Spin Tensor and Pseudo-Gauges in Relativistic Nuclear Collisions

Spin tensor and pseudo-gauges in relativistic nuclear collisions

Nora Weickgenannt

NW, E. Speranza, X.-l. Sheng, Q. Wang. D. H. Rischke, arXiv:2005.01506 (accepted for PRL) E. Speranza, NW, EPJA 57 (2021) 5, 155

Workshop on QGP phenomenology | May 31, 2021

Spin hydrodynamic equations of motion CRC - TR

I Spin hydrodynamics is based on conserved quantities:

energy-momentum tensor µν ∂µT = 0

+ total angular-momentum tensor

λ,µν µ λν ν λµ λ,µν J = x T − x T + ~S orbital part spin tensor

λ,µν λ,µν [νµ] ∂λJ = 0 =⇒ ~∂λS = T a[µbν] ≡ aµbν − aν bµ

I Given a Lagrangian density L, how to calculate T µν and Sλ,µν ?

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Textbook: general relativity CRC - TR

I Physical objects determined by space-time geometry

I Energy-momentum tensor: variation with respect to metric tensor δL T µν = −2 δgµν

I Spin tensor: ?

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Textbook: quantum eld theory CRC - TR

I Conserved currents obtained from Noether's theorem, consider spinor elds

I Energy-momentum tensor: invariance under space-time translations ∂L ∂L T µν = ∂ν ψ + ∂ν ψ¯ − gµν L ∂(∂µψ) ∂(∂µψ¯)

I Spin tensor: invariance under rotations ∂L ∂L Sλ,µν = f µν ψ − ψf¯ µν ∂(∂λψ) ∂(∂λψ¯) generators of rotation: µν i µν f = − 2 σ

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Dirac Lagrangian CRC - TR

I Apply Noether's theorem to Dirac Lagrangian: i ←→ L (x) = ~ ψ¯(x)γ · ∂ ψ(x) − m ψ¯(x)ψ(x) D 2

I Spin tensor: 1 Sλ,µν = − λµναψγ¯ γ ψ C 2 α 5

I Energy-momentum tensor: i ←→ T µν = ~ ψγ¯ µ ∂ ν ψ C 2 =⇒ not symmetric in µ and ν!

I Why is this inconsistent with general relativity?

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Pseudo-gauge transformations CRC - TR

I Denition of T µν and Sλ,µν depends on choice of pseudo-gauge. F. W. Hehl, Rept. Math. Phys. 9, 55 (1976) F. Becattini, W. Florkowski, and E. Speranza, PLB789, 419 (2019) E. Speranza, NW, EPJA 57 (2021) 5, 155 L. Tinti, W. Florkowski, arXiv:2007.04029 S. Li, M. Stephanov, H.-U. Yee, arXiv: 2011.12318 I Pseudo-gauge transformation:

T 0µν = T µν + ~ ∂ (Φλ,µν + Φν,µλ + Φµ,νλ), C 2 λ 0λ,µν λ,µν λ,µν µν,λρ S = SC − Φ + ~ ∂ρZ =⇒ equations of motion invariant 0µν 0λ,µν ∂µT = 0 ∂λJ = 0 and global charges invariant Z Z ν µν 0µν P ≡ dΣµT = dΣµT Σ Σ Z Z µν λ,µν 0λ,µν J ≡ dΣλJ = dΣλJ Σ Σ Pseudo-gauge transformations change global spin Dierent splitting into spin and orbital angular momentum

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Canonical vs. Belinfante pseudo-gauge CRC - TR

I Currents derived from Noether's theorem: Canonical energy-momentum tensor µν and spin tensor λ,µν TC SC

I Apply pseudo-gauge transformation with

λ,µν λ,µν µν,λρ Φ = SC ,Z = 0 =⇒ Belinfante currents i ←→ ←→ T µν = ~ ψγ¯ µ ∂ ν ψ + ψγ¯ ν ∂ µψ B 4 λ,µν SB = 0

I Spin tensor vanishes, full angular momentum contained in orbital part

I Belinfante energy-momentum tensor couples to gravity in conventional general relativity =⇒ Traditionally considered as "physical" energy-momentum tensor =⇒ "Physical" pseudo-gauge in quantum theory?

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Observable in heavy-ion collisions CRC - TR

I "Physical" currents → enter observables

I Observable for polarization in heavy-ion collisions:

Pauli-Lubanski vector for particles with momentum p F. Becattini, V. Chandra, L. Del Zanna, E. Grossi, AP338, 32 (2013) F. Becattini, arXiv:2004.04050 E. Speranza, NW, EPJA 57 (2021) 5, 155 L. Tinti, W. Florkowski, arXiv:2007.04029 Z 1 ν λ,αβ Πµ = − µναβ p dΣλJ 2m Σ total angular momentum

=⇒ Form of Pauli-Lubanski vector independent of pseudo-gauge

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Are all pseudo-gauges equally "physical"? CRC - TR

I Πµ independent of pseudo-gauge

I However: need to calculate ensemble average of operator hΠµi ≡ Tr(ρ Πµ)

I Density matrix ρ not known

I Method to (approximately) calculate ensemble average can give dierent results for dierent pseudo-gauges F. Becattini, W. Florkowski, and E. Speranza, PLB789, 419 (2019) F. Becattini, NPA 1005 (2021) 121833 E. Speranza, NW, EPJA 57 (2021) 5, 155 K. Fukushima, S. Pu, PLB 817 (2021) 136346 A. Das, W. Florkowski, R. Ryblewski, R. Singh, PRD 103 (2021) 9, L091502

I hΠµi dependent on pseudo-gauge

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Spin tensor in hydrodynamics I CRC - TR

I Hydrodynamics: eective theory, never contains all microscopic information

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Spin tensor in hydrodynamics II CRC - TR

I Obtain dynamics of densities from equations of motion =⇒ Dierent dynamical quantities for dierent pseudo-gauges =⇒ Dynamics depends on pseudo-gauge

I Main idea of spin hydrodynamics: Promote spin tensor to additional dynamical variable

W. Florkowski, B. Friman, A. Jaiswal, and E. Speranza, PRC 97, no. 4, 041901 (2018) W. Florkowski, B. Friman, A. Jaiswal, R. Ryblewski, and E. Speranza, PRD 97, no. 11, 116017 (2018) W. Florkowski, F. Becattini, and E. Speranza, APB 49, 1409 (2018)

I Obviously not possible for Belinfante spin tensor =⇒ Spin dynamics cannot be described

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Canonical spin tensor CRC - TR

µν not symmetric for free elds I TC =⇒ canonical spin tensor not conserved

Physical picture: spin density changed only by interactions

Canonical global spin Z µν λ,µν SC ≡ dΣλSC Σ no Lorentz tensor

I In general: Hypersurface-integrated quantities transform as tensors under Lorentz transformations only if integrand conserved For proof see e.g. E. Speranza, NW, EPJA 57 (2021) 5, 155

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Hilgevoord-Wouthuysen (HW) pseudo-gauge CRC - TR

I Idea for free elds: Apply Noether's theorem to Klein-Gordon Lagrangian for spinors

J. Hilgevoord and S. Wouthuysen, Nuclear Physics 40, 1 (1963) 1 L = ( 2∂ ψ∂¯ µψ − m2ψψ¯ ) KG 2m ~ µ

I Result: 2 T µν = ~ ∂µψ∂¯ ν ψ + ∂ν ψ∂¯ µψ − gµν L HW 2m KG i 2 ←→ Sλ,µν = ~ ψσ¯ µν ∂ λψ HW 4m

I Energy-momentum tensor symmetric for free elds.

I Conserved (nonzero) spin tensor.

I There exists pseudo-gauge transformation from canonical to HW tensors

I Physical interpretation?

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Pseudo-gauge and frame choice I CRC - TR

Nonrelativistic spin operator given by Pauli matrices: 1 I 2 σ

I How to generalize to relativistic theory?

I Spin vector S connected to global spin by Sij = ijkSk. Obviously no Lorentz tensor.

I Make this covariant: µν µναβ Sn = − nαSβ µ µν Spin dened in the frame moving with four-velocity n ⇐⇒nµSn = 0.

I Dierent choices of pseudo-gauge: dierent choices of frame vector. M. H. L. Pryce, Proc. Roy. Soc. Lond., A195:6281, 1948 C. Lorcé, Eur. Phys. J. C (2018) 78:785 E. Speranza, NW, EPJA 57 (2021) 5, 155

I One preferred reference frame for massive particles: rest frame.

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Pseudo-gauge and frame choice II CRC - TR

Global spin from spin tensor: Z Sµν ≡ d3x S0,µν

I Canonical choice: =⇒ Spin tensor not conserved for free elds =⇒ Global spin no Lorentz tensor =⇒ Equal to nonrelativistic spin in any frame,

0ν µ SC = 0, nC = (1, 0, 0, 0).

I HW choice: =⇒ Spin tensor conserved for free elds =⇒ Global spin is Lorentz tensor =⇒ Equal to nonrelativistic spin in rest frame, 1 p Sµν = 0, nµ = pµ. µ HW HW m

Nora Weickgenannt Spin tensor and pseudo-gauges 14 / 22

Spin kinetic theory CRC - TR

NW, E. Speranza, X.-l. Sheng, Q. Wang, and D.H. Rischke, arXiv:2005.01506, 2103.04896

I Framework: Quantum eld theory =⇒ Wigner function =⇒ kinetic theory =⇒ hydrodynamics See talk by Enrico Speranza

I Phase-space distribution function f(x, p, s), depends on spin variable sµ

I Boltzmann equation p · ∂f(x, p, s) = C[f]

I Nonlocal collision term Z 0 C[f] = dΓ1dΓ2dΓ W [f(x +∆ 1, p1, s1)

0 0 0 × f(x +∆ 2, p2, s2) − f(x +∆ , p, s)f(x +∆ , p , s )]

Particle positions displaced by ∆µ

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Canonical currents from kinetic theory CRC - TR

I Canonical energy-momentum tensor Z T µν = dΓ pν pµ + ~ Σµλ∂ f(x, p, s) + O( 2) C 2 s λ ~

I Canonical spin tensor Z λ,µν λ µν µ νλ ν λµ SC = dΓ p Σs + p Σs + p Σs f(x, p, s)

Dipole-moment tensor 1 Σµν ≡ − µναβ p s s m α β I Global equilibrium: 2 Z λ,µν [µν] 1 ~ [ν µ]λ ρ −β·p 3 ~∂λSC,eq = −TC,eq = − dP p $ p $λρe + O(~ ) (2π~)3 4 Thermal vorticity $µν

Canonical spin tensor not conserved in global equilibrium Inconsistent with physical picture

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HW currents from kinetic theory CRC - TR

NW, E. Speranza, X.-l. Sheng, Q. Wang, and D.H. Rischke, arXiv:2005.01506

I Generalize HW pseudo-gauge transformation to interacting case: I Choice of Φλ,µν : RecoverHW tensors for zero interactions. Obtain physically meaningful equations of motion (see next slide).

I Result: Z µν µ ν 2 THW = dΓp p f(x, p, s) + O(~ ) , Z 1 Sλ,µν = dΓpλ Σµν − ~ p[µ∂ν] f(x, p, s) + O( 2) . HW 2 s 4m2 ~

I Global equilibrium: Sλ,µν = ~ uλ$µν n(0) HW,eq 4 Particle density n(0) Form of spin tensor widely used in eective approaches W. Florkowski, B. Friman, A. Jaiswal, and E. Speranza, PRC 97, no. 4, 041901 (2018) K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, H. Taya, PLB 795 (2019) 100-106 S. Li, M. Stephanov, H.-U. Yee, arXiv: 2011.12318 A.D. Gallegos, U. Gürsoy, A. Yarom, arXiv: 2101.04759

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Equations of motion with collisions CRC - TR

I Using Boltzmann equation Z µν ν ∂µTHW = dΓ p C[f]=0 , Z ∂ Sλ,µν = dΓ ~ Σµν C[f]= T [νµ] . ~ λ HW 2 s HW

I Collisional invariant: pµ

=⇒ Energy-momentum conserved in a collision

Collisional invariant: total angular momentum [µ ν] µν I ∆ p + (~/2)Σs

Spin not conserved in nonlocal collisions [νµ] at 2 =⇒ ⇔ THW 6= 0 O(~ ) =⇒ Conversion between spin and orbital angular momentum

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Dynamics of spin tensor CRC - TR

[νµ] I THW = 0 (i) for local collisions (∆µ = 0), as spin is collisional invariant (ii) in global equilibrium, as collision term vanishes

I With nonlocal collisions out of global equilibrium: dynamics dissipative

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Comparison to nonrelativistic case: micropolar uids CRC - TR

I Simple model with many applications: spintronics, chiral active uids,... R. Takahashi, M. Matsuo, M. Ono, K. Harii, H. Chudo, S. Okayasu, J. Ieda, S. Takahashi, S. Maekawa, and E. Saitoh, Nature Physics 12, 52 (2016) D. Banerjee, A. Souslov, A. G. Abanov, and V. Vitelli, Nature communications 8, 1 (2017) I Fluid of rigid, randomly oriented particles with internal angular momentum `. I Mass density ρ, uid velocity u, non-symmetric stress tensor T ij . I Conservation of total angular momentum d Z Z d3x ρ(`i + ijkxj uk) = dΣl(Cli + ijkxj T lk) dt Ω(t) ∂Ω(t) Change in volume element given by surface ow described by stress for momentum, "couple stress" for internal angular momentum. I After short calculation: ρ ∂0 + uj ∂j `i = ∂j Cji + ijkT jk

I Gain or loss of internal angular momentum: couple stress tensor and antisymmetric part of stress tensor!

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Nonrelativistic limit CRC - TR

µ µν ijk k I p → m(1, v), Σs → s D E D E T [ji] = mijk∂0 ~ sk + mijk∂l vl ~ sk HW 2 2 √ with h...i ≡ (m2/2π 3) R d3v d3s δ(s2 − 3) (...)f I Agreement with phenomenological result of nonrelativistic kinetic theory. S. Hess and L. Waldmann, Zeitschrift für Naturforschung A 26, 1057 (1971) I Comparison with micropolar uids G. Lukaszewicz, Micropolar Fluids, Theory and Applications (Birkhäuser Boston, 1999) ρ ∂0 + uj ∂j `i = ∂j Cji + ijkT jk

=⇒ Internal angular momentum D E ρ `i = m ~ si , 2 =⇒ Couple stress tensor D E D E Cji = − ~ sipj + m ~ si uj . 2 2

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Conclusions CRC - TR

I Denition of energy-momentum and spin tensor depends on choice of pseudo-gauge =⇒ Splitting of total angular momentum into orbital and spin part not unique

I Pseudo-gauge choice can aect results of calculations

I Issue about "physical" choice of pseudo-gauge long-standing, not nally answered =⇒ Need calculations in dierent pseudo-gauges and comparison to experiment =⇒ "Physical" choice may depend on context of application

I Canonical currents: derived directly from Noether's theorem =⇒ Spin tensor not conserved for free elds

I Belinfante currents: couple to gravity in conventional general relativity =⇒ Spin tensor vanishes

I HW currents: derived from Klein-Gordon Lagrangian for spinors =⇒ Spin tensor conserved for free elds =⇒ Covariant generalization of canonical global spin =⇒ Transparent interpretation of equations of motion in kinetic theory =⇒ Nonrelativistic limit

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