Low-Momentum Pion Enhancement from Schematic Hadronization of a Gluon-Saturated Initial State
Round 1
Reviewer 1 Report
The submitted manuscript deals with the subject of low pt pion enhancement using kinetic approach by especially stressing the gluon-saturation initial state together with direct gluon-to-pion hadronisation scheme. The authors couple the Boltzmann equations for gluons and pions with simple constant interaction matrix element |M| under uniform and static system assumptions, then they demonstrate that with an over-populated gluonic initial condition the system can evolve ultimately to a thermal pion system showing clear low-momentum enhancement, which can be explained as a precursor of Bose condensation. In my view, the present authors have made an innovative even though very simple scheme for the low momentum pion enhancement subject.
So I think the paper is warranted consideration for publication here. With that said, in following I'd like to point out a few suggestions and also questions for the authors that might improve the manuscript.
(1) in line 17, 'and hadronic rescattering in the final stage [6]', I didn't found the correspondence of the text ('hadronic rescattering..') and the Ref[6], maybe one needs to change the corresponding description.
(2) in the main part (part 2), I recommend the authors put clear their setup firstly or somewhere else to let the readers know without confusion. Because the current description in the manuscript is potentially misleading, like: in line 45/46, 'For the masless gluon case we compare with the study...', in fact the present study take gluon with a mass m=0.7 GeV by hand but not massless; in line 46/47, 'where the transition amplitude is taken for one type of gluon ... Eq.(2)', in fact for simplicity and also as first step, the present calculation only considered constant matrix element for all processes here.
(3) The fact that only elastic scatterings are taken into account is actually esential for the phenomenon of Bose condensation due to number conservaton, so I want to ask the authors what's their expectation if inelastic scatterings are included for gluons?
(4) Did the authors looked at the self-similar behaviors with power law scaling distributions in their simulation? can see Ref[15] for non-relativistic boson system and Phys.Rev.D96 (2017) no.1, 014020 for relativistic gluon system. it would very intersting to know the relevant scaling behavior with coupled kinetic equations in the present study. Another important and intersting issue concerns how different scales evolve during the evolution, like the debye scale, the UV scale and IR scale, se Refs[6,13].
(5) One thing puzzels me here is when should the convertion of gluon-to-pion process be switched on. This invovles the phase transition dynamics which is beyond the scope of the present study, but still it might be kind of consistent to use the point when effective temperature approaching pure gluon system's 1st order phase transition temperatre. Otherwise it allows a mixture degree of freedom for gluons and pions even in early stage. For a dynamical scheme of this issue can see Ref: Phys.Rev. C95 (2017) no.2, 024907.
Author Response
We would like to thank the referee for reviewing our paper and the criticism and suggestions that helped us to prepare the revised version. We have carefully considered all the comments and applied the changes to the original version in order to address all the issues raised. Detailed responses to all the comments (in black color) can be found below (in red color), where the line numbers and references stated in the reply are from the revised manuscript, unless written otherwise.
The submitted manuscript deals with the subject of low pt pion enhancement using kinetic approach by especially stressing the gluon-saturation initial state together with direct gluon-to-pion hadronisation scheme. The authors couple the Boltzmann equations for gluons and pions with simple constant interaction matrix element |M| under uniform and static system assumptions, then they demonstrate that with an over-populated gluonic initial condition the system can evolve ultimately to a thermal pion system showing clear low-momentum enhancement, which can be explained as a precursor of Bose condensation. In my view, the present authors have made an innovative even though very simple scheme for the low momentum pion enhancement subject.
So I think the paper is warranted consideration for publication here. With that said, in following I'd like to point out a few suggestions and also questions for the authors that might improve the manuscript.
(1) in line 17, 'and hadronic rescattering in the final stage [6]', I didn't found the correspondence of the text ('hadronic rescattering..') and the Ref[6], maybe one needs to change the corresponding description.
We have changed the corresponding description in the first paragraph of the Introduction to:
“One of the issues which can be addressed by the kinetic approach is the question of a low-momentum pion enhancement in heavy ion collisions [1]. There are several solutions proposed to explain thiseffect as, e.g., the hadronization and freeze-out in a chemical non-equilibrium [2–4], the separate freeze-out for strange particles [5], Bose-Einstein condensate of pions [6–10], established by elastic rescattering in the final stage [10,11]. However, none of them is commonly accepted yet [8]. We believe, an explanation linked to the presence of non-equilibrium physics and a precursor of pion condensation in heavy ion collisions should be the favorable one, especially after the recent analysis of particle correlations performed by the ALICE collaboration is showing a coherent fraction of charged pimeson emission that is reaching 23% [1,9]. Such formation of Bose condensate is usually described by the introduction of additional non-equilibrium parameters to the statistical approach [10,12], see also [2,8,13].”
We added also the new references [2,3]
(2) in the main part (part 2), I recommend the authors put clear their setup firstly or somewhere else to let the readers know without confusion. Because the current description in the manuscript is potentially misleading, like: in line 45/46, 'For the massless gluon case we compare with the study...', in fact the present study take gluon with a mass m=0.7 GeV by hand but not massless; in line 46/47, 'where the transition amplitude is taken for one type of gluon ... Eq.(2)', in fact for simplicity and also as first step, the present calculation only considered constant matrix element for all processes here.
Following the reviewer’s recommendation we restructured the section 2 of the manuscript to clearly explain the setup used in the paper. Furthermore, we now start the section 2 from a more general equation (Eq. (1) in the new version), so that the old Eq. (1) appears in the new version of the paper as Eq. (4), after the clarification of the restrictions we make for the studied case. The description of the non-constant matrix element for the massless gluon case was removed completely from the manuscript, in order to avoid any confusion. In the section 4 we added a short discussion of the possible comparison to the Ref. [16] for the massless gluon case (lines 134-138 in the new version):
“Here it would be beneficial to make a comparison with the study in Ref. [16], where a system of massless gluons undergoes the evolution due to elastic scattering with similar restrictions as used in the current paper. However, the equation (7) will no longer be valid in the case of non-constant matrix elements and angle-dependence, and thus will need to be rederived.”
(3) The fact that only elastic scatterings are taken into account is actually essential for the phenomenon of Bose condensation due to number conservation, so I want to ask the authors what's their expectation if inelastic scatterings are included for gluons?
This is indeed an interesting matter. At the moment, we do not wish to speculate on what might happen in the case where we take into account inelastic scatterings for gluons. However, this will be definitely studied in the future within systematic extensions of our approach.
(4) Did the authors looked at the self-similar behaviors with power law scaling distributions in their simulation? can see Ref[15] for non-relativistic boson system and Phys.Rev.D96 (2017) no.1, 014020 for relativistic gluon system. It would be very interesting to know the relevant scaling behavior with coupled kinetic equations in the present study. Another important and interesting issue concerns how different scales evolve during the evolution, like the Debye scale, the UV scale and IR scale, see Refs. [6,13].
Unfortunately, at the moment our model is too simple to discuss scaling of the momentum dependence as shown in Ref. [Phys. Rev. D96 (2017) no.1, 014020]or in Refs. [7,11,16], because the model at this stage neglects any momentum/angle dependencies.
We added a corresponding statement with references in the first paragraph of the Discussion section: "Such an improved model would allow us to discuss thedifferent scales and their evolution, e.g., the Debye scale, the UV and IR scale, see Refs. [7,11,16,27]."
(5) One thing puzzles me here is when should the conversion of gluon-to-pion process be switched on. This involves the phase transition dynamics which is beyond the scope of the present study, but still it might be kind of consistent to use the point when effective temperature approaching pure gluon system's 1st order phase transition temperature. Otherwise it allows a mixture degree of freedom for gluons and pions even in early stage. For a dynamical scheme of this issue can see Ref: Phys. Rev. C95 (2017) no.2, 024907.
We added the following discussion at the end of section 2:
We keep our model simple and therefore do not introduce an extra timescale for the start of hadronization. However,we keepin mind that the underlying microphysical process is, e. g., a quark-box diagram, which consists of the Breit-Wheeler type process of 2g → q qand subsequent hadronization cross section q q→ ππ . In the future we plan to investigate the problem of the gluon-to-pion conversion in detail, for instance by applyingthe Nambu—Jona-Lasinio model [Rehberg et al. (1995) and (1998),Friesen et al. (2013), Marty et al. (2015)]and/or by exploiting dynamical schemes of hadronization that would address the confinement aspect as well[Feng& Greiner (2017), Florkowskiet al. (2015) and (2016)].
Reviewer 2 Report
The manuscript presents the kinetic approach to explain enhancement of the low-momentum pions recently observed in collisions of heavy ions at the LHC energy. Despite the fact that manuscript provides a clear microscopic picture of the above phenomenon, it requires elucidation of several questions.
Below are some typos that should be corrected and remarks that should be addressed in detail by the authors before this work can be considered for publication in the Particles.
Typos:
1) Line 4 'overpopolation' should be replaced by 'overpopulation'.
2) In line 9 'freezeout' should be replaced by 'freeze-out' as in the rest of the text.
3) In line 20 '...fraction in charged...' should be replaced by '...fraction of charged...'.
4) In line 32 '... to a. precursor' should be replaced by '... to a precursor'.
5) In the line 43 'Ff=...' should be replaced by 'F(f)=...'.
6) In the caption below Fig.1 'f (p)' should be replaced by 'f(p)'.
7) On the Fig.2 and its caption matrix elements defined as |M| and M, correspondingly. It has to be unified.
8) In the caption below Fig.3 'f (p)' should be replaced by 'f(p)'.
9) In Eq. A2 star symbol should be replaced by \cdot.
10) In Eq. A7 star symbol should be replaced by \cdot.
11) In Eq. A9 it is worth to use \left( and \right) brackets instead on the normal ones.
Questions and remarks:
1. Initial assumption of the manuscript is related to the (almost) pure glue state existing at early stages of heavy ion collisions. Recently this picture was criticized because of rather hight temperature of the deconfinement phase transition being significantly higher compared to the one of the hadronic chemical freeze-out. Additional justification of the pure glue scenario would be beneficial for the paper.
2. The Boltzmann-Nordheim kinetic equation (1) ignores inhomogeneity and expansion of a system under consideration. Isn’t it too rough approximation while speaking about heavy ion collisions?
3. Pre-equilibrium conversion of gluons to pions implies that within the present picture all pions are produced before the hadronic chemical freeze-out. This, however, contradicts to results of thermal models showing that significant amount of pions is produced in decays of heavier hadrons. An explanation of this contradiction is required.
4. All hadronic states except pions are neglected in the present picture. While, an interaction with these states scatter pions out from the low-momentum modes washing out effect of their enhancement. In such a case the effect of low-momentum enhancement won`t be so prominent. Please provide an evaluation of such and effect.
5. From the paper it is not clear how relations between the gluon-gluon, pion-pion and gluon-pion amplitudes affect evolution of the system. How equilibrated distribution of pions depends on these ratios? Is the low-momentum bump in its behavior preserved for any values of these ratios?
In addition, several rather technical questions are worth to be answered in the manuscript.
6. Are two-to-two reactions enough to catch (even schematically) all the necessary dynamics of the system under consideration?
7. It would be useful for readers if an order at which the amplitude (2) is calculated was explicitly noted in the text?
8. How the small-angle approximation is consistent with low momenta of scattering particles?
9. In the Eq.2 authors use variable g which was not defined.
Author Response
We would like to thank the referee for reviewing our paper and the criticism and suggestions that helped us to prepare the revised version. We have carefully considered all the comments and applied the changes to the original version in order to address all the issues raised. Detailed responses to all the comments (in black color) can be found below (in red color), where the line numbers and references stated in the reply are from the revised manuscript, unless written otherwise.
The manuscript presents the kinetic approach to explain enhancement of the low-momentum pions recently observed in collisions of heavy ions at the LHC energy. Despite the fact that manuscript provides a clear microscopic picture of the above phenomenon, it requires elucidation of several questions.
Below are some typos that should be corrected and remarks that should be addressed in detail by the authors before this work can be considered for publication in the Particles.
Typos:
1) Line 4 'overpopolation' should be replaced by 'overpopulation'.
Change made.
2) In line 9 'freezeout' should be replaced by 'freeze-out' as in the rest of the text.
Change made.
3) In line 20 '...fraction in charged...' should be replaced by '...fraction of charged...'.
Change made.
4) In line 32 '... to a. precursor' should be replaced by '... to a precursor'.
Change made.
5) In the line 43 'Ff=...' should be replaced by 'F(f)=...'.
Change made.
6) In the caption below Fig.1 'f (p)' should be replaced by 'f(p)'.
Change made. Also changed to math-mode, to have it in agreement with the equations in the paper.
7) On the Fig.2 and its caption matrix elements defined as |M| and M, correspondingly. It has to be unified.
Change made. Nowthe unified definition throughout the paper is|M|.
8) In the caption below Fig.3 'f (p)' should be replaced by 'f(p)'.
Change made. Also changed to math-mode, to have it in agreement with the equations in the paper.
9) In Eq. A2 star symbol should be replaced by \cdot.
Change made.
10) In Eq. A7 star symbol should be replaced by \cdot.
Change made.
11) In Eq. A9 it is worth to use \left( and \right) brackets instead on the normal ones.
Change made.
Questions and remarks:
1. Initial assumption of the manuscript is related to the (almost) pure glue state existing at early stages of heavy ion collisions. Recently this picture was criticized because of rather high temperature of the deconfinement phase transition being significantly higher compared to the one of the hadronic chemical freeze-out. Additional justification of the pure glue scenario would be beneficial for the paper.
In our current approximation, we use the pure gluonstate as ansatz for the initial state,with the gluon-to-pion conversion process switched on from the very beginning, so that the system getsquickly enriched with pions and becomes a coupledsystem of gluons and pions. This “quarkless hadronization” which omits the explicit discussion ofthe quark-antiquark kinetics should be justified. If we assume that the process q q→ π πis fast enough for quarks to be considered virtual, then we will have to deal with a gluon system that is immersed with pion “impurities” and thus should not necessarily showa1storder phase transition that was obtained in simulations of the thermalized pure glue system on the lattice. However, in the further studies we would like to resolve the conversion matrix element as a quark-box diagram and realize a dynamical confinement scheme for the hadronization process.
We added a discussion of these issues including references at the end of Section 2.
2. The Boltzmann-Nordheim kinetic equation (1) ignores inhomogeneity and expansion of a system under consideration. Isn’t it too rough approximation while speaking about heavy ion collisions?
We restructured the section 2 of the manuscript to clearly explain the setup used in the paper. The section 2 now starts from a more general equation (Eq. (1) in the new version), so that the old Eq. (1) appears in the new version of the paper as Eq. (4), after the clarification of the restrictions we make for the studied case. The description of the non-constant matrix element for the massless gluon case was removed completely from the manuscript, in order to avoid any confusion.
Admittedly, the approximation is rough and should be abandoned in the next step of our work, so that the process of isotropization could be studied.So farthis manuscript presents the initial exploratory results of our study.
3. Pre-equilibrium conversion of gluons to pions implies that within the present picture all pions are produced before the hadronic chemical freeze-out. This, however, contradicts to results of thermal models showing that significant amount of pions is produced in decays of heavier hadrons. An explanation of this contradiction is required.
We do not claim that all the pions are produced from such gluon-to-pion conversion as we study in the presented work. Our current model is very schematicas it omits higher hadronic states than pions at all. Once this initial approximation will be lifted, one can discuss the fraction of pions produced due to the resonance decays.
4. All hadronic states except pions are neglected in the present picture. While, an interaction with these states scatter pions out from the low-momentum modes washing out effect of their enhancement. In such a case the effect of low-momentum enhancement won`t be so prominent. Please provide an evaluation of such and effect.
We plan to include further hadron species in the improved version of our model. For the moment, we enjoy the beauty of such simplified model, which nevertheless bears an important message, namely that some parts of the low-p pion enhancement could arise from the quick conversion of gluon state, with low-momentum enhancement due to fast gg-rescattering.
5. From the paper it is not clear how relations between the gluon-gluon, pion-pion and gluon-pion amplitudes affect evolution of the system. How equilibrated distribution of pions depends on these ?ratios? Is the low-momentum bump in its behavior preserved for any values of these ratios?
It would be interesting to study various patterns of momentum distributions dependency on the amplitudes, e. g., if the pion-pion rescattering matrix element would be smaller, then there would appear a peak at non-zero momentum. For the present study, however, all the matrix elements in the coupled system of equationswere chosen equal and rather large. The further work will include a more sophisticated choice of the matrix elements as well as a thorough sampling ofparameter sets.
In addition, several rather technical questions are worth to be answered in the manuscript.
6. Are two-to-two reactions enough to catch (even schematically) all the necessary dynamics of the system under consideration?
We include a discussion of both possible extensions of our model (further hadronic species and their mutual interactions, including beyond decays also the 3M ↔BBreactions)in the Discussion section:
The model can be extended towards a more realistic description of a hadronizing gluon-dominated initial state in high-energy heavy-ion collisions by including more hadronic species as they are observed in those experiments in good agreement with the thermal statistical model [Andronicet al. (2018)].This calls then for an extension of the collision integrals in our kinetic model to other classes of processes than just 2 → 2 processes as, e.g., the three-meson conversion to a baryon-antibaryon pair and its reverse [Seifert& Cassing (2017)].
7. It would be useful for readers if an order at which the amplitude (2) is calculated was explicitly noted in the text?
The description of the non-constant matrix element for the massless gluon case was removed completely from the manuscript, in order to avoid any confusion, as in our current setup we used only constant transition amplitudes.The amplitude (Eq. (2)) was taken from the Ref. [16], where the authors calculated it in the leading order of the small scattering approximation for the case of massless gluons. In the section 4 we added a short discussion of the possible comparison to the Ref. [16] for the massless gluon case (lines 134-138in the new version):
“Here it would be beneficial to make a comparison with the study in Ref. [16], where a system of massless gluons undergo the evolution due to elastic scattering with the similar restrictions as used in the current paper. However, the equation (7) will no longer be valid in the case of non-constant matrix elements and angle-dependence, and thus will need to be rederived.”
8. How the small-angle approximation is consistent with low momenta of scattering particles?
In our study we did not use the small-angle approximation. The discussion of this approximation was connected to the Ref. [16] and has been removed from the manuscript to avoid confusion.
9. In the Eq.2 authors use variable g which was not defined.
It is the coupling constant (g^2/ 4pi = alpha_s). However, now we removed description of the non-constant matrix element for the massless gluon case from the manuscript.
Round 2
Reviewer 1 Report
The authors have replied to my comments and incorporated the answers into the paper, I now recommend publishing here.
Reviewer 2 Report
The authors have addressed all the comments and remarks. The
manuscript 'Low-momentum pion enhancement from schematic hadronization of a gluon-saturated initial state' is worth to be published in Particles.
