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$\beta$function of the levelzero GrossNeveu model
by Dmitri Bykov
Submission summary
Authors (as registered SciPost users):  Dmitri Bykov 
Submission information  

Preprint Link:  scipost_202303_00022v1 (pdf) 
Date accepted:  20230418 
Date submitted:  20230320 02:05 
Submitted by:  Bykov, Dmitri 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We explain that the supersymmetric $CP^{n1}$ sigma model is directly related to the levelzero chiral GrossNeveu (cGN) model. In particular, beta functions of the two theories should coincide. This is consistent with the oneloopexactness of the $CP^{n1}$ beta function and a conjectured allloop beta function of cGN models. We perform an explicit fourloop calculation on the cGN side and discuss the renormalization scheme dependence that arises.
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
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), 379384
whereas the MOMscheme expression is given in
J.A. Gracey, ``Three loop QCD MOM betafunctions,'' Phys. Lett. B \textbf{700} (2011), 7985
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 4loop 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 GrossNeveu (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 quantummechanical sigma model with target space CP^{n1}, 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 et.al. have been inserted in the historical section, since these seem to be the first papers where the oneloop beta functions of sigma models were calculated.