Hydrodynamics of spin currents
HHydrodynamics of spin currents
A.D. Gallegos and U. G¨ursoy
Institute for Theoretical Physics, Utrecht University,Leuvenlaan 4, 3584 CE Utrecht, The Netherlands
A. Yarom
Department of Physics, Technion, Haifa 32000, Israel (Dated: January 14, 2021)We study relativistic hydrodynamics in the presence of a non vanishing spin chemicalpotential. Using a variety of techniques we carry out an exhaustive analysis, and iden-tify 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 deriva-tives. We then solve the equations of motion in a perturbative setup and find surprisinglygood agreement with measurements of global Λ-hyperon polarization carried out at RHIC.
I. INTRODUCTION
The hydrodynamic behaviour of a spin current hasbeen playing an increasingly prominent role in a vari-ety of physical systems ranging from heavy ion collisionsto condensed matter experiments. In particular, the re-cent observation of global spin polarization of the Λ andΛ particles in heavy-ion collisions at RHIC [1, 2] and theexperimental realization of spin currents induced by vor-ticity in liquid metals [3] have aroused strong interestin the subject, calling for a theoretical underpinning ofhydrodynamics in the presence of a spin current.The derivation of a complete and consistent set of con-stitutive relations in spin hydrodynamics is lacking inthe literature. The main goal of this work is to providethe tools for carrying out such an analysis and to usethese tools to obtain the constitutive relations for a par-ity invariant and conformal fluid to subleading order ina derivative expansion.Recall that hydrodynamics is a universal low energy ef-fective field theory of many body, finite temperature sys-tems. The equations of motion of hydrodynamics consistof local conservation laws (e.g., energy momentum con-servation or charge conservation). The dynamical vari-ables are given by a temperature field T , a velocity field u µ (which we normalize such that u µ u µ = − µ ab . We denote the vielbein by e µa and will use itto convert spacetime indices to tangent bundle indices.For a system where the only conserved charge is thespin current and energy momentum tensor, and in the absence of anomalies, one finds˚ ∇ µ T µν = 12 R ρσνλ S ρλσ − T ρσ K νab e ρa e σb ˚ ∇ λ S λµν = 2 T [ µν ] − S λρ [ µ e ν ] a e ρb K λab , (1)where A [ αβ ] = ( A αβ − A βα ), R αβγδ is the Riemanntensor and K µab is the contorsion tensor, related tothe spin connection, ω µab via ω µab = ˚ ω µab + K µab where ˚ ω µab = e νa (cid:16) ∂ µ e νb + ˚Γ νσµ e σb (cid:17) with ˚Γ αβγ = g αδ (cid:0) ∂ [ β g γ ] δ − ∂ δ g αβ (cid:1) . That is, ringed connectionsand derivatives denote expressions evaluated using theChristoffel connection. Of course, the torsion and curva-ture should be set to zero when considering, e.g., heavyion collisions.To obtain the explicit form of the equations of mo-tion for the hydrodynamic variables one needs a set ofconstitutive relations whereby all the conserved chargedensities (including the energy momentum tensor) areexpressed in terms of T , u µ , the relevant chemical poten-tials and their derivatives. These constitutive relationsmust satisfy certain criteria which have been shown tobe captured by the second law of thermodynamics, atleast to leading order in a derivative expansion [4–10].Often, such constitutive relations are expressed in termsof a truncated expansion in derivatives of the hydrody-namic variables. As we will discuss at length shortly, anunusual feature of hydrodynamics with a spin current isthat the spin chemical potential is naturally associatedwith terms which are first order in derivatives.In this work we compute the constitutive relations forthe stress tensor T µν and spin current S µαβ of a par-ity invariant conformal theory in 3 + 1 dimensions, in aflat, torsionless background geometry, including all termswhich contribute to the equations of motion expanded tosecond order in derivatives. We find a r X i v : . [ h e p - t h ] J a n T ( µν ) = (cid:15) T u µ u ν + 13 (cid:15) T ∆ µν − η T σ µν + T ( µν ) BR + O ( ∂ ) T − T [ µν ] =∆ β [ µ u ν ] (cid:16) (cid:96) D α σ αβ + (cid:96) D α ˆ M αβ (cid:17) + (cid:96) ∆ ρ [ µ ∆ ν ] σ ˚ ∇ ρ ˆ m σ + (cid:96) u [ µ σ ν ] ρ ˆ m ρ + (cid:96) u [ µ ˆ M ν ] ρ ˆ m ρ + (cid:96) u [ µ M ν ] ρ ˆ m ρ + (cid:96) σ [ µρ ˆ M ν ] ρ + (cid:96) σ [ µρ M ν ] ρ + (cid:96) ˆ M [ µρ M ν ] ρ − T − S [ µν ] ρ (cid:18) a ρ −
13 Θ u ρ (cid:19) + T − (cid:18) S ρρ [ µ a ν ] −
13 Θ S ρρ [ µ u ν ] (cid:19) + T [ µν ] BR T − S µνρ = 8 ρ u λ M νρ + 2 s u λ u [ ν ˆ m ρ ] + 2 s u λ ˆ M νρ + S BRλνρ (2) with T µνBR = 12 ˚ ∇ λ (cid:0) S BRµνλ + S BRνµλ − S BRλνµ (cid:1) S BRλµν =2 T χ ∆ λ [ µ u ν ] + 2 T χ M λ [ µ u ν ] + 2 σ T σ λ [ µ u ν ] + 2 σ T ˆ M λ [ µ u ν ] + 2 σ T ∆ λµ ˆ m ν ] (3)where we have decomposed the spin chemical potentialinto transverse components, µ ab = 2 u [ a m b ] + M ab (4)with m a u a = 0 and M ab u b = 0 and definedΘ = ˚ ∇ λ u λ , a µ = u α ˚ ∇ α u µ , ∆ µν = g µν + u µ u ν , Ω µν = ∆ µα ∆ νβ ˚ ∇ [ α u β ] ,σ µν = ∆ µα ∆ νβ ˚ ∇ ( α u β ) −
13 ∆ µν Θ , (5)which correspond to expansion, acceleration, the trans-verse projector, vorticity and the shear tensor, respec-tively. Caligraphic derivatives and hatted quantities aregiven by D α σ αβ = ˚ ∇ α σ αβ − a α σ αβ , D α ˆ M αβ = ˚ ∇ α ˆ M αβ − a α ˆ M αβ , ˆ m µ = m µ − a µ , ˆ M µν = M µν + Ω µν , (6)and circular brackets denote a symmetrized decomposi-tion of indices, viz., T ( µν ) = ( T µν + T νµ ).Adding terms of the form (3) to the stress tensor andcurrent is usually referred to as a Belinfante Rosenfeldtransformation [11, 12]. Such terms will not modify theequations of motion and are often used to generate a sym-metric stress tensor and vanishing spin current from anasymmetric stress tensor and its associated spin current.As we will see shortly, such terms should not be removed.While they do not contribute to the equations of motion,they do contribute to the expectation value of the stresstensor and current. This was first remarked on in [13].Spin current hydrodynamics may be relevant for thestudy of hyperon polarization measurements in heavyion collisions. The prediction of global spin polariza-tion in heavy-ion collisions, based on perturbative QCD, was initiated in [14–18]. The need for a hydrodynamictreatment of spin polarization in heavy-ion collisions waslater emphasized and discussed in [19, 20] (see [21] fora review). This thread was continued by studies of en-tropy production in [22] and a classification of spin cur-rent sources and hydrodynamic constitutive relations in[23]. The latter work also pioneered a holographic studyof spin transport. The recent [24] introduces an alterna-tive approach to spin hydrodynamics.In section IV we carry out a simple analysis of per-turbed Bjorken flow associated with the constitutive re-lations (2). Using some coarse approximations we find asimple one parameter model that nicely fits the experi-mental results. See figure 1. II. SPIN CURRENT HYDROSTATICS
In the presence of time independent sources such asan external metric or gauge field, the fluid is expected toreach a time independent hydrostatic equilibrium config-uration whereby Euclidean correlators of the theory de-cay exponentially. This exponential decay implies thatmomentum space correlation functions at zero frequencyare analytic in the spatial momenta implying that theirassociated generating function will be a local function ofthe background fields. Such a generating function wascomputed explicitly in [4, 5]. In what follows we use thesame technique to study hydrostatically equilibrated spincurrent dynamics.The sources which couple to the energy momentumtensor and spin current are the vielbein e aµ and spinconnection ω µab , δS = (cid:90) d x (cid:112) | e | (cid:18) T µa δe aµ + 12 S λab δω λab (cid:19) , (7)where the integral is over all space dimensions and a com-pact Euclidean time direction with parameteric length T − . In a hydrostatic setting the sources will be timeindependent, viz. £ V e aµ = 0 , £ V ω abµ = 0 (8)where V µ points in the time direction and £ V denotesits associated Lie derivative. The generating functionfor hydrodynamics with a spin current will be given bya local diffeomorphism and Lorentz invariant expressionconstructed out of the sources e aµ and ω µab , and the timedirection V µ .With some prescience let us denote T = T √− V , u µ = V µ √− V , µ ab = ω µab V µ √− V . (9)These quantities will correspond to the hydrostatic tem-perature, velocity field and spin chemical potential re-spectively. To see this we consider the most general gen-erating function which will lead to constitutive relationswhich contain no derivatives of the parameters (9),ln Z i = W i = (cid:90) d x | e | P ( T, M , m · ˜ M , m ) (10)where M = 2 M µν M µν and m · ˜ M = m α M βγ u δ (cid:15) αβγδ .We will refer to a fluid whose constitutive relations arecompletely determined by (10) as an ideal fluid.The current and stress tensor associated with (10) aregiven by T αβi = (cid:15)u α u β + P ∆ αβ − (cid:18) ∂P∂m + 4 ∂P∂M (cid:19) u α M βγ m γ S λi αβ = u λ ρ αβ , (11a)with (cid:15) = − P + ∂P∂T T + 12 ρ ab µ ab ,ρ αβ =8 ∂P∂M M αβ + ∂P∂m · ˜ M (cid:16) m αβ − u α ˜ M β + ˜ M α u β (cid:17) + 2 ∂P∂m ( u α m β − m α u β ) , (11b)once we restrict ourselves to a flat, torsionless geometry.There are several lessons to be learnt from (8) through(11). First, note that the identifications (9) yield the ex-pected Gibbs Duhem relations once we identify P withthe pressure and s = ∂P/∂T with the entropy density s . As we will show in the next section, the entropy cur-rent J µ = su µ is conserved for the ideal fluid, once theequations of motion are satisfied.Second, since all sources are time independent, we findthat (9) imply the hydrostatic relations k ab ≡ u µ K µab = µ ab + e aµ e bν (cid:16) Ω µν − u [ µ a ν ] (cid:17) ,T e ρa e σb ˚ ∇ λ µ ab T = R ρσλα u α , a µ = − ˚ ∇ µ TT , (12)The first equality in (12) has been mentioned in [18, 25]in the absence of torsion. It implies that a non vanishingspin chemical potential must be supported by fluid vor-ticity or by acceleration (or, alternatively, temperaturegradients) in order to maintain thermal equilibrium. In the absence of torsion, a non flat metric, or other externalforces, the fluid will eventually settle down to a thermallyequilibriated steady state in which the velocity field andtemperature are covariantly constant. The first equalityin (12) implies that the spin chemical potential must van-ish in such an equilibrated state. Therefore, if we wishto construct a gradient expansion around an equilibriumconfiguration we must count the spin chemical potentialas first order in derivatives.Classifying the spin chemical potential as a first or-der in derivatives term implies that the timelike compo-nents of the torsion tensor, k ab , are also first order inderivatives. It remains to classify the transverse compo-nents of torsion, κ νab = ∆ µν K νab . In what follows weconsider κ µab as first order in derivatives, but it shouldbe possible to set κ µab to be zeroth order in derivativesyielding torsio-hydrodynamics, an analog of magnetohy-drodynamics.Before proceeding with higher order corrections to theideal fluid, we remark that (12) implies that in the ab-sence of torsion, M µν + Ω µν = 0 and m µ − a µ = 0.Thus, there is an ambiguity in determining the constitu-tive relations (11). Of course, such an ambiguity will beresolved once non hydrostatic corrections are taken intoaccount. In what follows we will consistently choose M µν and m µ as hydrostatic variables in the absence of torsionover Ω µν and a µ .The hydrostatic gradient corrections to the constitu-tive relations can be obtained by expanding the hydro-static generating function, W , in a derivative expansion.In this work we are interested in the equations of mo-tion expanded to second order in derivatives. Since theantisymmetric components of the stress tensor sourcesthe divergence of the spin current, we must expand theconstitutive relations associated with the antisymmetriccomponent of the stress tensor to second order in deriva-tives and the remaining constitutive relations to first or-der in derivatives. Therefore, to compute all possible cor-rections to the ideal fluid constitutive relations, we mustclassify all possible first order in derivative scalars whichcan contribute to the hydrostatic generating function, W ,and all possible second order in derivative scalars whichcan contribute to the antisymmetric components of thestress tensor. In order to limit the number of such termsand also simplify future expressions we assume that thesystem is invariant under parity and also conformally in-variant. A full analysis of the constitutive relations whichare not restricted by symmetry will be discussed in a fu-ture paper.The Weyl transformation of the spin connection asso-ciated with the Christoffel connection, ˚ ω abµ can be deter-mined from the Weyl rescaling of the vielbein, e aµ → e φ e aµ . In what follows we will assume that the spin con-nection transforms in the same way as ˚ ω µab . Alternately,that the contorsion tensor is inert under Weyl rescalingsof the metric. One can argue that if the contorsion ten-sor transforms non trivially under Weyl rescalings thenits transformation properties are such that a vanishingcontorsion tensor is conformally equivalent to a non van-ishing one [26, 27].Using (1) we find that the change in the stress tensorand spin current due to an infinitesimal Weyl rescaling isgiven by δT µν = − φT µν − S µνρ ∂ ρ φ + S λλµ ∂ ν φδS λµν = − φS λµν (13a)Using (7) we find that tracelesness is replaced by T µµ = ˚ ∇ µ S λλµ . (13b)We defer an extensive discussion of conformal invariancein the presence of torsion, and the recovery of the canoni-cal transformation laws for the stress tensor in its absenceto future work.It follows that the transverse part of the spin chemicalpotential, M µν , transforms homogenously under Weyl rescalings while m α does not. Thus, in a conformallyinvariant theory the pressure P in (10) can depend onlyon T and M . Counting M as second order in deriva-tives implies that (cid:15) = (cid:15) T , P = 13 (cid:15) T , ∂P∂M = ρ T , (14)up to second order in derivative corrections.It is now straightforward, though somewhat tedious toargue that the most general correction to W , W h , at theorder we are interested in is given by W h = (cid:90) d x | e | (cid:16) χ (1) T κ + 2 χ (2)1 T κ µνA M µν + 2 χ (2)2 T K µν M µν (cid:17) , (15) where χ (1) and the χ (2) j ’s are numbers, κ = u a e µb K µba , κ ανA = u β ∆ µ [ ν e α ] a K abµ e βb and K αβ = u µ ∆ αγ ∆ βδ e γc e δd K µcd . The stress tensor and spin cur-rent derived from (15) are given by T ( µν ) h =4 T χ (1) u ( µ m ν ) + O ( ∂ ) ,T [ µν ] h = T χ (1) (cid:16) u [ µ m ν ] − M µν (cid:17) + 4 T ( χ (2)1 − χ (2)2 ) u [ µ M ν ] α m α + 2 T χ (2)1 ˚ ∇ α M α [ ν u µ ] S λh ab =2 T χ (1) ∆ λ [ a u b ] − T χ (2)1 M λ [ a u b ] + 4 T χ (2)2 u λ M ab , (16)once we set the torsion to zero.The contribution of the term associated with χ (2)2 tothe constitutive relations is identical to that of an ideal,conformal fluid at second order in the derivative expan-sion. At least as far as the antisymmetric part of thestress tensor and the spin current are concerned. There-fore, we may, without loss of generality remove the former by an appropriate shift of the latter. III. SPIN CURRENT HYDRODYNAMICS
The remaining contributions to the stress tensor andcurrent, T µνr and S λr ab contain all possible expressionswhich do not vanish in equilibrium, are parity invariant,and satisfy (13). We find T − T ( µν ) r = − σ σ µν + (cid:16) σ − χ (1) (cid:17) u ( µ ˆ m ν ) − χ (1) Θ∆ µν − χ (1) Θ u µ u ν ,T − T [ µν ] r = T (cid:16) σ − χ (1) (cid:17) u [ µ ˆ m ν ] + T σ ˆ M µν + ∆ β [ µ u ν ] (cid:16) λ D α σ αβ + λ D α ˆ M αβ (cid:17) + λ ∆ ρ [ µ ∆ ν ] σ ˚ ∇ ρ ˆ m σ + λ u [ µ σ ν ] ρ ˆ m ρ + λ u [ µ ˆ M ν ] ρ ˆ m ρ + ( λ − χ (2)1 + 8 ρ ) u [ µ M ν ] ρ ˆ m ρ + λ σ [ µρ ˆ M ν ] ρ + λ σ [ µρ M ν ] ρ + λ ˆ M [ µρ M ν ] ρ + 23 χ (2)1 M µν Θ − T − S [ µν ] ρr (cid:18) a ρ −
13 Θ u ρ (cid:19) + T − (cid:18) S ρr ρ [ µ a ν ] −
13 Θ S ρr ρ [ µ u ν ] (cid:19) T − S λr ab =2 σ σ λ [ a u b ] + 2 σ ˆ M λ [ a u b ] + 2 σ ∆ λ [ a ˆ m b ] + 2 σ u λ u [ a ˆ m b ] + 2 σ u λ ˆ M ab , (17)A few comments are in order. To help the reader iden-tify the role of the various terms in (17) we have labeled coefficients associated with first order in derivative termsby σ i and coefficients associated with second order inderivative terms by λ i . While one often denotes the shearviscosity by η we have refrained from doing so for rea-sons that will become clear shortly. The χ (1) and χ (2) i dependent terms appearing in (17) have been introducedin order to ensure that (13) are satisfied out of equilib-rium. The same goes for the last two terms on the righthand side of the expression for T [ µν ] .Also, we have written (17) in what is usually referredto as the Landau frame where u µ is an eigenvector of thestress tensor with negative eigenvalue. Frame transfor-mations offer an additional freedom in redefining the spinchemical potential which we avoid using at this order inthe derivative expansion. The hydrostatic stress tensorand spin current, T µνi + T µνh are written in a hydrostaticframe which is more natural from the point of view ofthe hydrostatic partition function.To further simplify (17) it is convenient to make theredefinitions σ = 2 η + χ (1) , σ = η + χ (1) ,λ = (cid:96) + σ , λ = (cid:96) + σ ,λ = (cid:96) + σ , λ = (cid:96) − σ ,λ = (cid:96) − σ , λ = (cid:96) + σ , (18) λ = (cid:96) − σ + σ , λ = (cid:96) − χ (2)1 + σ ,λ = (cid:96) + 2 χ (2)1 − σ . With these redefinitions, the terms in the spin current as-sociated with χ (1) , χ (2)1 , and σ i with i = 1 , . . . , σ = 0 and η = 0.Combining (11), (16), (17) and (18), removing χ (2)2 following the discussion after (16), and setting σ = 0and η = 0 yields (2).We note in passing that the terms associated with σ and σ can also be packaged as a BelinfanteRosenfeld term by adding a λ T u λ u [ µ D λ ˆ m ν ] and a λ T u λ D λ ˆ M µν term to the antisymmetric part of thestress tensor and then redefining λ = (cid:96) + σ and λ = (cid:96) + σ . (with u λ D λ ˆ m µ = u λ ˚ ∇ λ + θ ˆ m µ − m · ˆ m + ˆ m · ˆ mu µ and u λ D λ ˆ M µν = u λ ˚ ∇ λ ˆ M µν + θ ˆ M µν +2 u [ µ ˆ M ν ] λ m λ − u [ µ ˆ M ν ] λ ˆ m λ ) The reason these last twoterms don’t appear in (17) is that we have substitutedthose expressions with their values under the equationsof motion.The various coefficients multiplying the tensor struc-tures in (2), e.g., η , are restricted by positivity of entropyproduction, unitarity of retarded correlation functions or unitarity of the Schwinger-Keldysh generating function[7–10]. In what follows we will study restrictions on thecoefficients in (2) coming from positivity of entropy pro-duction. A study of the restrictions on coefficients viaother methods is left for future work.Following [28] we posit the existence of an entropy cur-rent J µS satisfying ˚ ∇ µ J µS ≥ J µS = su µ with s = ∂P/∂T . For a non ideal fluid we take J µS = su µ + O ( ∂ )where O ( ∂ ) denotes corrections to the entropy currentcoming from explicit derivative terms appearing in theconstitutive relations. Thus, the most general entropycurrent we may construct, to first order in derivatives isgiven by J µS = J µc + ( s Θ u µ + s a µ + s m µ ) T . (19)where J µc = su µ − u ν T ( T µν − T µνi ) − µ ab T ( S µab − S µi ab ) (20)is referred to as the canonical part of the entropy current.In a conformal theory the s i are constant.When expanding the entropy current to first order inderivatives, the divergence of the entropy current is asecond order in derivatives scalar. It is useful to classifythe latter into two categories. The first are independentsecond order scalars, these are scalars which can not bewritten as products of first order scalars. The secondincludes products of first order scalars. All independentsecond order scalars appearing in the divergence of theentropy current must vanish on account of the positivitycondition. For the same reason all products of first orderscalars must arrange themselves into complete squares orvanish.It is straightforward to show that˚ ∇ µ J µc = − ˚ ∇ µ (cid:16) u ν T (cid:17) ( T µν − T µνi ) −
12 ˚ ∇ µ (cid:18) µ ab T (cid:19) ( S µab − S µi ab ) − µ ab T T ab . (21)Inserting (19) into ˚ ∇ µ J µS ≥ s = − χ , s = χ , s = − χ . η ≥ . (22)Let us make the following remarks. Since the spinchemical potential is first order in derivatives the n − n ’th order entropycurrent. Thus, the first order entropy current can onlyconstrain the first order energy momentum tensor andzeroth order spin current. In practice, it constrains only η , the shear viscosity.To determine constraints on the first order terms in thespin current one would need to go to second order in theentropy current. While we have not carried out such ananalysis, we note that, at least for spin-less charged flu-ids, all constraints from the entropy current which implyequality type relations among transport coefficient are al-ready implemented from the partition function. Further,all inequality type constraints appear at leading order inthe entropy current [6].Another somewhat unusual feature of hydrodyanmicswith a spin current is that the coefficient of the shearterm in T µν is − ( η + χ ) T , c.f, (18). Nevertheless, itis η that is constrained to be positive which is perhapscompatible with the fact that χ does not enter into theequations of motion. We have checked that positivity of η also follows from positivity of the approprtiate stresstensor correlator. Note that a computation of two pointfunctions of the stress tensor require knowledge of theexpectation value of the stress tensor in the presence ofa background metric and spin connection which we havenot presented here. IV. AN APPLICATION TO HEAVY IONCOLLISIONS
In this short letter we do not presume to carry out a fullfledged analysis of heavy ion collision experiments withpossible spin currents manifesting during the short colli-sion period. Instead, we consider a perturbed solution tothe hydrodynamic equations of motion in the presence ofspin with an underlying Bjorken ( SO (1 , × ISO (2) × Z )symmetry. We then attempt to relate the dependence ofthe spin chemical potential on the initial temperatureto the dependence of the average hyperon polarizationvector on the beam energy. Of course, a complete analy-sis, which we do not carry out in this short letter, shouldinclude a proper treatment of initial conditions, a full hy-drodynamic simulation, and a comprehensive treatmentof hadronization of the quark gluon plasma before reach-ing the detector.Consider a collision of two gold ions of radii R initiallymoving with a relativistic velocity directed along a Carte-sian ‘ z ’ coordinate. Let’s assume that the fluid formedafter the collision has Bjorken symmetry, that is, it isinvariant under boosts along the beam direction, trans-lations and rotations along the ‘ x ’ and ‘ y ’ directions, andunder z/t → − z/t . Going to a Milne coordinate system, ds = − dτ + τ dη + dx + dy where τ = √ t − z and η = arctanh( z/t ) are proper time and pseudo-rapidityrespectively, we find that u τ = 1 , T = T (cid:16) τ τ (cid:17) − η (cid:15) τ , (23)with all other components of u µ and µ ab vanishing, solvethe equations of motion. Here T is the temperature atthe initial time τ when the fluid description is a viableone. Note that the shear viscosity to entropy ratio, η/s ,satisfies η/s = 3 η / (cid:15) .Since the spin chemical potential vanishes on accountof Bjorken symmetry, let’s consider linear perturba-tions of Bjorken flow which break transverse transla-tions and axial rotation, T → T + (cid:82) d qδT e i ( q x x + q y y ) , u µ → u µ + (cid:82) d qδu µ e i ( q x x + q y y ) , and µ ab → µ ab + (cid:82) d qδµ ab e i ( q x x + q y y ) . To mimic the experiment, we con-sider a peripheral collision with impact parameter b alongthe ‘ x ’ axis. Glancing beams are expected to create anon-trivial velocity gradient in the x direction at initialproper time τ at which we assume hydrodynamics be-comes applicable. To this end, we consider an initial ve-locity profile where δu η ( τ ) ∝ bq x , and other componentsof the perturbations to the velocity vanish. As a result,we find that δm η , δM ηx = δM η iq x and δM ηy = δM η iq y are non zero while the temperature perturbations and allother components of the spin chemical potential vanish.To solve the equations of motion we will usethe Floerchinger-Wiedemann (FW) approximation [29],where η (cid:15) T τ is perturbatively small but q τ η (cid:15) T τ (with q = q x + q y ) is finite. In this approximation, only theleading term for the temperature in (23) becomes rele-vant, the velocity field perturbations take the form δu η = iu b q x τ − e − q η τ T (cid:15) (cid:16) ττ (cid:17) , (24)and δM η and δm η are determined algebraically from δu η and its derivatives.Presumably, the stress tensor and spin current willevolve according to hydrodynamic theory from an ini-tial Bjorken time τ to a final time τ f when matterhadronizes, T ( τ f ) = T f (cid:39) M eV . The hadrons yieldis then collected by the detector which measures its prop-erties. Converting a hydrodynamics spin current andenergy momentum tensor to a Hadron distribution isfrought with difficulty. One often used prescription fordoing so works under the assumption that the particledistribution after hadronization follows a thermal distri-bution with temperature, velocity and chemical poten-tial of the hydrodynamic configuration leading to it [30].Within this framework the polarization vector readsΠ µ ( p ) = − (cid:15) αρσβ p β m (cid:82) d Σ λ p λ Bµ ρσ (cid:82) d Σ λ p λ n F (25)where (cid:82) d Σ µ is an integral over the hadronization sur-face, d Σ µ = τ δ τµ dηdxdy in Bjorken coordinates, p µ is theparticle momentum, m its mass, n F is the Fermi Diracdistribution and B is an additional distributional quan-tity that depdends on u µ and T . See, e.g., [18, 31] fordetails.It is tempting to use our solution to evaluate (25) andcompare to data. However, one should keep in mind thatour hydrodynamic solution is rather simple minded, in-volving a linear perturbation on Bjorken symmetry ontop of which we used the FW approximation. This per-turbation should presumably capture a non vanishingimpact parameter. Realistic collisions at mid central-ity have an impact parameter of order of the nucleussize and are unlikely to resemble Bjorken flow. Theyshould generate a large enough vorticity for a non triv-ial spin current to be generated which makes the validityof our linearized approximation somewhat suspect. Still,we have at our disposal an analytic solution to the hy-drodynamic equations of motion with spin and it is hardto resist the temptation to compare it with the exper- imental results using (25). Hence, throwing caution tothe wind, and inserting the perturbed Bjorken solutioninto (25), we findΠ µ ( p ) = 16 be − T f (cid:15) x T η τ u π T f x (cid:15) ( (cid:96) + (cid:96) )Erf (cid:32)(cid:114) T f (cid:15) y T η τ (cid:33) mT η (cid:96) τ × − p y I (1) I (2) (26)where now x = R − b , y = (cid:113) R − b and we haveintegrated over the range − x < x < x and − y 52. See figure 1. V. DISCUSSION In this paper, we initiated a fully fledged study of rel-ativistic spin hydrodynamics. The hydrodynamic consti-tutive relations, relating the spin current and the stresstensor to fluid velocity, temperature, spin chemical po-tentials and their derivatives, can be found in (2). Forconsistency, the constitutive relations for the symmetricpart of the stress tensor and spin current were expandedto first order in derivatives while those for the antisym-metric part of the stress tensor were expanded to secondorder in derivatives. This mismatch in the derivative ex-pansion is a result of the hydrostatic equilibrium relation(12) between torsion, spin chemical potential, vorticityand acceleration which implies that the spin chemicalpotential and the longitudinal component of the torsionmust be first order in derivatives.In a background with vanishing torsion, such as theone discussed in this work, it is sensible to choose thetransverse components of the torsion tensor to be firstorder, as is the case with the longitudinal components.However, in certain condensed matter systems such asgraphene torsion is used as the long-wavelength descrip-tion of dislocations and disclinations in the atomic struc-ture [33, 34]. In these systems a sensible choice might beto keep the spatial torsion at zeroth order in derivativesalong with temperature and velocity. It would be inter-esting to develop such a torsio-hydrodynamic theory fur-ther, following the route we outlined here. Recent works[35–39] strongly suggest that graphene, as well as certainclean Dirac and Weyl semimetals, are well described byhydrodynamic theory which further motivates this study.Another possible extension of our current work is toextend our results to non parity invariant and conformalinvariant theories. Indeed, Bayesian analysis of heavy- ion data suggest that bulk viscosity plays an impor-tant role in the hydrodynamic description of heavy ioncollisions[40, 41]. Finally, for off-central heavy ion col-lisions, which is the regime where spin hydrodynamicsis most relevant, it is desirable to employ a more real-istic hydrodynamic configuration in which the rotationsymmetry around the beam axis is broken. VI. ACKNOWLEDGEMENTS We thank F. Becattini, K. Fukushima, G. Torrieri andJ. Zaanen for useful discussions. DG and UG are par-tially supported by the Delta-Institute for TheoreticalPhysics (D-ITP) funded by the Dutch Ministry of Ed-ucation, Culture and Science (OCW). 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