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I was reading the article "Dynamical Supersymmetry Breaking" by Yael Shadmi and Yuri Shirman. In particular i was studing the relation between the presence of a global continus spontaneusly broken symmetry and SUSY breaking. All it's clear, except for this statement (on page 28-29):

"If the global symmetry is spontaneously broken, there is a massless scalar field, the Goldstone boson, with no potential. With unbroken supersymmetry, the Goldstone boson is part of a chiral supermultiplet that contains an additional massless scalar, again with no potential. ... "

I don't get who is the scalar companion of the Goldstone boson: if i think of the simple N=1 susy the massless chiral supermultiplet contains only one scalar and one Weyl Fermion.

Any suggestion?

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The Goldstone boson of a spontaneously global symmetry is a real scalar. This is in contrast to the complex scalar that is part of a chiral supermultiplet.

When a global symmetry is broken in the supersymmetric limit, the real scalar Goldstone boson has another real scalar partner so that they can together form a complex scalar. As you note, there is also a Weyl fermion.

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  • $\begingroup$ Sorry, but where does the Weyl fermion come from? Moreover, it's mentioned in Bertolini's notes on SUSY that the second scalar creates a flat direction by changing the value of the VEV. Can you please elaborate on your answer? $\endgroup$ Sep 11, 2022 at 20:22
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    $\begingroup$ The complex scalar is composed of the Goldstone and its real scalar partner. Think of this as the Goldstone direction and the Higgs/radial direction of complex scalar whose vev breaks a U(1) symmetry. The scalar partner is protected from having a mass because the Goldstone is protected from having a mass. In ordinary global symmetry breaking, the mass of the radial mode stabilizes the vev to take a particular value. Because the radial scalar does not have a mass, its potential is flat and it can take any vev. The Weyl fermion is the fermionic partner of this complex scalar. $\endgroup$ Feb 26 at 4:49

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