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Polarization of fermions in a vortiular fluid

Xin-Nian Wang CCNU/LBNL

AGS/RHIC Users meeting June 7, 2016

Fang, Pang, Wang & XNW arXiv:1604.04036 Pang, Petersen, Wang & XNW arXiv:1605.04024

Global Orbital Angular Momentum

Large v1 at large rapidity

Transverse gradient of longitudinal fluid velocity in a Landau fireball

,P TpartdNdx

Distribution of number of participant projectile or target nucleons

( , )2

( )

Ppart

Pp

Tpart

Tpartart

z

dNdxd

dsp x b

N

NdxdNdxx

sd

c

−=

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠

Collective longitudinal momentum per produced parton

x

z b

Momentum conservation à BJ scaling violation

Local Orbital Angular Momentum

zz

dpp xdx

Δ = Δ

2 zy z

dpL x p xdx

= −Δ Δ = −Δ

Δx GeV/fm68.0/2 0 ≈ARp

GeVscsp 22.2)(2/0 ≈=

~! = ~r⇥ ~v ⇠ �yh 1E

dpz

dx

i

Quark Polarization

2 202 2 2 4 ( )T s T

T T T

d d d C K xd x d x d xσ σ σ

α µ+ −= + =

2 ( )qq

pPE E m

σ µπ

σΔ

≡ = −+

Unpolarized cross section:

dΔσd 2xT

=dσ

+

d 2xT−dσ

−

d 2xT= −µ

!p ⋅ ( xT ×!n)

E(E +mq )4CTαs

2K0 (µxT )K1(µxT )

Polarized cross section:

xT

p

n

pf

Liang & XNW, PRL 94 (2005) 102301

2 22

2 2 2 2 2,

1 1 ( )(2 ) 2 ( )

i

TT T

d gM q Cd q qλ λ

σπ µ

= =+∑

spin polarization in A+A

Spin-orbital coupling

nonrelativistic limit: (mq ≫ p,µ)

Pq ≈ −πµ p4mq

2

4 ( )qq

pPE E m

σ µπ

σΔ

≡ = −+

xT

p

n

pf

Liang & XNW, PRL 94 (2005) 102301

⇠ �!/m

Transverse momentum dependence

Pf = Pi �(1� P 2

i )⇡µp

2E(E +m)� Pi⇡µp

Huang, Pasi & XNW PRC 84 (2011) 054910

Pf (qT ) =⇡E(E +m)Pi � p

pq2T + µ2K(qT /

pq2T + µ2)

⇡E(E +m)� Pipp

q2T + µ2K(qT /pq2T + µ2)

Polarization rate Eq.

Consequences in A+A collisions

Globally Polarized thermal dilepton, J/Ψ, Hyperons and vector mesons

Liang & XNW, PRL 94 (05) 102301 Liang & XNW, PLB 629(05)20 Gao et al, PRC 77 (08) 044902

Spin polarization in equilibrium

Becattini & Ferroni, EJPC 52 (2007) 597, Betz, Gyulassy & Torrieri, PRC 76 (2007) 044901, Becattini, Piccinini & Rizzo, PRC 77 (2008) 024906, Beccatini, Csernai & Wang, PRC 87 (2013) 034905, Xie, Glastad & Csernai, PRC 92 (2015) 064901, Deng & Huang, arXiv 1603.06117

Spin: vorticity coupling

Dirac Eq.

Magnetic coupling

�Es =~2n · !

[�µ(i@µ + eqAµ)�m] (x) = 0

eq~n ·BEp

+

⇡Z

d3p

(2⇡)3�Es

@f(Ep)

@Ep

⇧ =1

2

Zd3p

(2⇡)3[f(Ep + �Es)� f(Ep + �Es)]

gradient expansion

CME/CVE and spin polarization

R (L) fermion’s spin parallel (opposite) to vorticity & magnetic field

Pu, Gao, Liang, Wang & XNW, PRL 109 (2012) 232301

Kharzeev, McLerran, Warringa (2008), Fukushima, Kharzeev, Warringa (2008)

Chiral Magnetic Effect Chiral Vorticity Effect Son, Surowka (2009), Kharzeev, Son (2011)

Quantum Kinetic Theory

Polarization density

+

µ chemical potential Polarization on the freeze-out surface:

~

Fang, Pang, Wang, XNW arXiv:1604.04036

Vorticity induced polarization per fermion

0

5

10

bEp

01

23

4

bm

0.0

0.1

0.2Pêr

0

5

10

bEp0

12

34

bm

0.0

0.1

0.2

Pêr

In the unit of !

fermion

anti-fermion

polarization fermion/anti-fermion

0

5

10

bEp

01

23

4

bm

0.0

0.5

1.0

R

R =[⇧/⇢]

fermion

[⇧/⇢]anti�fermion


Transverse gradient of longitudinal fluid velocity in the HIJING model

Approximate BJ scaling at y=0, no transverse gradient in longitudinal fluid velocity, BJ scaling violation at large y or lower collision energy

Gao, Chen, Deng, Liang, Q. Wang, XNW, PRC 77 (2008)044902

!y ⇠ sinh(Y )@Y

@x

Vorticity from HIJING model

Deng & Huang, arXiv:1603.06117

Global vorticity increases with impact-parameter and decreasing colliding energy

Features of Global Hyperon Polarization

•  Hyperons & anti-hyperons are similarly polarized •  PH=0 in central and increases with b •  Increases with y at fixed collision energy •  Increase at lower energy at fixed y

Talk by M. Lisa Workshop on Vorticity & magnetic field in heavy-ion collisions @UCLA, 2016

Transverse vorticity: Toroidal structure

Toroidal structure of transverse fluid vorticity, in addition to the global net vorticity

x

Pang, Petersen, Wang & XNW, arXiv:1605.04024

(vx

, v⌘

)

Transverse spin correlation

Pb + Pb @ 2.76 TeV (20-30%)


Transverse spin correlation


Longitudinal vorticity: vortex-pairing

Vortex Pairing


~v?

Longitudinal spin correlation

Pb + Pb @ 2.76 TeV (20-30%)

Pang, Petersen, Wang & XNW arXiv:1605.04024 ,

Longitudinal spin correlation


Summary

•  fluid velocity gradient or fluid vorticity leads to global spin polarization

•  Spin structure in QGP in A+A collisions –  Hyperon polarization, vector meson spin alignment,

contribution to vn (small) •  Convective longitudinal flow induces toroidal structure

of transverse vorticity à transverse spin-spin correlation

•  Convective radial flow of hot spots induces longitudinal vortex-pairing à longitudinal spin-spin correlation

•  Study of spin polarization can shed light on transport properties of QGP and hadronization mechanism

Longitudinal structure in A+A

E-by-E 3+1D ideal hydro + HIJING initial condition

Pang & XNW (2013)

Longitudinal decorrelation of vn

Fluctuation and twist of the event plane

Pang, Petersen, Qin, Roy & XNW, EPJA 52(06) 97

Vorticity in convective fluid

Constraints via v2

2 000 ( 1/ 3)v ρ< >

200 001 (3 1)cos

cosdNd

ρ ρ θθ= − + −

2 000 ( 1/ 3)v ρ> <

pT

pT

Spin Alignment of Vector Mesons

00

1 13 3

q q

q q

P PP P

ρ−

= ≤+ 11

(1 )(1 )3q q

q q

P PP P

ρ+ +

=+1 1

(1 )(1 )3q q

q q

P PP P

ρ− −

− −=

+

Recombination model

2

00 2

1 13 3

q

q

PP

αρ

α

+= ≥

−

Fragmentation

First rank: q-bar anti-parallel to leading q 0e e Z qq Xρ+ − →→ → → +

Other effects

1( ) (( () () ) )2f fM u p kS q V qk u pπΛ

⎡ ⎤/ /= + +⎢ ⎥⎣ ⎦

PΛ>0 PΛ<0 Hadronic scattering: Λ+π à Λ π

Quark polarization due to Chromo-magnetic field? (Kharzeev’04)

No global chromo-magnetic field ß color correlation length

L ~ rN inside nuclei, L ~ 1/µ in QGP

L ~ 1/Qs in CGC

Towards a numeric estimation at RHIC

0 0.34 GeV/fm2 ( ) A

dp sdx c s R

= ≈

0.1 GeV zp µΔ : =

“Small angle approximation” is NOT valid at RHIC.

( ) 45 at RHIC (assume parton/hadron=2)c s =

1 0.5 GeVx

µΔ

: :

Semi-central: b=RA

We need to go beyond small angle approximation

QK equation for massive fermions

Spin quantized in a given direction n in the fermion’s rest frame


Quantum Kinetic approach to CME/CVE and spin polarization

Pu, Gao, Liang, Wang & XNW, PRL 109 (2012) 232301

Wignar function:

Quantum Kinetic equation

Gradient expansion:

Quark Scattering with fixed Ly

M λ ,λi(!qT ) =

−ig2Euλ ( pf ) /A(qT )uλi ( p)

02 2

1( )TT

A qq µ

=+

Screened static potential model With small angle approximation: qT<<p

Iλ !g 2

2A0 (qT )A

0 (kT ) 1− iλ(!qT −

!kT ) ⋅ (

!n × !p)2E(E +mq )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

: final quark spin λ

xT

p

n

pf

dσλ

d 2xT=CT

d 2qT(2π )2∫

d 2kT(2π )2

ei (!kT −!qT )i!xT 12

M λ ,λi(!qT )

λi

∑ M λ ,λi

† (!kT )

Polarization in unpolarized collisions

Transverse polarized hyperons in unpolarized p+p, p+A collisions.

p Λ

1 (1 cos )cos 2dN Pd

α θθ Λ= + θ : angle of decay products

in rest fame relative to the polarization direction αpπ-=0.642

Hyperon Polarization in p+p, p+A

•  PΛ,,PΞ, Panti-Ξ, <0, PΣ , Panti-Σ >0.

•  P=0 for anti-Λ, Ω and anti-Ω. •  increase with xf and pT

•  Weak A-dependence in p+A collisions. •  Measured in A+A at AGS: vanishes

p

h

Connection to single-spin asymmetry and spin-orbital angular momentum coupling

φ

Constituent Quark Recombination

PΛ= Ps P

Σ=4Pq − Ps −3PsPq

2

3− 4PqPs + Pq2

PΞ=4Ps − Pq −3PqPs

2

3− 4PqPs + Ps2

2

2

53(1 )

ss

s

PP PPΩ

+=

+

Polarization of quarks

Polarization of hyperons: ( )( )( )

0

0

1/ 2

1/ 6 2

1/ 6 2

u d u d s

s s u s s s s u

u d s u d u d s

↑ ↑ ↓ ↓ ↑ ↑

↑ ↑ ↑ ↓ ↑ ↓ ↓ ↑ ↑

↑ ↑ ↑ ↓ ↑ ↓ ↓ ↑ ↑

Λ = −

⎡ ⎤Ξ = − +⎣ ⎦

⎡ ⎤Σ = − +⎣ ⎦

Constituent quark model:

(l=0, color anti-symmetric)

Fragmentation Scenario

Λ

φq

0e e Z qq X+ − →→ → →Λ +

Model: Only first rank hadron (containing q with the same initial polarization) is polarized

2s

ss s

nP Pn fΛ = + ( ) 14

3( 2 )s q s ss s

P f P n Pn fΣ = −+

( ) 143(2 )s s q q

s s

P n P f Pn fΞ = −+

/ 3sP PΩ =

ns, fs strange quark abundance relative to u, d in QGP and fragmentation

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