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

Preprint Link:  https://arxiv.org/abs/2209.10502v1 (pdf) 
Date submitted:  20220927 16:04 
Submitted by:  Bykov, Dmitri 
Submitted to:  SciPost Physics 
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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.
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Anonymous Report 2 on 2023221 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2209.10502v1, delivered 20230221, doi: 10.21468/SciPost.Report.6771
Report
The paper is devoted to the investigation of quantum corrections in the chiral GrossNeveu models and, in particular, the higher order contributions to the βfunction. The author performs the explicit calculation of the fourloop βfunction in the levelzero chiral GrossNeveu model and compares the result with the earlier proposed exact expression. It is demonstrated that in momentum subtraction scheme the exact expression is not valid, but the disagreement can be attributed to the scheme dependence of the βfunction. The difference of the fourloop result and the corresponding part of the exact expression is proportional to ζ(3). The author notes that this is analogous to the situation with the exact NSVZ βfunction in D=4 N=1 supersymmetric electrodynamics, where such a disagreement arises in three loops and also appears due to the scheme dependence. The calculation made in the paper is technically complicated, its result seems very interesting, and the conclusions are quite reasonable. That is why I recommend to publish the paper in its present form.
Anonymous Report 1 on 202323 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2209.10502v1, delivered 20230203, doi: 10.21468/SciPost.Report.6670
Report
The paper has a clear formulation of the problem and solution argumentation appears to be complete within the framework provided. It possesses very well established structure along with necessary derivations that makes the paper selfconsistent overall.
The computation of observables in sigma models and their deformations is an important task, that interrelates several subjects of string and field theory. In particular, the renormalisability and allloop exactness of $ \beta $function in 2 and 4dim supersymmetric gauge theories is a crucial longstanding problem. In the present work the author not only addresses the problem of $ \beta $function oneloop exactness and its computation for $ k = 0 $ GrossNeveu model, but provides a novel formalism for studying different types of observables arising in Sigma and Field theory models. The last is realised through the mapping relation of supersymmetric $ \mathbb{CP}^{n1} $ model to chiral GrossNeveu model, which allows to approach the problem by means of field theoretic apparatus and gain understanding on both sides of correspondence.
Additional useful extensions and comments might include (not affecting recommendation for publication):
In the computation of 4loop $ \beta $function, the MOMscheme is always dependent on the configuration of momenta as of (5.22) (no matter if asymmetric or a different momenta configuration is chosen), which consequently leads to oneloop exactness breaking. Might it be possible to implement analogous scheme in other spaces, where $ \zeta(3) $ transcendentality would not emerge and supersymmetry is respected? Is it understood if in MOMscheme $ \zeta $dependence is universal or how does it develop at higher orders? [ Possibly, understanding of the last or its resummation would lead to consistent correlation with GerganovLeClairMoriconi proposal. ]
In relation to similar properties of the 4dim supersymmetric theories, e.g. $ \mathcal{N} = 1, 2 $ or SQED, where an analogy in transcendental dependence occurs. Are there other reasons for this to hold, i.e. apart from the fact that $ \mathbb{CP}^{n1} $ is a theory with $ N_{f} $ matter degrees (eventually leading to similar class of integrands)? Likewise, are there physical reasons for corrections arising in $ \mathcal{N}=2 $ $ \beta_{\,\text{4loop}}^{\,\text{NSVZ}} $ and $ \beta_{\,\text{4loop}}^{\,\text{cGN}} $ ? Is it known if in 2dim higher covariant derivatives (through Slavnov procedure) would solve regularisation/renormalisation dependence or problem would remain like in 4dim case?
If it is possible to embed present formalism for systems where left/right propagating sector is absent (like $ (0,2) $model), is it known whether GN type/Sigma model mapping would still hold for these classes? In this respect, since in GN formalism it is proper to consider phase space formulation, might analogous technique work for a system with orthosymplectic space (e.g. bow quiver variety)?
The work results are original and form important contribution in the field of integrable systems and Sigma models.