of the two-loop diagram) is that the one-loop cosmological constant must also be suppressed, by Bose-Fermi degeneracy in the massless spectrum. We comment on the effects of the residual exponentially suppressed one-loop dilaton tadpole, with the conclusion that the remaining instability would be under perturbative control in a generic phenomenological construction. We remark that theories of this kind, that have continued exponential suppression to higher orders, can form the basis for a string implementation of the “naturalness without supersymmetry” idea.

Figure 1

The spectrum of theories that satisfy Bose-Fermi degeneracy with a standard model-like light sector (reproduced from [1]). As the standard model does not have Bose-Fermi degeneracy a cancelling hidden sector is inevitable, but note there is no supersymmetry in the spectrum. Models off this kind were constructed in [1].

Figure 2

Canonical homology basis for genus 2.

Figure 3

Generic sunset diagram for the two-point function.

Figure 4

The Feynman diagrams for the two-loop cosmological constant in the effective $\mathcal{N}=2$ field theory of the untwisted sector with dashed lines indicating scalar components of hypermultiplets, solid lines fermionic components. Likewise “photon” lines represent the bosonic component of the gauge supermultiplet (i.e., vector plus scalar adjoint), while the gaugino lines represent the $\mathcal{N}=2$ gauginos. Leading order corrections (i.e., not exponentially suppressed) contributions are proportional to the sum over all these coefficients in the entire theory. In a supersymmetric theory the contributions vanish line by line as they should. In a Scherk-Schwarzed theory, only those diagrams with all masses unshifted count (twice) towards the cosmological constant. Cancellation in a nonsupersymmetric theory can be achieved by choosing field content.

Figure 5

The separating degeneration limit.