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$\beta$-function of the level-zero Gross-Neveu model

by Dmitri Bykov

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Submission summary

Authors (as registered SciPost users): Dmitri Bykov
Submission information
Preprint Link: scipost_202303_00022v1  (pdf)
Date accepted: 2023-04-18
Date submitted: 2023-03-20 02:05
Submitted by: Bykov, Dmitri
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We explain that the supersymmetric $CP^{n-1}$ sigma model is directly related to the level-zero chiral Gross-Neveu (cGN) model. In particular, beta functions of the two theories should coincide. This is consistent with the one-loop-exactness of the $CP^{n-1}$ beta function and a conjectured all-loop beta function of cGN models. We perform an explicit four-loop calculation on the cGN side and discuss the renormalization scheme dependence that arises.

Author comments upon resubmission

I would like to thank the referees for a careful reading of the manuscript.

Below I will try to respond to the questions raised in Report 1:

  • The MOM scheme is substantially more complicated than $\bar{MS}$. This can be seen on the example of the QCD beta function at three loops: the $\bar{MS}$ expression can be found in

T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, ``The Four loop beta function in quantum chromodynamics,'' Phys. Lett. B \textbf{400} (1997), 379-384

whereas the MOM-scheme expression is given in

J.A. Gracey, ``Three loop QCD MOM beta-functions,'' Phys. Lett. B \textbf{700} (2011), 79-85

Already from this example it is clear that transcendentality is likely to grow tremendously at higher loops in the MOM scheme.

The example of N=1 SQED in 4D was given for a similar reason: here one has expressions for the beta function in both schemes, and one can compare between the two. On the other hand, I am not aware of any 4-loop calculations in 4D gauge theories in the MOM scheme.

  • Regarding bow varieties, perhaps such generalizations could be possible but this goes far beyond the scope of the present paper.

List of changes

Minor changes have been implemented, mostly in an attempt to answer other points raised by the referee:

- A comment on the higher derivative regularization has been added, explaining the various complications that might arise as well as the potential resolutions of these issues.

- Regarding the Gross-Neveu (GN) formulation of (0,2) theories, at present this has not been worked out in concrete terms. Nevertheless, in my recent work with A. Smilga I considered the N=2 SUSY quantum-mechanical sigma model with target space CP^{n-1}, which may be seen as the dimensional reduction of a (0,2) theory. In this context we also discussed the potential GN formulation of (0,2) theories, whose structure is partially visible in the reduction. I have thus inserted a reference to this paper.

- Two references to the work of J.Honerkamp have been inserted in the historical section, since these seem to be the first papers where the one-loop beta functions of sigma models were calculated.

Published as SciPost Phys. 15, 127 (2023)

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