Musings

We divide the chiral multiplets of the theory into the visible sector fields, $C^{\alpha n}$, which are charged under the Standard Model gauge group and the moduli, which are neutral. Here, $\alpha$ runs over irreps of $SU(3)\times SU(2)\times U(1)$ and $n$ is a flavour index; for brevity, I will sometimes denote the pair $(\alpha,n)$ by $A$. The moduli are further divided into two sets, $\Psi_i$ and $\Phi_i$.

• The Kähler potential for the moduli is assumed to take the form $$\begin{aligned} K &= \hat{K} + K_{\text{matter}}\\ \hat{K} &= K_1(\Psi+\overline{\Psi}) + K_2(\Phi,\overline{\Phi}) \end{aligned}$$
• The gauge kinetic functions are linear functions of the $\Psi$s $$f_a(\Psi) = \sum_i \lambda_{a i} \Psi_i$$

Together, these two statements amount to saying that the $Im \Psi_i$ are axions, and that, at least on the level of the Kähler potential, the Peccei-Quinn symmetry is unbroken by the VEVs of the $\Phi_i$.

• The matter superpotential is independent of the $\Psi_i$ $$W = \hat{W}(\Psi,\Phi) + {\mu(\Phi)}_{A B} C^A C^B + {Y(\Phi)}_{A B C}C^A C^B C^C + \dots$$

• The matter Kähler potential $$K_{\text{matter}} = h_{\alpha \overline{\alpha}}(\Psi+\overline{\Psi}) k_{m n}(\Phi,\overline{\Phi}) C^{\alpha m} \overline{C^{\alpha n}} + (Z_{A B}(\Psi,\overline{\Psi},\Phi,\overline{\Phi}) C^A C^B + h.c.) + \dots$$ also leads to a flavour-diagonal Kähler metric for the quarks and leptons which, moreover, respects the Peccei-Quinn symmetry.

• The scalar potential for the moduli $$\hat{V} = e^{\hat{K}}\left(\hat{K}^{i\overline{\jmath}} D_i \hat{W} \overline{D_j \hat{W}}-3 |\hat{W}|^2\right)$$ is minimized with $$D_{\Phi_i} W = 0,\qquad D_{\Psi_i} W \neq 0$$ where $D$ is the Kähler covariant derivative, $D_{\Phi_i} W= \partial_{\Phi_i} W + \partial_{\Phi_i} K W$.

That is, supersymmetry-breaking takes place in the $\Psi_i$ sector, but their couplings to the visible sector are flavour-blind. The couplings of the $\Phi_i$ to the visible sector have nontrivial flavour structure, but the $\Phi_i$ are stabilized “supersymmetrically.”

This structure of the supergravity Lagrangian was cooked up to solve the FCNC and CP problems. As such, from the low-energy field theorist’s perspective, it seems rather ad-hoc. However, as Conlon points out, it pops out quite naturally in certain classes of string compactifications.

In both Type IIA and Type IIB, the moduli come in two type: complex structure and Kähler moduli. And, at large radius, the moduli space factorizes, in pretty much this fashion and, on one side or the other, one has a a Peccei-Quinn symmetry. Depending on the details, however, this structure may not be preserved. In the presence of D3-brane moduli (for instance), the Peccei-Quinn symmetry of the Kähler potential, in IIB, gets messed up. So not all of the popular scenarios for moduli stabilization will be compatible with mirror mediation.

But, it’s tantalizing that, at least in some circumstances, this structure pops out “for free.”