## October 9, 2007

### Mirror Mediation

I’ve been planning to write some things about recent developments in string phenomenology. Unfortunately, the list of papers I want to talk about seems to grow faster than my ability to keep pace. So, rather than being as systematic as I would like, I’ll just plunge in and talk about a recent paper by Joseph Conlon. The subject is supersymmetry-breaking, and a mechanism he calls “mirror-mediation.”

First-off, I have to say that I hate the nomenclature of this sub-field, where a profusion of fanciful (and, usually, not terribly descriptive) names, of the form …-mediation, are attached to various restrictions on the form of the supergravity Lagrangian. I realize it’s hard to stop, once such a convention is established, but I think it tends to obscure more than it illuminates.

Anyway, the key problems that any of these mediation mechanisms needs to solve is to somehow avoid that the soft SUSY-breaking terms induced for the MSSM fields lead to flavour-changing neutral currents and/or large CP-violation.

Conlon points to a mechanism which might look a little ad-hoc, from the perspective of low energy effective field theory, but which is quite natural in certain classes of string vacua.

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.”

Posted by distler at October 9, 2007 11:51 AM

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### Re: Mirror Mediation

Hi Jacques,

Speculation:

Rather than symmetry breaking, could symmetry be bent or folded in a manner that might give an illusion of breaking?

Posted by: Doug on October 9, 2007 9:59 PM | Permalink | Reply to this

### Re: Mirror Mediation

Is it obvious that in the typical string theory construction either the Kaehler or the complex structure moduli would couple to the gauge fields in a flavour blind way? Probably depends on how you realise the flavour symmetry but for example in intersecting D-brane models where the families arise as multiple brane intersection (multiple intersection numbers of the cycles the branes are wrapped).

Posted by: Robert on October 10, 2007 9:09 AM | Permalink | Reply to this

### Gauge kinetic functions

I’m not quite sure what that question means. There’s no flavour-structure to the gauge kinetic functions, $f_a$. Maybe what you meant to ask is: why should the gauge kinetic functions depend on just one type of moduli, $f_a(\Psi)$, instead of on both, $f_a(\Psi,\Phi)$?

The answer, in most cases, is that the gauge kinetic functions depend on the volumes of some cycles on which there are wrapped branes, and these volumes (depending on whether we are in IIA or IIB) are controlled by the complex structure or the Kähler moduli, respectively.

Posted by: Jacques Distler on October 10, 2007 9:30 AM | Permalink | PGP Sig | Reply to this