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Hydrodynamics of spin currents
by Domingo Gallegos, Umut Gursoy, Amos Yarom
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Authors (as registered SciPost users):  Umut Gursoy · Amos Yarom 
Submission information  

Preprint Link:  https://arxiv.org/abs/2101.04759v2 (pdf) 
Date submitted:  20210415 01:55 
Submitted by:  Gursoy, Umut 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We study relativistic hydrodynamics in the presence of a non vanishing spin chemical potential. Using a variety of techniques we carry out an exhaustive analysis, and identify the constitutive relations for the stress tensor and spin current in such a setup, allowing us to write the hydrodynamic equations of motion to second order in derivatives. We then solve the equations of motion in a perturbative setup and find surprisingly good agreement with measurements of global $\Lambda$hyperon polarization carried out at RHIC.
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Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 202175
(Invited Report) Cite as: Anonymous, Report on arXiv:2101.04759v2, delivered 20210705, doi: 10.21468/SciPost.Report.3194
Report
The paper “hydrodynamics of spin currents” performs a study of the firstorder theory of hydrodynamics that results from including spin currents in the effective gradient expansion of relativistic hydrodynamics. The study is well motivated from at least two points of view. Firstly it is always of interest to formulate effective descriptions of symmetryallowed quantities, we never know when and in what context they become useful. Secondly, a potentially useful applications of this theory is in the context of heavyion physics and spin polarization (of e.g. the Lambda hyperon). Indeed the authors include a preliminary analysis of their hydrodynamic theory to this case of interest.
Technically, while involved, the paper follows the standard route of constructing such hydrodynamic theories, as described clearly in Landau and Lifshitz, and the results appear trustworthy. As the authors note themselves it would be desirable in the future to use perhaps more modern approaches (e.g. coupling to arbitrary backgrounds) in order to obtain potentially further constraints on the transport coefficients appearing in this work.
The only minor complaint I have (and I would make addressing this optional) is presentational and notational. Firstly, in a paper with as many symbols and indices as this one, I would prefer it if every symbol were defined right after its first appearance. As obvious as it might seem to the authors, this is for example not done for the spin current which first appears in Eq. (1) and is not formally identified until much later.
Secondly, I would recommend denoting the ‘ideal’ part of the stress tensor and the spin current not by the subscript `i’, but in some other way (e.g {\rm id} or {\rm ideal}.) as it is too easily confused with one of the many indices.
Typo: below equation (7) the authors refer to ‘parameteric’, when I think they mean ‘parametric’
Report 1 by Michal P. Heller on 2021529 (Invited Report)
 Cite as: Michal P. Heller, Report on arXiv:2101.04759v2, delivered 20210529, doi: 10.21468/SciPost.Report.2992
Report
Hydrodynamics with spins is one of the emerging subjects in the topics of heavyion collisions  the primary subject of the present manuscript  with strong relevance also for condensed matter physics.
The authors provide a comprehensive analysis of hydrodynamic with spins from the point of view of effective field theory and characterize contributions to hydrodynamic constitutive relations up to and including first order in derivatives. Subsequently, they perform a phenomenological analysis and, despite using a rather simplistic model, they manage to fit experimental data upon fitting a single parameter, which I find impressive.
The paper is well written and comprehensive and I will be happy to endorse it for publication upon I hear from the authors their response on the following, mostly presentational points:
1) Introduction section contains many flashforwards to the results, so in my view it would be more appropriate to call it an introduction and summary.
2) I am not sure I get why the authors a dot over nabla to denote a covariant derivative. They use regular partial derivatives symbol and they I believe regular nabla would be enough. I am writing this because the paper is heavy in notation and steps to lighten it would be welcome.
3) The original Bjorken paper on the boostinvariant flow is not cited anywhere in the paper (might be relevant for the condensed matter audience, although the flow itself is well introduced in the paper).
4) Since by now there are many works on spin hydrodynamics and the paper aims to be comprehensive, it would be very useful for a reader (including myself) to see which parts/aspects of the constitutive relations the authors derive were already known and why are genuinely new.
5) Regarding the phenomenological analysis the authors did, it would be likewisely interesting to know if such an analysis relies on the fact that they added some new terms to the constitutive relations that improve the agreement with the data or simpler constitutive relations would do or Bjorken flow + linearization are just sensitive to some parts of the constitutive relations.
6) In practical applications to heavyion collisions one would need to use equations of motions that are hyperbolic in nature akin to IsraelSteward formulation of relativistic viscous hydrodynamics or the more recent BemficaDisconziNoronha/Kovtun formulation. It would be useful if authors make this point apparent in their study.
Author: Umut Gursoy on 20210621 [id 1514]
(in reply to Report 1 by Michal P. Heller on 20210529)
We thank the referee for a careful reading of the manuscript and insightful comments. We will respond to the comments in the same order as given by the referee:
 We will modify the title of the first section as requested.
 The difference between a covariant derivative with and without ring is that the former refers to derivative with Christoffel connection (no torsion) and the latter, one with torsion. The referee is right that we do not use the latter in this short paper as we expand expressions in torsion. However, we prefer to keep the notation with the ringed nabla in face of our upcoming works where we do need to make the distinction between the two clear. In particular, the torsionful derivative becomes highly useful in a hydrodynamic expansion where torsion is kept finite.
 We will add a reference to Bjorken's original paper.
 As far as we know the full consistent constitutive relations were not derived anywhere in the literature so the bulk of the results presented in the draft are new. We will add a sentence explaining this in the paragraph before the last in section I.
 Our phenomenological analysis crucially depend on the new terms added to the constitutive relations. In particular Bjorken + fluctuation would result in vanishing spin current without the transport coefficients $l_1$ and $l_2$. In this sense, these terms certainly affect the agreement with data. We will add a sentence in the last paragraph of section 4 to make this clear.
 We agree. We will add a sentence in the Discussion to address the referee's comment on the BDN/K frame.
We hope that we successfully addressed the comments of the referee with the aforementioned modifications (which will be listed in a separate accompaniment) and that our modified draft can be granted publication.
Author: Umut Gursoy on 20210722 [id 1605]
(in reply to Report 2 on 20210705)We thank the referee for the detailed reading and insightful comments. Indeed our paper is technical. While we agree with the referee that defining every symbol after its first appearance is helpful, we prefer not to do this with the spin current below Eq. (1) because it is clearly mentioned in the paragraph just above that equation. Instead, we went through the draft to check if there are any symbols left undefined and discovered one: $\tilde m_{ab} = 1/2 \epsilon_{abcd} u^c m^d$ in eq. (11b) which we define in the new version below that equation. We agree that using subscript "i" denoting ideal currents is confusing hence we replaced it with "id" throughout. We also corrected the typo mentioned by the referee.