Kinetic Approach to Pair Production in Strong Fields—Two Lessons for Applications to Heavy-Ion Collisions
Round 1
Reviewer 1 Report
The paper "Kinetic approach to pair production in strong fields" by Blaschke et al.
is scientifically sound and I recommend it for publication, but only after a revision.
Let me give some examples I found that need to improved upon.
The definite and indefinite articles, "the" and "a", are missing at various places.
Some equations are missing the "," or "." at places, e.g. Eqs. 10, 12, 15, 16 etc.
L41: should be
in terms of the transverse energy and ...
L42:
Here p_\perp ... perpendicular to the field vector and p_\parallel = p_3 is the momentum component parallel to the field.
There is no "p" only "\tilde{p}" in the previous definitions, so that should be corrected. Second, which field we are
talking about ? p_\parallel = p_3, is this always true or only when the system is rotated as such or when p_\perp = 0.
What is "m" ? what is "e" ?
As the initial condition ... => For the initial condition we choose ...
L47: we can take the integration step ..
L52: leads to a system of ..
L53: The above system is much simpler to solve...
The next sentence tells the reader that, these auxiliary functions "u" and "v" describe vacuum polarization effects.
Shouldn't this be better after Eq. 9 ?
L57: when the applied external electric field, $E << E_c$, is considerably smaller than the critical field strength,
(note that, then is about time)
After Eq. (17) the mass of the electrons is defined, but that should have been right after Eq. (5).
L59: should read something like
When the electric field, $E$ is small we expect the probability of pair creation to be small $f << 1$, hence
$1 - f \approx 1$.
Consequently, the source term .. must have the following form.
L64: Most things are obvious, i.e. Markovian limit, hence no need to explain "if we do it" the way.
We can avoid the time integration of the whole pre-history of the distribution function, and obtain the kinetic equation
L69: this is true but needles to say again, since $E << E_c$ was given at the beginning of this section.
L71: standard textbook trick ? If proven mathematical identities are called tricks what are other mathematical tools called ?
L72: Firstly is usually followed by secondly and then finally, so why not write something like:
A further simplification of (24) can be obtained by expanding the dispersion relation with respect to a small external field $A$ \appox 0$
L88: Where exactly is the transition domain between the Quasi particle and Residual plasma, on the figures ?
L96: The first two asymptotics follow from Eq. (33) so that should be written.
L104: than f and f^0
L108: After two ions collided or After the collision of two ions GCC is likely formed, ...
L111:
Which subplots and lines need to be compared to pinpoint the claim that the particle production is greatly enhanced ?
What is the reason for the parton number increase with rapid collisions ?
Certainly the number of partons is defined with respect to an energy scale or momentum transfer but how is that related to "nuclei collide rapidly".
The presentation of Figs. 2 and 3 are simply unacceptable!
There are no labels on the y-axis and no units, no plot legend and it is advised to use different line style, so the
plots are understandable even in black and white.
L129: There in no such thing as thermal spectra in heavy-ion collisions, hence there is no contradiction!
There are of course so called thermal fits with an inverse slope parameter, but all current fluid dynamical
models go beyond local thermal equilibrium, at least up to 2-nd order in p_T..
The best fits are given by power-law distributions, which are non-thermal per se, but also have a slope parameter.
L176:
The density of produced particles was not calculated, but n ~ E^2 was given both here and in the abstract.
The authors should make an effort to calculate it before discussing it, for example in the low field limit in Eq. (32)
there is an exact result that can be integrated, as as well as in general.
That's all for now.
Author Response
We are grateful to the reviewer for providing us with insightful comments and questions as well as a detailed list of typos and corrections which helped us to prepare the present, revised version of the manuscript. Below, we provide a point-by-point response (in red color) to all comments (in black color).
The paper "Kinetic approach to pair production in strong fields" by Blaschke et al. is scientifically sound and I recommend it for publication, but only after a revision.
Let me give some examples I found that need to improved upon.
The definite and indefinite articles, "the" and "a", are missing at various places.
Some equations are missing the "," or "." at places, e.g. Eqs. 10, 12, 15, 16 etc.
changes done
L41: should be
in terms of the transverse energy and ...
changes done
L42:
Here p_\perp ... perpendicular to the field vector and p_\parallel = p_3 is the momentum component parallel to the field.
There is no "p" only "\tilde{p}" in the previous definitions, so that should be corrected. Second, which field we are
talking about ? p_\parallel = p_3, is this always true or only when the system is rotated as such or when p_\perp = 0.
What is "m" ? what is "e" ?
As the initial condition ... => For the initial condition we choose …
changes done
L47: we can take the integration step ..
L52: leads to a system of ..
L53: The above system is much simpler to solve...
The next sentence tells the reader that, these auxiliary functions "u" and "v" describe vacuum polarization effects.
Shouldn't this be better after Eq. 9 ?
changes done
L57: when the applied external electric field, $E << E_c$, is considerably smaller than the critical field strength,
(note that, then is about time)
After Eq. (17) the mass of the electrons is defined, but that should have been right after Eq. (5).
changes done
L59: should read something like
When the electric field, $E$ is small we expect the probability of pair creation to be small $f << 1$, hence
$1 - f \approx 1$.
Consequently, the source term .. must have the following form
changes done
L64: Most things are obvious, i.e. Markovian limit, hence no need to explain "if we do it" the way.
We can avoid the time integration of the whole pre-history of the distribution function, and obtain the kinetic equation
changes done
L69: this is true but needles to say again, since $E << E_c$ was given at the beginning of this section.
done
L71: standard textbook trick ? If proven mathematical identities are called tricks what are other mathematical tools called ?
changed to mathematical identity
L72: Firstly is usually followed by secondly and then finally, so why not write something like:
A further simplification of (24) can be obtained by expanding the dispersion relation with respect to a small external field $A$ \appox 0$
changed according to the suggestion
L88: Where exactly is the transition domain between the Quasi particle and Residual plasma, on the figures ?
When the distribution function $f(t)$ follows the trend of $ E(t)^2$/16 we are dealing with quasi-particle electron-positron plasma (QEPP). However, when $f(t) = const$, reaching the asymptotic (residual) value, a real particle electron-positron plasma is created (REPP).
In-between there is the transition region of fast oscillations which divides the system evolution into QEPP and REPP domains. See Fig.1.
(we included the above text fragment into manuscript)
L96: The first two asymptotics follow from Eq. (33) so that should be written.
L104: than f and f^0
L108: After two ions collided or After the collision of two ions GCC is likely formed, ...
fixed
L111:
Which subplots and lines need to be compared to pinpoint the claim that the particle production is greatly enhanced ?
What is the reason for the parton number increase with rapid collisions ?
Certainly the number of partons is defined with respect to an energy scale or momentum transfer but how is that related to "nuclei collide rapidly".
In Fig. 2 from the upper left to the lower right panel the pulse duration increases but the residual (asymptotic) value of the distribution function decreases.
If the Sauter profile can be viewed as the shape of a strong (color) field pulse, then the short duration (small tau) means a rapid collision.
The presentation of Figs. 2 and 3 are simply unacceptable!
There are no labels on the y-axis and no units, no plot legend and it is advised to use different line style, so the plots are understandable even in black and white.
fixed
L129: There is no such thing as thermal spectra in heavy-ion collisions, hence there is no contradiction!
There are of course so called thermal fits with an inverse slope parameter, but all current fluid dynamical models go beyond local thermal equilibrium, at least up to 2-nd order in p_T..
The best fits are given by power-law distributions, which are non-thermal per se, but also have a slope parameter.
According to this remark, we changed the wording in Section 5 and Conclusions.
L176:
The density of produced particles was not calculated, but n ~ E^2 was given both here and in the abstract.
The authors should make an effort to calculate it before discussing it, for example in the low field limit in Eq. (32)
there is an exact result that can be integrated, as as well as in general.
changed to f(t) ~ E^2
Reviewer 2 Report
The authors use a proven technique in a new innovative way. They employ kinetic theory to study the influence of external fields in several limits of a collisionless plasma. The time evolution of different pulses is studied. We do recommend the work for publication after minor corrections to be taken into account.
Please check this list of suggested improvements:
1) * section 3
There is no point in having footnotes for two acronyms that are never reused later. Please expand the footnotes directly into the text without acronyms for better readability.
2) * section 3
The very weak and the weak field case is characterized by the same amount $10^{-2}. I would assume the very weak field case $E_0$ to be smaller. This is probably just an copy and paste oversight.
3) When citing flux tubes 16-18 add citation to the percolation model of Phys.Lett. B287 (1992) 154-158, DOI: 10.1016/0370-2693(92)91892-D
when citing recent work 22-25 add most recent work Phys.Rev. D98 (2018) no.1, 014006, DOI: 10.1103/PhysRevD.98.014006, arXiv:1804.01966.
4) Mention that isotropization with the chromo-Weibel instability needs also quite some time Phys.Rev. D87 (2013) no.2, 025010, DOI: 10.1103/PhysRevD.87.025010, arXiv:1207.5795.
5) Explain difference and similarities to the findings of Phys.Rev. C92 (2015) 064904, DOI: 10.1103/PhysRevC.92.064904, arXiv:1505.08081.
6) Figure 1
A bigger font for the legends inside the plot would make it easier to read.
7) Figure 2
Please add the plot legends.
An added bonus for figure 1 and 2 would be to use full, dotted, dashed and dot-dashed lines for all four/three curves, helping on black and white prints to identify the curves.
8) Suggested wording improvements, please change proverbial wording: "fastly" (rapidly?), "nonthermal" (not thermal?),
9) Please fix the following typos: "When E is smalle", "presenting behavior od Sauter"
Author Response
We are grateful to the reviewer for providing us with insightful comments and questions as well as a detailed list of typos and corrections which helped us to prepare the present, revised version of the manuscript. Below, we provide a point-by-point response (in red color) to all comments (in black color).
The authors use a proven technique in a new innovative way. They employ kinetic theory to study the influence of external fields in several limits of a collisionless plasma. The time evolution of different pulses is studied. We do recommend the work for publication after minor corrections to be taken into account.
Please check this list of suggested improvements:
1) * section 3
There is no point in having footnotes for two acronyms that are never reused later. Please expand the footnotes directly into the text without acronyms for better readability.
change is done
2) * section 3
The very weak and the weak field case is characterized by the same amount $10^{-2}. I would assume the very weak field case $E_0$ to be smaller. This is probably just an copy and paste oversight.
Yes, it was a copy & paste mistake; fixed
3) When citing flux tubes 16-18 add citation to the percolation model ofPhys.Lett. B287 (1992) 154-158, DOI: 10.1016/0370-2693(92)91892-D
when citing recent work 22-25 add most recent work Phys.Rev. D98 (2018) no.1, 014006,DOI: 10.1103/PhysRevD.98.014006, arXiv:1804.01966.
References added
4) Mention that isotropization with the chromo-Weibel instability needs also quite some time Phys.Rev. D87 (2013) no.2, 025010,DOI: 10.1103/PhysRevD.87.025010, arXiv:1207.5795.
Reference and remark added
5) Explain difference and similarities to the findings of Phys.Rev. C92 (2015) 064904, DOI: 10.1103/PhysRevC.92.064904, arXiv:1505.08081.
We thank the referee for pointing out this interesting reference which we have now included. We added the following paragraph (lines 163-165) with a footnote in Section 5:
A recent study of the thermalization and isotropization question in the early stages of heavy-ion collisions [Ruggieriet al. (2015)]by solving a relativistic Boltzmann transport equation with a Schwinger source term for particle production from flux-tube decay goes beyond Ref. [Ryblewski& Florkowski (2013)]by taking into account viscosity effects and $2\to 2$ collisions in the gluon sector.
This study finds that for ideal fluid conditions with a minimal viscosity at the KSS bound $ \eta = s /(4 \pi)$ [Kovtunet al. (2004)]already at a timescale below 1 fm/c the ratio of longitudinal to transverse pressure approaches unity with oscillations being damped out and the transverse momentum spectrum shows thermal behavior $dN/(p_T^2 dp_T dy) \propto \exp(-\beta p_T)$ with an inverse slope parameter fulfilling the ideal gas relationship $\beta^{-1}=T_{\rm eff} \propto \varepsilon_{\rm kin}^{1/4}$, where $\varepsilon_{\rm kin}$ is the kinetic energy density.
It is interesting to note that this feature is reproduced by the much simpler model considered here which neglects collisions, spatial evolution and finite size as well as backreaction of the produced particles on the field.
It has, however, the advantage of being particularly suitable for discussing the temporal evolution (pulse shape) of the flux-tube field with special emphasis on subcritical field strengths\footnote{We remind that the account for confining boundary conditions in a flux tube of finite radial extension $r_0$ gives rise to an $r_0$-dependent suppression of the Schwinger pair production rate [Pavel& Brink (1990)].}.
6) Figure 1
A bigger font for the legends inside the plot would make it easier to read.
done
7) Figure 2
Please add the plot legends.
An added bonus for figure 1 and 2 would be to use full, dotted, dashed and dot-dashed lines for all four/three curves, helping on black and white prints to identify the curves.
done
8) Suggested wording improvements, please change proverbial wording: "fastly" (rapidly?), "nonthermal" (not thermal?),
We decided to use “non-thermal” ;
“fastly” changed to “rapidly”
9) Please fix the following typos: "When E is smalle", "presenting behavior od Sauter"
fixed
Round 2
Reviewer 1 Report
Better now.
Reviewer 2 Report
All minor remarks are fully taken into account in the updated version. The publication is very much recommended.
