# Terminology about chiral supermultiplet and vector supermultiplet

I have an issue with the terminology about chiral supermultiplet and vector supermultiplet.

In the Pierre Ramond's Journeys Beyond The Standard Model p.286, he used the phrase

chiral vector multiplet or chiral vector supermultiplet.

What does it mean exactly? Is that a chiral supermultiplet or a vector supermultiplet? See below the paragraph image.

My understanding about the convention is that:

• spin-1/2 fermions and superpartners form chiral supermultiplet.

• spin-1 vector gauge bosons and superpartners form vector supermultiplet.

• spin-0 Higgs and superpartners form chiral supermultiplet.

Question: So where and how does Pierre Ramond define chiral vector multiplet or chiral vector supermultiplet? Can you clarify whether my list above of supermultiplet is complete or not?

All gauge bosons of the standard model are now part of a gauge multiplet, described (in the Wess-Zumino gauge) by a chiral vector multiplet $$\mathcal{W}$$. As a result, all have their spin one-half gaugino superpartners, distinguished from their spin one partners by a twiddle. We have then, eight gluinos, $$\tilde{\mathbf{g}}$$ one Bino, $$\widetilde{B}^{0},$$ one charged and one neutral Wino, $$\widetilde{W}^{\pm},$$ and $$\widetilde{W}^{0} .$$ Every quark and lepton of the standard model is viewed as the spinor component of a Wess-Zumino chiral superfield, generically denoted by $$\Phi$$, \begin{align*} L \rightarrow \Phi_{L}, \quad \mathbf{Q} \rightarrow \Phi_{\mathbf{Q}}, \quad \overline{\mathbf{u}} \rightarrow \Phi_{\overline{\mathbf{u}}}, \quad \overline{\mathbf{d}} \rightarrow \Phi_{\overline{\mathbf{d}}}, \quad \bar{e} \rightarrow \Phi_{\bar{e}} ; \end{align*}

• Pierre Ramond said "chiral vector multiplet" in his second line of my quoted paragraph from his book. Did you see ? Should gauge bosons be just that of "vector multiplet" not "chiral vector multiplet" as Pierre Ramond said? – ann marie cœur Mar 14 at 18:17
• Also what is the chiral Wess-Zumino multiplet? – ann marie cœur Mar 14 at 18:18
• I honestly have no idea. If he's talking about the MSSM ($\mathcal N = 1$) here, then what you've written is correct: the gauge bosons + gauginos form a vector multiplet, I don't know why he's conferred the moniker "chiral" (unless it has something to do with the imposition of the WZ gauge, but I don't know what that has to do with chirality). Also, "Wess-Zumino multiplet" is a synonym of "chiral matter multiplet", his usage is somewhat redundant. – Nihar Karve Mar 15 at 4:31
• Thanks - What is the relation between Wess-Zumino gauge and "Wess-Zumino multiplet"? – ann marie cœur Mar 18 at 3:09
• Nothing except the name :) The WZ multiplet is just a chiral multiplet, while the WZ gauge is used to kill off most of the degrees of freedom of a vector superfield – Nihar Karve Mar 18 at 4:44

## What is a supermultiplet?

Let's get some of the terminology straight first: a supermultiplet is an irreducible representation of the supersymmetry algebra (in the same sense that a particle is an irrep of the Poincaré algebra). However, SUSY representations furnish reducible Poincaré representations, so supermultiplets in general correspond to multiple particles having the same mass, which are related by supersymmetry transforms. In this context, the broader term "multiplet" is used interchangeably with "supermultiplet".

Specifically, the vector multiplet in the massless $$\mathcal N = 1$$ sector corresponds to a spin-1 gauge boson and its superpartner Weyl fermion, the gaugino, i.e. $$\left(-1, -\frac12\right)\oplus\left(\frac12,1\right)$$ where we have added in the CPT conjugate for invariance. The other multiplets allowed in the $$\mathcal N = 1$$ extension of the Standard Model are the chiral multiplet (for quarks/leptons/Higgsino + squarks/sleptons/Higgs) and if gravity is considered, then the gravitino + gauge boson and graviton + gravitino multiplets are also allowed, provided that they appear together. The hypermultiplet is a different beast - it's the $$\mathcal N = 2$$ SUSY matter multiplet with $$\left(-\frac12,0,0,\frac12\right)$$ + CPT conjugate. However, the SM is chiral, so massless $$\mathcal N = 1$$ is more important to us while constructing the MSSM since we can use the Higgs mechanism to give masses to the particles. In summary, your list of multiplets is essentially complete:

Multiplet Particle Content
Chiral/Matter $$\left(-\frac12,0\right)\oplus\left(0,\frac12\right)$$
Vector/Gauge $$\left(-1,-\frac12\right)\oplus\left(\frac12,1\right)$$
Gravitino $$\left(-\frac32,-1\right)\oplus\left(1,\frac32\right)$$
Gravity $$\left(-2,-\frac32\right)\oplus\left(\frac32,2\right)$$

## What is a superfield?

Analogously to how we "embed particles in fields" in the Standard Model, we seek representations of the SUSY algebra on superfields, fields which are a function of not only spacetime $$x^\mu$$, but also of the Grassman number coordinates $$\theta_\alpha$$ and $$\bar\theta_\dot\alpha$$ (which, among other things, allows us to easily construct supersymmetric Lagrangians). Each superfield must have exactly the number on-shell degrees of freedom as the multiplet it is trying to embed - this is enforced using SUSY-invariant projections of the most general superfield in combination with gauge fixing. Importantly, while a chiral multiplet can be embedded in a "chiral superfield", this is not its only purpose - the chiral superfield can also be used to embed composite operators, serve as a gauge parameter, etc.

Vanilla QFT SUSY
Particle Supermultiplet
Spacetime $$x^\mu$$ Superspace $$\left(x^\mu, \theta_\alpha, \bar\theta_\dot\alpha\right)$$
Field $$\psi^{\{(a)\}}(x^\mu)$$ Superfield $$\Psi^{\{(a)\}}(x^\mu, \theta_\alpha, \bar\theta_\dot\alpha)$$

where $$\{a\}$$ are internal indices.

## Ramond's terminology

The only thing the Wess-Zumino gauge and the Wess-Zumino multiplet have in common is the name. The WZ gauge is a supersymmetry-breaking gauge which is used to set many of the DoF of a vector superfield to zero, while a "Wess-Zumino multiplet" is simply a historical synonym for a chiral/matter multiplet, i.e. $$\left(-\frac12,0\right)\oplus\left(0,\frac12\right)$$. Ramond's usage of "chiral Wess-Zumino multiplet" thus seems redundant or archaic.

As for the term "chiral vector multiplet", it is not mainstream terminology: as I've mentioned, in all $$\mathcal N = 1$$ extensions of the Standard Model, including the MSSM, the gauge boson $$\oplus$$ gaugino multiplet is a vector multiplet, period.

The only time you will ever see the phrase "chiral vector [...]" in basic SUSY$$^\dagger$$ is this: just like gauge fields in QFT that have a field strength tensor, vector superfields $$V$$ in SUSY have a gauge-invariant field strength $$W_\alpha\equiv-\frac14\bar D\bar D D_\alpha V + (\text{non-abelian part})$$, which is actually a chiral superfield. So $$W_\alpha$$ is a "chiral vector multiplet field strength superfield" (not to be confused with the superpotential, which is also denoted by $$W$$).

This object is what must appear in the super-Yang-Mills Lagrangian, and so combined with the sloppy lack of distinction between supermultiplets and superfields, one says "gauge bosons are described by a chiral vector multiplet". A more precise version of this statement is:

Gauge bosons and their superpartner gauginos form a $$\left(-1,-\frac12\right)\oplus\left(\frac12,1\right)$$ multiplet, which is embedded in a vector superfield. This vector superfield has an associated chiral superfield as its gauge-invariant field strength. This object, fully contracted, enters the super-YM Lagrangian, which describes the dynamics of the particles in the multiplet.

$$^\dagger$$The reason I say "basic" SUSY is that "chiral vector multiplets" are actually a thing in $$(2,2)$$ SUSY, and can be interpreted fairly unambiguously in $$\mathcal N = 2$$ SUSY. This is not related to what Ramond is talking about, however.