# Can we have supersymmetry using real scalar instead of complex scalars?

We will have to shut down our server temporarily for maintenance. The downtime will start at Wednesday, 27. January 2021 at 12:00 GMT and have a duration of about two hours. Please save your edits before this time. Thanks for your patience and your understanding.

+ 2 like - 0 dislike
227 views

I am aware that a suersymmetric theories containing a complex scalar a Weyl fermion and an auxiliary field exist.

I was wondering if we can have something analogous using real and not complex scalar fields, and some kind of real Weyl fermions (if such a thing does indeed exist).

+ 3 like - 0 dislike

A spinor satisfying a reality condition is a Majorana spinor and the question is about the existence of a Majorana-Weyl spinor. But in four spacetime dimensions (in Lorentzian signature), no Majorana-Weyl spinor exists, the Weyl spinor is the smallest and so the minimal supersymmetry in four dimensions has four real supercharges and the scalar superpartner of a Weyl spinor has to be a complex scalar.

It is possible to find something analogous in three spacetime dimensions. In this case, the smallest spinor is a Majorana spinor, with two real components, and so the minimal supersymmetry in three dimensions has two real supercharges (half the number of the minimal supersymmetry in four dimensions) and the scalar superpartner of a Majorana spinor in three dimensions is a real scalar.

answered Aug 15, 2015 by (5,120 points)

A hands-on (but not rigorous and definitive) approach would be to investigate the supermultiplets (irreducible representations) arising in the $N=1$ superfield. There, the chiral and antichiral superfields completely cover the scalar+fermion degree of freedom and what is left is only the vector+bispinor degrees of freedom (which is what you get from the "real superfield" under the Wess-Zumino gauge). I am not sure, however, whether it is not possible to get a longitudinal component of the vector field ($\sim$ spin 0 because $\partial_\mu \phi \leftrightarrow A_\mu$) from an appropriately constructed Lagrangian of the "real superfield". @silvrfuck

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysic$\varnothing$OverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.