Musings

The starting point is the Ferrara-Zumino multiplet, a real vector-valued superfield, $\mathcal{J}_{\alpha\dot{\alpha}}$, which satisfies $$\overline{D}^{\dot{\alpha}}\mathcal{J}_{\alpha\dot{\alpha}} = D_\alpha X$$ where $X$ is a chiral superfield. The Ferrara-Zumino multiplet contains the supercurrent, the stress tensor, and the R-current (which, depending on the model, may be explicitly non-conserved). But these field form reducible multiplets of the Lorentz group, and the chiral field $X$ contains $$X = x + \frac{2}{3}\theta^\alpha \sigma^\mu_{\alpha\dot{\alpha}} S^{\dagger \dot{\alpha}}_\mu + \theta^2(\frac{2}{3} \tensor{T}{_\mu_^\mu} + i \partial^\mu j_\mu)$$ where $\partial^\mu j_\mu$ is the divergence of the R-current. When $X=0$, the theory is not merely supersymmetric, it is superconformal.

But we are interested in the case where supersymmetry is broken. In that case, the $\theta$ component of $X$ is $f \psi$, where $f$ is the Goldstino decay constant and $\psi$ is an interpolating field for the Goldstino (not quite the same field as in Akulov-Volkov, but they agree to linear order). It’s convenient to define the canonically-normalized field, $X_{NL}= X/f$.

Nati’s prescription for the nonlinear realization is to write down the most general superspace Lagrangian you can, in terms of $X_{NL}$ and $X_{NL}^\dagger$ and then impose the constraint $$X_{NL}^2 = 0$$ A-priori, the lowest component of $X_{NL}$ was an independent complex scalar field. This constraint kills that mode, leaving only the Goldstone fermion (and an auxiliary field). The solution is $$X_{NL} = \frac{\psi\psi}{2F} +\sqrt{2} \theta \psi + \theta^2 F$$

The lowest order Lagrangian is

Integrating out the auxiliary field, $F$, $$F = -f + \dots$$ and we obtain the Akulov-Volkov Lagrangian, with the expected vacuum energy density, $f^2$.

Note that this lowest order Lagrangian has an accidental conserved R-symmetry. Not every microscopic Lagrangian has such an R-symmetry. Thus the Akulov-Volkov Lagrangian must have higher-order corrections which explicitly violate the R-symmetry. These exist, but are very painful to construct by the methods of Akulov and Volkov. In Nati’s approach, it’s easy to write down higher-order corrections to (1) which do not respect the R-symmetry. For instance, $$\int d^2\theta {(\partial X_{NL})}^2 + \text{h.c.}$$

When you want to couple to matter, just couple these superfield to ordinary matter superfield. Essentially, it’s the same method used in the spurion approach to writing down the soft supersymmetry-breaking terms in the action, with “$X_{NL}$” playing the role of the spurion. Because it has a nonzero $F$-component, it is the spurion, but it also contains couplings to the Goldstino.

There’s even a formalism to write down the effective Lagrangian below the mass scale of the superpartners, thus capturing the coupling of the light fields to the Goldstino. In theories, such as gauge mediation, where the gravitino is the lightest superpartner, these Goldstino couplings give the leading term in the coupling of gravitinos to ordinary matter (at energies well above the gravitino mass, but well below the masses of the other superpartners).

For instance, for ordinary quarks and leptons, denoted by a chiral superfield, $Q$, take the constrained field to be $Q_{NL}$, where $$X_{NL} Q_{NL}=0$$ This removed the scalar superpartner, keeping the fermion. If, instead, you want to keep the complex scalar (e.g., the Higgs), and integrate out its superpartner, take $$X_{NL} H_{NL}^\dagger = \text{chiral}$$ For a gauge field, take $$X_{NL} W_{\alpha NL} = 0$$ Real Goldstone bosons and R-axions satisfy $$X_{NL} (A_{NL} - A_{NL}^\dagger) = 0$$ where, for an R-axion, you need to ensure that the shift symmetry of $A$ is realized as an R-symmetry, by adding to the action terms like $$\int d^4 \theta\, A_{NL}^\dagger A_{NL} + \int d^2\theta\, \tilde{f} e^{2i A_{NL}} + \text{h.c.}$$ If you look in the literature, there’s considerable confusion on how to do this example correctly. But, in this formalism, it is more or less trivial.

Update: Fayet-Iliopoulos terms

This whole story connects up nicely with the issue of whether there is D-term breaking (i.e., the Fayet-Iliopoulos mechanism) in supergravity. In our paper on gauge mediation, Dan and I sweated a bit over the section on D-term breaking. The analysis wasn’t complicated, but the result was ugly. Fortunately, according to Nati and Zohar, it’s also superfluous.

For the Fayet-Iliopoulos model, $$L = \frac{1}{4g^2} \int d^2\theta\, W_\alpha^2 + \text{h.c.} + \int d^4\theta\, \xi V$$ the Ferrara-Zumino multiplet $$\begin{gathered} \mathcal{J}_{\alpha\dot{\alpha}} = -\frac{4}{3} W_\alpha W^\dagger_{\dot{\alpha}} -\frac{2}{3} \xi [D_\alpha,D_{\dot{\alpha}}] V\\ \xi = -\frac{xi}{3} \overline{D}^2 V \end{gathered}$$ isn’t gauge-invariant. Indeed, that’s just symptomatic of the fact that integrand of the Fayet-Iliopulos term isn’t gauge-invariant (though the integral is).

This isn’t really a problem in global supersymmetry, but does vastly complicate the coupling to supergravity. The upshot of a rather complicated literature is that the Fayet-Iliopoulos model can be coupled to supergravity iff

1. The global theory has an exact $U(1)$ R-symmetry.
2. The $U(1)$ that is gauged in the supergravity theory is a certain linear combination of the original $U(1)$ and the $U(1)_R$, such that the charges of the fields are shifted $$q_i \to q_i + r_i \xi/{M_{\text{PL}}}^2$$where $r_i$ is the R-charge.
3. In particular, this means that the gravitino is charged, so we are doing gauged supergravity.
4. When the dust settles, there remains an exact global $U(1)$ (which you can, at your pleasure, take to be an R-symmetry, or not.
5. Since sensible theories of quantum gravity can’t have exact continuous global symmetries, this possibility is never actually realized.

Note that this does not exclude so-called “field-dependent” FI terms. For instance, if the $U(1)$ is everywhere Higgsed (by a chiral field, $\Phi$, of charge $q$), we can write a term like $$\int d^4 \theta\, \xi (V + \tfrac{1}{q} \log {|\Phi|}^2)$$ This is gauge-invariant, and makes perfect sense, as long as the point $\Phi=0$ (where the $U(1)$ gauge symmetry would be restored) is infinitely far away in field space. But, in that case, there’s no sharply-defined meaning to the FI coefficient, $\xi$. As another example, if $\Phi$ is a neutral chiral field, with a non-vanishing F-component (breaking supersymmetry), we can write $$\int d^4\theta\, {|\Phi|}^2 D^\alpha W_\alpha$$ Again, with this “field-dependent” FI term, the integrand is perfectly gauge-invariant, and there’s no obstruction to coupling to supergravity. It’s also a theory with “F-type”, rather than “D-type” supersymmetry breaking.