What is meant by the phrase "this operator does not renormalize this other operator", and how can one understand it using diagrammatic arguments?

Question

I am trying to understand some sentences in a paper http://arxiv.org/pdf/1412.7151v2.pdf. In section two  the following theory of a (complex) massless scalar coupled to a $U(1)$ gauge boson is introduced
$$\cal{L}_4=-|D_{\mu}\phi|^2-\lambda_{\phi}|\phi|^4-\frac{1}{4g^2}F_{\mu\nu}^2\qquad{}\cal{L}_6=\frac{1}{\Lambda^2}[c_rO_r+c_6O_6+c_{FF}O_{FF}]$$
where $\Lambda$ is an energy scale suppresing the dimension 6 operators and
$$\cal{O}_r=|\phi|^2|D_{\mu}\phi|^2\qquad{}O_6=|\phi|^6\qquad{}O_{FF}=|\phi|^2F_{\mu\nu}F^{\mu\nu}$$
What I want to understand is what is meant in the last paragraph of the same page 

>"Many of the one-loop non-renormalization results that we discuss can be understood from arguments based on the Lorentz structure of the vertices involved. Take for instance thenon-renormalization of $\cal{O}_{FF}$ by $\cal{O}_R$."

my first question is, what is exactly meant when they say that an operator doesn't renormalize the other? ( I somehow suspect this has something to do with the renormalization group but since my knowledge on this matter is very recent I would like as explicit an explanation as possible)

the paragraph continues

>"Integrating by parts and using the EOM, we can eliminate $\cal{O}_r$ in favor of $\cal{O}'_r=(\phi{}D_{\mu}\phi^*)^2+h.c..$ Now it is apparent that $\cal{O}'_r$ cannot renormalize $\cal{O}_{FF}$ because either $\phi{}D_{\mu}\phi^*$ or $\phi^*{}D_{\mu}\phi$ is external in all one-loop diagrams, and these Lorentz structures cannot be completed to form $\cal{O}_{FF}$."

This whole part confuses me. I want to know how do these diagrammatic arguments arise in this context and how can I learn to use them (it would be nice also if someone pointed out which are "all one-loop diagrams" that are mentioned").

Comments

This paper: http://arxiv.org/abs/1505.01844, is more worthy of study on these issues.

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The paper cited in the comment of anonymous cites the paper in the OP with the comment "Interestingly, the superfield formalism offers an enlightening albeit partial explanation of these cancellations [16]".

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The complete reference to the previous work is in fact this:

"Interestingly, the superfield formalism offers an enlightening albeit partial explanation of these cancellations [16] as well as analogous effects in chiral perturbation theory [17]. These results are clearly connected to our own via the well-known “effective” supersymmetry of tree-level QCD [18], and so merits further study."

As Savas Dimopoulos likes to say: "don't be a hater".

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For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled. --

--Richard P. Feynman
 

http://arxiv.org/abs/1505.01844  has the title "Non-renormalization Theorems without Supersymmetry"

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Precisely so: cancellations without supersymmetric spectrum. That's the point. And still supersymmetry helps your understanding, same way it helped with spinor-helicity cancellations in non-supersymmetric QCD. Facts, not PR.

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Certainly there is no PR here. None whatsoever.

Another point of view is that supersymmetry supplies a way to misunderstand the actual physics at work quite profoundly.

Here is another fact. A relevant one. The word "helicity" never appears in http://arxiv.org/pdf/1412.7151.pdf.

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Answers turned into comments since they dont answer the question.