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Relativistic hydrodynamics with spin in the presence of electromagnetic fields

Rajeev Singh

a,b,c,, Masoud Shokric, S. M. A. Tabatabaee Mehrd a Institute of Nuclear Physics Polish Academy of Sciences PL-31-342 Krakow Poland

bCenter for Nuclear Theory Department of Physics and Astronomy Stony Brook University Stony Brook New York 11794-3800 USA

cInstitute for Theoretical Physics Goethe University Frankfurt Max-von-Laue-Strasse 1 D-60438 Frankfurt am Main Germany

dSchool of Particles and Accelerators Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5531 Tehran IranAbstract

We extend the classical phase-space distribution function to include the spin and electromagnetic fields coupling and derive the

modified constitutive relations for charge current, energy-momentum tensor, and spin tensor. Because of the coupling, the new

tensors receive corrections to their perfect fluid counterparts and make the background and spin fluid equations of motion commu-

nicate with each other. We investigate special cases which are relevant in high-energy heavy-ion collisions, including baryon free

matter and large mass limit. Using Bjorken symmetries, we find that spin polarization increases with increasing magnetic field for

an initially positive baryon chemical potential. The corrections derived in this framework may help to explain the splitting observed

in Lambda hyperons spin polarization measurements.1. Introduction Over the last few decades, relativistic heavy-ion collision ex- periments at RHIC and LHC have provided a unique oppor- tunity for the study of the properties of hot and dense rela- tivistic nuclear matter under extreme conditions [ 1 ]. The sys- tem produced by the colliding nuclei at high collision energies has been shown to quickly evolve from the initial nonequilib- rium glasma state to an equilibrated quark-gluon plasma (QGP) phase and eventually recombine into hadrons below the freeze- out temperature [ 2 3 4 5 ]. Using the theory of relativistic hy- drodynamics it was shown that QGP matter produced this way forms the smallest fluid droplets ever observed which exhibit a nearly perfect fluid behavior [ 6 7 8 9 10 ]. The success of hy- drodynamic description of relativistic heavy-ion collisions has opened new avenues in theoretical studies of relativistic matter, including the development of the theory of hydrodynamics and its oequilibrium applications [11]. Particularly interesting di- rections of study which are recently being followed include in- vestigations of spin polarization phenomena in relativistic nu- clear matter [ 12 13 ] and its dynamics under the influence of electromagnetic fields (EM) [ 14 15 16 ]. Recent spin polariza- tion measurements of emitted particles provided a new probe for studying QGP [ 17 18 19 20 21
22
23
24
] and triggered many theoretical developments [ 25
26
27
28
29
30
31
32
33
34
], see also Refs. [ 35
36
37
38
39
40
41
42
43
44
Being oriented along the direction of the global angular mo- mentum of the colliding system, microscopically, the spin po- larization is believed to arise because of the spin-orbit cou- pling [ 45
]. On the other hand, since at the macroscopic level,

Corresponding author

Email addresses:rajeev.singh@stonybrook.edu(Rajeev Singh ), shokri@itp.uni-frankfurt.de(Masoud Shokri),tabatabaee@ipm.ir (S. M. A. Tabatabaee Mehr)the QGP admits close-to-equilibrium dynamics, it is believed that the spin degrees of freedom undergo the thermalization as well. This may allow generation of spin polarization through the coupling between the fluid vorticity and spin of its con- stituents [ 46
]. While the agreement between the global polar- ization measurements and "spin-thermal" models supports the hypothesis of polarization-vorticity coupling [ 47
48
49
50
51
52
], the same theories do not quite explain the measure- ments related to dierential observables [48,51 ,53 ]; though there are some recent advances in this direction [ 54
55
56
57
Discrepancies between the theoretical predictions and the ex- perimental data indicate that the current theoretical under- standing of spin polarization dynamics in heavy-ion collisions is incomplete. If the spin degrees of freedom are thermal- ized, they should be incorporated into the hydrodynamic for- malism on the same footing as the other macroscopic quan- tities, allowing for their non-trivial dynamics. To this end, several frameworks considering spin as a dynamical quantity 58
], per- fect fluid spin hydrodynamics [ 25
26
], eective action ap- proach [ 59
38
41
60
61
], method of entropy current analy- sis [ 35
36
37
62
63
64
], statistical operator formalism [ 65
nonlocal collisions [ 66
67
68
39
69
42
70
71
72
73
], ki- netic theory for massless fermions [ 74
75
76
66
77
], holo- graphic method [ 78
79
40
41
80
81
], anomalous hydrody- namics [ 82
83
], and Lagrangian method [ 84
85
]. Also see

Refs. [

86
87
88
89
90
91
] for the studies related to helicity polarization. Relativistic hydrodynamics with spin proposed in

Refs. [

25
26
] was further developed [ 27
28
29
30
31
32
] and also extended to dissipative systems in Refs. [ 33
34
]. In its cur- rent form, the spin hydrodynamics framework does not include interactions with the electromagnetic field which may possibly be present in the early-time evolution of QGP. Such a coupling may be crucial for the explanation of the splitting inand¯ Preprint submitted to Nuclear Physics AApril 20, 2023 spin polarization signal observed in the experiments [17,92 ], which may arise because of opposite magnetic moments of these particles [ 47
], for other approaches, see Refs. [ 93
88
]. In this work, we extend the spin hydrodynamics framework devel- opedinRefs.[ 27
28
29
]andincludethecouplingbetweenspin and electromagnetic fields in the phase-space distribution func- tion of the constituent particles. We obtain modified constitu- tive relations for the baryon charge current, energy-momentum tensor, and spin tensor arising because of the coupling. Then, we investigate special cases of these quantities for the baryon- free system and the large mass limit, which are relevant to the physics of ultra-relativistic heavy-ion collisions. Finally, we study equations of motion in the case of Bjorken symmetry in the ideal MHD limit and obtain the evolution of the spin com- ponent. Although the qualitative features of spin polarization remains same as in Ref. [ 29
] due to strong symmetries assumed herein, however, we find that magnetic field positively enhances the spin polarization for an initial positive baryon chemical po- tential. We think that in more realistic setup our framework will prove to be crucial in understanding the splitting betweenand¯spin polarization.

2. Conventions

We use the mostly-minus Minkowski metric signature which readsg=diag(+1;1;1;1). As a result, the fluid four- velocity is normalized asUU=1. The operator= g

UUprojects tensors on the space transverse toU.

A tensorMcan be decomposed into a symmetricM()

12 M +Mand an asymmetricM[]12 M

Mpart.

WedenoteLevi-Civitasymbolas

whichistotallyantisym- metric and use the convention of0123=0123=1. Euclidean three-vectors are denoted with boldface, likeB, as opposed to four-vectors. For the scalar and Frobenius product we use the notationababandA:BAB, respectively. Through- out the paper we assume natural unitsi.e. c=~=kB=1, unless stated otherwise.

3. One-particle distribution function: coupling spin to EM

fields In this section we extend the classical phase-space spin dis- tribution function [ 28
33
] by introducing a term coupling the spin to the external EM fields, in a way suggested in Ref. [ 12 For the classical treatment of particles having spin one half and massm, phase-space distribution function with spin reads [ 28
33
f

0(x;p;s)=f0(x;p)exp"12

!(x) :s(p)# ;(1) where!(x) is the spin polarization tensor ands(p) repre- sents the particle internal angular momentum expressed with spin four-vectorsand four-momentumpas s =1m p s:(2)In Eq. (1),f0(x;p)=expp(x)(x)is the J¨uttner distri- bution, with(x) being the ratio of the baryon chemical poten- tial(x) over temperatureT(x),==T, and(x) is the ratio of the fluid four-velocityU(x) to temperature,=U=T.

Note that, the classical distribution function (

1 ) is valid only for the case of local collisions between particles, however, it can be extended to include nonlocal eects through the gradients of f

0(x;p;s) [28].

We generalize the phase-space distribution function ( 1 ) to a case of interaction between the particle magnetic moment and the external EM field by introducing the modified distribution function in the form f s(x;p;s)=f0(x;p;s)exp[M(x)F(x) :s];(3) whereFis the Faraday tensor expressed in terms of electric E and magneticBfour-vectors as F =EUEU+UB;(4) with E

FU;B12

FU:(5)

In Eq. (

3 ),M=M=TwhereM=gQNis the magnetic mo- ment of the quasiparticles andNbeing the nuclear magneton. In this work, for simplicity, we assume that the quasiparticles arehyperons, withg=0:61380:0047 [94]. However, it should be mentioned that, a more realistic setup needs multi- ple quark-like quasiparticles, with constitutive masses, see for example [ 31
quotesdbs_dbs29.pdfusesText_35

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