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Fourquark scatterings in QCD II
by Weijie Fu, Chuang Huang, Jan M. Pawlowski, Yangyang Tan
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Submission summary
Authors (as registered SciPost users):  Chuang Huang 
Submission information  

Preprint Link:  https://arxiv.org/abs/2401.07638v2 (pdf) 
Date submitted:  20240201 13:07 
Submitted by:  Huang, Chuang 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Phenomenological 
Abstract
In [1], we initiated a program for the quantitative investigation of dynamical chiral symmetry breaking and resonant bound states in QCD with the functional renormalisation group, concentrating on the full infrared dynamics of fourquark scatterings. In the present work we extend this study and take into account a threemomentum channel approximation ($s,t,u$channel) for the Fierzcomplete fourquark vertices. We find that the fourquark vertex in this approximation is quantitatively reliable. In particular, we have computed the pion pole mass, pion decay constant, BetheSalpeter amplitudes, the quark mass function and wave function. Our results confirm previous findings that low energy effective theories only reproduce QCD quantitatively, if initiated with a relatively low ultraviolet cutoff scale of the order of 500 MeV. The quantitative description set up here paves the way for reliable quantitative access to the resonance structure in QCD within the fRG approach to QCD.
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Reports on this Submission
Strengths
1) This work is part of a series of articles aimed at describing the emergence of mesonic bound states from the infrared dynamics of fourquark scatterings in QCD, employing the functional renormalisation group (fRG).
2) The manuscript is solid, the organization of the article is good, and the presentation of the material is coherent.
3) The results presented here set the stage for the final step towards quantitative predictions of the QCD resonance structure and general timelike correlation functions in firstprinciples functional QCD using the fRG.
Report
The present manuscript is part of a series of articles in which the authors study fourquark scatterings in QCD, employing the functional renormalisation group (fRG) approach. More specifically, they have computed the fourquark dressings, including the s, t, and u channels. They find that, in this improved approximation, the fourquark vertex can be accurately estimated. In particular, they computed the pion pole mass, pion decay constant, BetheSalpeter amplitudes, and the quark mass and wave functions.
The general idea and the results presented in the manuscript are quite interesting. As the authors emphasized, this work sets the stage for the final step towards quantitative predictions of the QCD resonance structure and general timelike correlation functions in firstprinciples functional QCD using the fRG.
From a technical standpoint, the paper is solid, the organization of the article is good, and the presentation of the material is coherent. For these reasons, the article merits publication in SciPost. However, before that, the authors might consider providing some additional details/discussion about the following points in the paper.
1) In Fig. 7, the authors compare their result for the dynamical quark mass with the lattice result of Ref. [37], which was obtained with a pion mass of 280 MeV. For this particular case, the current quark mass employed in that lattice simulation was $m_q=6.2$ MeV, whereas the authors fixed their current mass at $m_q=19.2$ MeV. As we know, the dynamical quark mass displays considerable dependence on the values of the current masses used as input. Could the authors comment on these differences in the current masses employed in this comparison, and how this would affect the agreement between the fRG and lattice results presented in this plot?
2) In the same Ref. [37], there are lattice results for the quark wave function, which is also computed in the present manuscript and shown in Fig. 17. However, the authors did not compare their prediction for this function with the lattice results. Could the authors comment on this? What are the main differences between both sets of results?
Requested changes
I have no additional modification requests, apart from the clarification points mentioned above
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
1The findings are important, in particular the identification of a reliable approximation scheme for the fRG computation of the fourquark vertex, and the scale below which the effective theory reproduces QCD;
2The approximations employed, such as restricting the dressing functions' kinematic dependence to the s, t, u channels, are clearly stated and wellmotivated;
3The errors introduced by these approximations are quantified, as much as possible;
4Ways forward are identified for followup research.
5The article is well structured;
Report
The manuscript details an approximation scheme for the computation of the fourquark vertex through the Functional Renormalization Group (fRG), through which dynamical chiral symmetry breaking and the formation of bound states can be studied quantitatively. The approximation scheme relies on three main steps: (i) dynamical hadronization, where effective fields are introduced to describe the interaction of composite states (sigma and pion, in the present case); (ii) the identification of a consistent subset of tensor structures for the vertex, which respects the symmetries of the theory and are expected to be dominant at low energies; (iii) the simplification of the kinematic dependence of the attending dressing functions to functions of the s, t, and u variables only, rather than the six variables allowed by Lorentz symmetry. The authors find that the t channel alone is not sufficient for quantitative reliability, but that the three channels s, t and u, suffice. Moreover, they demonstrate that the sigmapion tensor structures dominate the infrared dynamics, and identify the cutoff scale, namely 500 MeV, beyond which QCD fails to be reproduced quantitatively, indicating that above this scale the inclusion of the gluon degrees of freedom is necessary.
These conclusions are firmly supported by explicitly estimating the errors introduced by these approximations through a series of numerical experiments.
These results are certainly an important step towards a full QCD description of the dynamical chiral symmetry breaking and bound state formation in the fRG framework. Moreover, the manuscript is well structured, and the presentation is quite clear. I thus recommend its publication in present form.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Chuang Huang on 20241010 [id 4854]
(in reply to Report 2 on 20240706)We are grateful to the referee for the positive comments on this manuscript. Here are our responses to the questions raised by the referee. We have also added relevant discussions after Eq.(37) on page 7 and Eq.(C3) on page 17 of the paper.
The referee writes:
Our response: For the first comment, we think there are two points that need clarification. Firstly, this work is done within the lowenergy effective theory of fourquark scattering, without considering the glue dynamics. Correspondingly, the cutoff scale $\Lambda=500\,\mathrm{MeV}$ is also lower than the QCD cutoff scale $\Lambda_{\mathrm{QCD}}\sim 2 5\,\mathrm{GeV}$. This is also why the initial quark masses are relatively large at the cutoff scale. In the third step paper, we will obtain quark masses from firstprinciples QCD and conduct a more precise comparison with lattice results. Secondly, in reference [37], $m_{q}=6.2\,\text{MeV}$ is the subtracted bare quark mass under the $\overline{\mathrm{MS}}$ renormalization scheme, while the fRG method corresponds to $\mathrm{MOM}^{2}$ scheme in the ultraviolet region, see in reference [39]. Therefore, the current quark mass at the cutoff scale is not a physical quantity that can be directly compared between the fRG method and the lattice method. In different approaches, only physical quantities that are almost independent of the renormalization scheme can be directly compared, such as momentumdependent quark mass function at the IR scale, meson masses, meson decay constants, and so on.
The referee writes:
Our response: Through simple analysis of the Dirac structure ($\gamma$ matrix) of the quark twopoint functions in QCD flow equations, it can be inferred qualitatively that the quark wave function is primarily dominated by the quark selfenergy diagrams which include the quarkgluon vertices. The fourquark vertices studied in this work mainly contribute to generating quark masses. In Fig.17 it is illustrated that even when considering the $s,t,u$channel momentum dependence, the quark wave function resulting from the fourquark interactions is still away from the unity very mildly, which confirms the simple analysis above quantitatively. Furthermore, it is worth mentioning that currently there are qualitative differences in the results of quark wave functions obtained by different lattice QCD collaboration groups, see in reference [37] and Phys.Rev.D 104 (2021) 9, 094509. Obviously, further studies of quark wave functions are required in QCD calculation. In the third paper in this series, we will calculate the quark wave functions from the firstprinciples QCD within the fRG method.