Four-quark scatterings in QCD I

We would like to thank the referee for his positive comments on this manuscript. There are the responses to the questions raised by the referee.

(1) In fact in the second paper of this series we have computed the pion decay constant within an improved truncation, where a momentum-dependent quark mass and three-momentum $s$-, $t$-, $u$-channel dependent four-quark vertices are used. The pion decay constant can be calculated via the definition as follows
\begin{align}
i P_{\mu}f_{\pi}\delta^{ab}&\equiv \langle{0}|J_{5\mu}^{a}(x)| \pi^{b }\rangle =\int \frac{d^{4}q}{(2\pi)^{4}} \mathrm{Tr}\Big{[}\gamma_{\mu}\gamma_{5}T^{a}G_{q}(q+P)h_{\pi}(q,P)\gamma_{5}T^{b}G_{q}(q)\Big{]}
\end{align}
with the Bethe-Salpeter amplitude of pion $h_{\pi}(q,P)$, which can be extracted from the residue of the four-quark coupling at the pole of pion mass. From the equation above, it is clear that in order to calculate $f_{\pi}$ one needs the momentum-dependent quark mass and Bethe-Salpeter amplitude.

Furthermore, the sigma mass is also calculated in Paper II, where the pole mass is calculated by means of analytic continuation based on the results in the Euclidean regime.

(2) It is indeed an interesting question. In principle, this approach can be used to investigate the dynamics of the diquark and vector channels. We also have the plan to investigate the $\rho$ meson within this approach. The off-shell dynamics of diquarks are also accessible with the Fierz-complete four-quark basis in the vacuum or at finite temperature and density, but we are not sure whether the diquark condensate can be calculated with the current setup. We will keep thinking about this question.

(3) As the schematic diagram of meson resonance in Fig.13 shows, when the momentum of pion is in the vicinity of its on-shell momentum, one has the four-quark coupling of the pion channel $\lambda_{\pi}\sim 1/(p^{2}+m_{\pi}^{2})$. In the chiral limit, the pion mass is vanishing, i.e., $m_{\pi} \to 0$, which leaves us with $\lambda_{\pi} \to \infty$ if the momentum dependence is neglected and assumed to be $p=0$. Consequently, the constituent quark mass is enhanced by the overestimated four-quark coupling. It is expected that this effect is especially obvious when the chiral limit is being approached. In Paper II we find the enhancement of the constituent quark mass near the chiral limit is significantly suppressed when more momentum dependence for the four-quark vertex is included.

As for the comparison to the dynamical hadronization, frankly speaking, we have no conclusion yet. We guess it may be attributed to the role played by the potential of mesons in the approach of dynamical hadronization.

(4) In fact, the extension of this approach to $N_{f}=2+1$ has always been in our minds. It is not difficult from the concept side, but obviously it is not readily accessible from the side of computation.

(5) Yes, we have a plan to extend this work to finite temperature and density.

Thanks again for the interesting questions. Moreover, we have corrected some typos in the manuscript in the revised version.

We would like to thank the referee for his comments on this manuscript. There is the response as follows.

We completely do not agree with the statement by the referee in Report 2, that in this work we just set up a framework and the progress is essentially incremental only. Yes, we have set up a framework, but it is not trivial. It has never been known before that the coupled system of the flow of the four-quark vertex and that of the quark two-point function can play the same role as the system of the Bethe-Salpeter equation and the quark gap equation. That is the reason why the referee in Report 1 says ''It is very interesting to me the conclusion that by computing the functional renormalisation group flows of the Fierz-complete four-quark interaction of up and down quarks with its t channel momentum dependence (in the isospin symmetric case, also including the flow of the quark two-point function) the system can be understood as the fRG analogues of the complete Bethe-Salpeter (BS) equations and quark gap equation.'' and ''I think this article will be a further step on the road to a better understanding of QCD.''

In fact, we found by accident that the quark mass production and the natural emergence of bound states can be well described by the language of the flows of four-quark and two-quark correlation functions. Therefore, in the first paper of this sequence we try to demonstrate clearly the underlying mechanism by employing the simplest truncation, that is, the momentum-independent quark mass and just one-channel momentum-dependent four-quark vertex. Even within this simplest truncation, one can see that the fRG analogues of BS and gap equations provide us with the necessary ingredients to study the physics related to the dynamical chiral symmetry breaking. Hence, the mechanism and feasibility of the framework are the central concerns in the first paper. We have no idea why the referee in Report 2 focuses on the stuffs of Minkowski space-time and 3d regulator. The reason why we use the 3d regulator in the work is that, the results of 3d regulator directly computed in the Minkowski space-time are in good agreement with those of analytic continuation from the Euclidean calculations as shown in the left panel of Fig.14, which shows that meson observables, e.g., the pole mass of pion, can be reliably extracted via analytic continuation based on results in the Euclidean regime. We certainly know that with more sophisticated truncations, such as the more momentum dependence for the four-quark and two-quark vertices in the second paper of this sequence, and encoding of the gluon dynamics in this framework in Paper III, one is not able to do the calculations directly in the Minkowski regime. But one can use the verified method of analytic continuation to extract observables in Minkowski regime.

Moreover, we have obtained some preliminary results in Paper II and Paper III. For example, in Paper II we employ a momentum-dependent quark mass and three-momentum $s$-, $t$-, $u$-channel dependent four-quark vertices, and obtain the poles masses of not only pion but also the scalar meson sigma, the pion decay constant, the Bethe-Salpeter amplitude of pion, etc. In Paper III, the flow of the four-quark vertex is included in QCD self-consistently, which receives contributions from not only the four-quark scattering but also the gluon exchanges. We hope we can report the relevant results in the near future.

Since this is the first paper of series, we would like to discuss more about the relation between the new framework with e.g., the dynamical hadronization. Therefore, we would like to keep it as it is.