Dear contact authors,

As a general comment we find it a pity that there is no numerical result for the
full p_T and y range of LHCb.
Tables 2 and 3 give a lot of information but does not support
the statement of "no significant polarization is observed" in the abstract.
This statement seems also different from that in the Conclusion.
 
- l.12:
Replace "The JPC of phi(2S) mesons is 1-- ands its polarization can ..." by
"The phi(2S) meson has quantum numbers JPC = 1-- and its polarization can ...".
Its polarization would refer to the grammatical subject JPC and not to the meson.
And JPC needs clarification.

- l.20-25: 
It seems that you use capitals for the X-, Y-, and Z-axis to avoid confusion with the rapidity, y. But is this so critical? 
>From the context a real misunderstanding does not seem possible.

- l.22: 
The beam axis is probably precise enough but still ambiguous.
Which beam is used in the angle?

- l.25: 
Change "is oriented to complete" to "is oriented such to complete".

- l.27:
Replace "Frame-invariant observables can also ... quantity being [5,6] " by
"The invariant variable lambda_inv, that is commonly used, can be reconstructed 
from the parameters lambda_theta and lambda-phi as (see [5,6]),"

- l.36-37:
Split the sentence "In the NRQCD model ... polarizations for quarkonia [9]" to
"In the NRQCD model quarkonium production is dominated by gluon fragmentation.
For large transverse momenta compared to the mass of the quarkonium the spin
of the almost on-shell gluon is close to unity.
This leads to the prediction of quarkonia with large transverse polarizations [9]"
This text avoids the comments:
- Add comma after "process".
- Add "(for c=1)" in "much larger (for c=1)".
- Why does it matter whether the gluon is on- or off-shell, since gluons are not 
final-state particles. 
Is this related to the spin of the gluon being (almost) constrained to one, 
in case it is (almost) on-shell? 
In that case, we believe it would be better to phrase it like above.

- l. 41-53: 
Could you define pt and y, being the transverse momentum and rapidity?

- l.94: 
Replace "of J/psi polarisation [20], the pseudo-decay-time significance S_tau" by
"of the J/psi polarisation [20], the significance S_tau of the reconstructed decay time". 
Add "the". "Pseudo" adds no information.

- l.99: 
Replace "transverse momentum and five y bins" by
"transverse momentum and five bins in rapidity"
Use either symbols or variable names. 

- l.101: 
Is "combination of two Crystal Ball functions" correct? 
Usually the CB has the tail to the left (see Ref. [32]). 
In case the 2nd CB has a tail to the right, please mention this in the text. 
In case both have a tail to the right, it is not convincing that you can 
actually fit the 2nd tail.

- l.144:
Replace "two-dimensional muon momentum, p_mu, and y_mu" by 
"two variables p_mu and y_mu".
Define p_mu to be the absolute muon momentum. 
And remove the comma after p_mu. 
Or do you mean a 2-D muon momentum in the p-y space?

- Fig. 2:
Unreadable in BW. Can you use another more printer friendly palette?
Or, if you think this is mostly useful for talks, put the figure in additional material 
and remove it from the paper.

- Fig. 4: 
Looks very fuzzy due to the many points per bin. 
We understand the problem, please try e.g. to make logarithmic plot 
to make the first bins wider.

- Fig. 5:
Add "CSM" to "coulour singlet model CSM [35]" in the caption.
With colours the model regions are difficult to distinguish;
in black-and-white even harder.

- l.216: 
Suggest to replace "our data" by "the LHCb data". 
In this paper "we" has not been used. 
And in line 232 you do refer to "the LHCb data".

- l.226:
Replace "does not exhibit either large transverse or longitudinal polarization"
by "does exhibit neither large transverse nor large longitudinal polarization".

- l.286
[16] collaboration