Some questions on a version of the O'Raifeartaigh model

Question

This form is taken from a talk by Seiberg to which I was listening to,

Take the Kahler potential ($K$) and the supersymmetric potential ($W$) as,

$K = \vert X\vert ^2 + \vert \phi _1 \vert ^2 + \vert \phi_2\vert ^2$

$W = fX + m\phi_1 \phi_2 + \frac{h}{2}X\phi_1 ^2$

• This notation looks a bit confusing to me. Are the fields $X$, $\phi_1$ and $\phi_2$ real or complex? The form of $K$ seems to suggest that they are complex - since I would be inclined to read $\vert \psi \vert ^2$ as $\psi ^* \psi$ - but then the form of $W$ looks misleading - it seems that $W$ could be complex. Is that okay?

Now he looks at the potential $V$ defined as $V = \frac{\partial ^2 K}{\partial \psi_m \partial \psi_n} \left ( \frac {\partial W}{\partial \psi_m} \right )^* \frac {\partial W}{\partial \psi_n}$

(..where $\psi_n$ and $\psi_m$ sums over all fields in the theory..)

For this case this will give, $V = \vert \frac{h}{2}\phi_1^2 + f\vert ^2 + \vert m\phi_1 \vert ^2 + \vert hX\phi_1 + m\phi_2 \vert ^2$

• Though for the last term Seiberg seemed to have a "-" sign as $\vert hX\phi_1 - m\phi_2 \vert ^2$ - which I could not understand.

I think the first point he was making is that it is clear by looking at the above expression for $V$ that it can't go to $0$ anywhere and hence supersymmetry is not broken at any value of the fields.

• I would like to hear of some discussion as to why this particular function $V$ is important for the analysis - after all this is one among several terms that will appear in the Lagrangian with this Kahler potential and the supersymmetry potential.

• He seemed to say that if *``$\phi_1$ and $\phi_2$ are integrated out then in terms of the massless field $X$ the potential is just $f^2$"* - I would be glad if someone can elaborate the calculation that he is referring to - I would naively think that in the limit of $h$ and $m$ going to $0$ the potential is looking like just $f^2$.

• With reference to the above case when the potential is just $f^2$ he seemed to be referring to the case when $\phi_2 = -\frac{hX\phi_1}{m}$. I could not get the significance of this. The equations of motion from this $V$ are clearly much more complicated.

• He said that one can work out the spectrum of the field theory by "diagonalizing the small fluctuations" - what did he mean? Was he meaning to drop all terms cubic or higher in the fields $\phi_1, \phi_2, X$ ? In this what would the "mass matrix" be defined as?

The confusion arises because of the initial doubt about whether the fields are real or complex. It seems that $V$ will have terms like $\phi^*\phi^*$ and $\phi \phi$ and also a constant term $f^2$ - these features are confusing me as to what diagonalizing will mean.

Normally with complex fields say $\psi_i$ the "mass-matrix" would be defined the $M$ in the terms $\psi_i ^* M_{ij}\psi_j$ But here I can't see that structure!

• The point he wanted to make is that once the mass-matrix is diagonalized it will have the same number of bosonic and fermionic masses and also the super-trace of its square will be $0$ - I can't see from where will fermionic masses come here!

• If the mass-matrix is $M$ then he seemed to claim - almost magically out of the top of his hat! - that the 1-loop effective action is $\frac{1}{64\pi^2} STr \left ( M^4 log \frac{M^2}{M_{cut_off}^2} \right )$ - he seemed to be saying that it follows from something else and he didn't need to do any loop calculation for that!

I would be glad if someone can help with these.

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Comments

A brief comment on your first question: the superpotential is a holomorphic function, hence complex unless (locally) constant. It appears in the lagrangian density as the real part of its integral over "half the superspace". Chiral superfields are therefore complex: indeed they define holomorphic coordinates on a Kähler manifold.

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Indeed, all the fields are complex. The rest is more or less standard in QFT, except for the minus sign error, which you got right.

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Jose - Thanks for the explanation. @Zohar Ko I can understand that in principle this is standard QFT but I can't exactly do what he seems to say should be done. Like in what sense is the potential just $f^2$ and how is the mass-matrix (with fermionic masses!) going to emerge here. I have done this kind of calculation elsewhere but here things don't seem to fit in. And the last expression for the 1-loop effective action looks as mysterious. It would be great if you can sketch out the calculation.

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The mass matrix of the bosons and fermions coincides if $f=0$. Hence, in the super-trace over M^4\log M^2 there would be perfect cancelation if $f=0$. The answer is nonzero if $f\neq0$ and it scales like f^2.

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@Zohar Can you give a reference for such an analysis about the supertrace of the mass-matrix of a theory?

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I believe that this is what Mr. O'R did in his original paper... The foundations of this approach were of course developed by Coleman-Weinberg, which is a very nice paper.

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Answer 1

I found Stephen P. Martin's review very useful for this analysis.

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Comments

@Tail, thanks for answering, but please provide more information (e.g. link to the review you refer to, and at least a sentence what is there).

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@Piotr Migdal I guess the review being referred to is this one. http://arxiv.org/pdf/hep-ph/9709356v6.pdf But I don't see this STr there which I have quoted. It would be great if Tali can point out where this is there in that large review.

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