Chiral Spin Matter

Kenji Fukushima
The University of Tokyo— Summer School 2021 —1

DISCLAIMER
Lecture is limited to an elementary level.
Textbook knowledge is only partially needed.
□ Quantization / Gauge Theories / Chiral Anomaly

Not much “Knowns” but much “unknowns”□ Knowns can be all found in published papers.
□ If unknowns are solved, new works can be published!

Expressions (coefficients/signs) might have typos.
□ Matter of convention…

Feedbacks are always welcome(taken into a forthcoming textbook)!
August 6, 2021 @ summer school online2

Symmetry
Noether Current<latexit sha1_base64="iqPasb2NrFtKQdSAr8xfCtq4sIo=">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</latexit>

Z⇠

Z

D e

iS[ ]

Theory has a symmetryunderZ<latexit sha1_base64="n+I5Psw0RtlAOqbEGoIO00Pv4tQ=">AAACpXichVFBSxwxGH1Oq9XV6ra9CL0EF6uiLFkpbSkUFnvppaCuuwo76zAzRo1mZsJMdtEu/gH/gAdPFtpS/BlePPRqwZ8gPVrw4qHfzg6IldYvJHl5+d6Xl8TTSiaG8/Me68HD3r5H/QO5waHHwyP5J09rSdSMfVH1IxXFK56bCCVDUTXSKLGiY+EGnhLL3vb7zv5yS8SJjMIls6tFI3A3QrkufdcQ5eTLlbqtN+VEg71jXdhg08yWoWFrO/YMs4VOpIrCyZ0pZms3NtJVjh002dYqjcQ6+QIv8jTYXVDKQAFZzEf5b7Cxhgg+mgggEMIQVnCRUKujBA5NXANt4mJCMt0X2EOOtE3KEpThErtN4wat6hkb0rpTM0nVPp2iqMekZBjnZ/w7v+Sn/Jhf8Ot/1mqnNTpedmn2ulqhnZH90crVvaqAZoPNG9V/PRus403qVZJ3nTKdW/hdfevTwWXl7eJ4+wX/zH+R/yN+zk/oBmHrt/9lQSweIkcfUPr7ue+C2myx9Ko4u/CyUJ7LvqIfzzGGSXrv1yjjA+ZRpXO/4gfO8NOasD5aS1atm2r1ZJpnuBWW8wfJB6Dt</latexit>

0S[ ] = S[ ] +Z<latexit sha1_base64="Eu9/D7sanzJvICgSo6Bt2Ukhy98=">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</latexit>'

DZ

0 iS[ 0 ]eZ

D eiS[ ] iZ<latexit sha1_base64="VaQBDRQfSEFY6Wj7tGhQHdrGR1o=">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</latexit>

!
µ

0

+✏

if Z unchanged.

D =D 0<latexit sha1_base64="EI9q5A+CYcfHc5/GXCsPK9Cfof4=">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</latexit>dx ✏(x)@µ j (x)andD eiS[ ]Conserved Currentdx ✏(x)@µ j µ (x) = 0
August 6, 2021 @ summer school online

h@µ j µ (x)i = 0<latexit sha1_base64="ZLH8ITkRg3bBQOHqVGQNej51WnM=">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</latexit>

3Symmetry
ExampleTranslational symmetry: xµ ! x0µ(x) ! 0 (x) + ✏µ @µ (x)<latexit sha1_base64="lH7NiAVI3Dq6Ce0cvjs+EuPULMU=">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</latexit>✏µ<latexit sha1_base64="v7LR7sOCj4uG6ps6ovVqgmPwdAE=">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</latexit><latexit sha1_base64="snNjMUuOwVfL8pFO6UU4LvepvaY=">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</latexit>S[ 0 ]ZZ

dx L[ 0 (x)]dx L[ (x)]
⇢
Z
@L ⌫
@L
' dx
✏ @⌫
@µ [✏⌫ @⌫ ]@@(@µ )Z@L⌫⌫= dx ✏ @⌫ L (@µ ✏ )@⌫@(@µ )Z@L⌫µ= dx ✏ @µ@⌫ Canonical⌫L+@(@µ )
Energy-Momentum
TensorS[ ] =

August 6, 2021 @ summer school online

4Symmetry
Fermionic Action in the presence of AZ
iS[ ¯, ,A]¯
L = ¯[i µ (@µ + ieAµ )Z[A] ⇠ D D e<latexit sha1_base64="L20NRvLDg9uYQbvia+nqfsRaH48=">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</latexit><latexit sha1_base64="NFJ2f/tutpSv4prUI3wFGk7TzTs=">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</latexit>m]

U(1) symmetry<latexit sha1_base64="Bs/D/G1oKIVwHyVP3G2BFDTlPtk=">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</latexit><latexit sha1_base64="2lBaXipZBBw/Ebhqjt0dHbBDEWI=">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</latexit>! ei✏

S[ ¯0 ,

0

]

0

'

0

+ i✏
Z



L is invariant under this transf.

@L
@L
¯
dx i✏
+
@µ ( i✏ )¯@(@µ )@Z@L= dx i✏ @µ@(@µ )S[ ¯, ] 'h@µ j µ i = 0<latexit sha1_base64="XiOA4WEReN97zbLdJhF6QQqqdSU=">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</latexit>

Number conservation

jµ = ¯ µAugust 6, 2021 @ summer school online

5Symmetry
Fermionic Action in the presence of AZ
iS[ ¯, ,A]¯
L = ¯[i µ (@µ + ieAµ )Z[A] ⇠ D D e<latexit sha1_base64="L20NRvLDg9uYQbvia+nqfsRaH48=">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</latexit><latexit sha1_base64="NFJ2f/tutpSv4prUI3wFGk7TzTs=">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</latexit>

U(1)A symmetry<latexit sha1_base64="vughBiICuEeLP3xjt/ClTHI1r/I=">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</latexit><latexit sha1_base64="RoQcZkH8eLowZ6/KdB+7iIZI3fM=">AAAD0XicjZHLbtNAFIaPYy4lXJrCBonNiKghiDSaVNyEhFTBhgWLtiFtpUwUjd2JO8r4gj2JCJElxBIegAUrkBBCPAYb1kgs+giIZZFgwYJjx82l5TaWPcf/nO/Mf2asQMlIU7pr5MwjR48dnzuRP3nq9Jn5wsLZjcjvhbZo2L7ywy2LR0JJTzS01EpsBaHgrqXEptW9m6xv9kUYSd97oAeBaLnc8WRH2lyj1F4wuvUms3g4ZEEkL8WVdGqRJTIlj9RWvsQi6YqHhElPk+1HzJKOo5pkSTKB68r3JgRzuOvy9jXWCbmNEg+15Iowl+sdm6vh/TgeqxMqJlfI/xCkvB+1mdtL0MsxmZbKE09kbCVJI6npsEUYy5dujzupkEkPlZlKB/z81k55FsjsTG27v2u7UKRVmg5yOKhlQRGyseoX3gKDbfDBhh64IMADjbECDhE+TagBhQC1FgxRCzGS6bqAGPLI9jBLYAZHtYtfB/+amerhf1IzSmkbd1H4hkgSWKSf6Tu6Rz/S9/QL/fnHWsO0RuJlgLM1YkXQnn92vv79n5SLs4adCfVXzxo6cDP1KtF7kCpJF/aI7z9+sVe/tb44LNHX9Cv6f0V36QfswOt/s9+sifWXkMcLqB087sPBxnK1dr26vHa1uHInu4o5uAAXoYznfQNW4B6sQgNs45PxI2fkcmbdHJhPzKej1JyRMedgZpjPfwEsjxOL</latexit>!e

S[ ¯0 ,

i✏ 5

0

]

0

'

0

m]

Explicit Breaking<latexit sha1_base64="p5JB8iYyP5YS79VyFZzUCiv+EdA=">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</latexit>+ i✏ 5Z

L!L

2i✏m ¯ 5@L@L¯dx i✏ 5 ¯ +@µ ( i✏ 5 )@(@µ )@Z@L= dx i✏ @µ
5@(@µ )S[ ¯, ] 'µChirality conservation h@µ jA i = 2m ¯i 5<latexit sha1_base64="6dkzsMP5+K2htoyUR0Fn1Y1A6dQ=">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</latexit>

August 6, 2021 @ summer school onlineµjA= ¯ µ 5
6

Chiral Anomaly
Fermionic Action in the presence of AZ
iS[ ¯, ,A]¯
L = ¯[i µ (@µ + ieAµ )Z[A] ⇠ D D e<latexit sha1_base64="L20NRvLDg9uYQbvia+nqfsRaH48=">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</latexit><latexit sha1_base64="NFJ2f/tutpSv4prUI3wFGk7TzTs=">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</latexit>

U(1)A symmetry<latexit sha1_base64="vughBiICuEeLP3xjt/ClTHI1r/I=">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</latexit>

i✏ 5

0

m]

Explicit Breaking

0<latexit sha1_base64="p5JB8iYyP5YS79VyFZzUCiv+EdA=">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</latexit>!e' + i✏ 5
D ¯D ! ei [A] D ¯0 D 0<latexit sha1_base64="TW04NlOumh4S7Zwm1+W0hLYPda0=">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</latexit>

L!L<latexit sha1_base64="ip8meJ3pZiC5MFhL94FVeWbVLs0=">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</latexit>[A] =Anomaly2i✏m ¯ 5Zdx ✏a[A]µh@µ jAi = a + 2m ¯i 5
Zdx a = (integer) = QW
Anomaly is “topological”

Chiral Anomaly<latexit sha1_base64="qdxCwupRzb8Hq0ouTPUF8N0sXn4=">AAACr3ichVHLahRBFD3p+IjjI2PcCG4Kh4ggDNWDURGEqBtxlYeTCaSTtrpTGSup6m6qawZiMz/gD7hwpSAibvwHN/6AQla6FZcR3Ljwdk+DaEi8TVede+qeW6eqokyr3HG+N+FNHjt+4uTUqcbpM2fPTTfPz6zk6cDGshunOrWrkcilVonsOuW0XM2sFCbSshft3C/Xe0Npc5Umj9xuJteN6CdqS8XCERU2HwZaJH0tWZAJ65TQYWAGbHuDxrAIrGF3Ryyw45I7TLBrrGOCSNgiyHI1UkFfGCPCuTILmy3e5lWwg8CvQQt1LKTNNwiwiRQxBjCQSOAIawjk9K3BB0dG3DoK4iwhVa1LjNAg7YCqJFUIYndo7FO2VrMJ5WXPvFLHtIum35KSYZZ/4m/5Pv/I3/Fv/NehvYqqR+lll+ZorJVZOP3s4vLP/6oMzQ5P/qiO9OywhVuVV0Xes4opTxGP9cOnz/eXby/NFlf4K/6d/L/ke/wDnSAZ/ohfL8qlF2jQA/j/XvdBsNJp+zfancXrrfl79VNM4RIu4yrd903M4wEW0KV93+MzvuCr53s9b8N7PC71JmrNBfwVnvoNDi2mkw==</latexit><latexit sha1_base64="cfeirjEVCGsIkPSUeRseK9lto9s=">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</latexit>

August 6, 2021 @ summer school online

7

Chiral Anomaly
Winding number and Chern-Simons current<latexit sha1_base64="5bevO+8tIek3DF0glPZS+JFlLeg=">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</latexit>

QW [A] =Z

dd x @µ K µWinding number is unchanged
under continuous A changes(changed under large A transf.)

To make the energy finite, A should go to a pure gauge
at infinite distance, i.e., F ~ 0.
In some gauges, spatial surface terms are dropped.

244

Tunneling and Classical Solutions in Quantum Mechanics

completely analogous to that undertaken in Section 11.1 and we shall notpresent it here.
We further note that calculation of the semiclassical exponential forthe splitting between levels in the quantum mechanics of many variables(see Problem 11) also reduces to finding an instanton with precisely theproperties described, and calculating its Euclidean action. The substantiation of this assertion is based on the results of Problem 11 andessentially repeats that given in Section 11.2.
<latexit sha1_base64="z0iqQ/yzMAu9eB0mvsf9EQPlxIw=">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</latexit>

QW [A] =
V (q)

Z

d⌧ @⌧

Z

dd 1 x K 0 = ⌫(⌧ = +1)

⌫(⌧ =

1)

See: Classical Theory of Gauge Fields
by V. Rubakov−a0aFigure 11.8.

2aqAugust 6, 2021 @ summer school online

8

Chiral Anomaly
Topological susceptibility and the q angle
If the theory has massless fermions, we can choose
axially rotated fields without affecting the classical action.
<latexit sha1_base64="vughBiICuEeLP3xjt/ClTHI1r/I=">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</latexit>! ei✏ 50

D ¯D ! ei<latexit sha1_base64="ip8meJ3pZiC5MFhL94FVeWbVLs0=">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</latexit>

'

0

[A]

+ i✏ 5D ¯0 D0

The theory has an ambiguity of adding dG ~ eQW
If the theory had an additional term:<latexit sha1_base64="QSOQwqtdQEcUVgDT08llFLr6Pic=">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</latexit>

Z[A; ✓] ⇠Z<latexit sha1_base64="iQ9+0nHwJqEyMyoNNM/eD4r3Q1I=">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</latexit>

D ¯D eiS+i✓QW@✓n Z[A; ✓]<latexit sha1_base64="+mTAbivxWGZsVTol91Ean+2r+4I=">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</latexit>

Topological susceptibility is zero:

✓=0

=0

2
2
⇠
hQ
i
⇠
@
Q
W
✓ ln Z

August 6, 2021 @ summer school online

9

Chiral Anomaly
For massless fermionsµ@µ jA= @µ K µ<latexit sha1_base64="Kj0DV+CTllC+XoedGVix+m7+y8o=">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</latexit>

It is said that U(1)A is “explicitly” broken, andh0 meson is not a Nambu-Goldsone boson anymore…
<latexit sha1_base64="R2C+7l4xZi8r/5amBrT5n28bQIs=">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</latexit>

@µ J

µ

µ
= @µ (jA

K µ) = 0

Conserved current???

If so, another (unobserved) massless boson cannot be avoided?
Let’s see a more concrete form of the CS current in the
Abelian and the non-Abelian theories.

August 6, 2021 @ summer school online

10

CS Current
Triangle (ABJ) Anomalies<latexit sha1_base64="F3raL3LkZG7fQ4ep71ylBhjCx0s=">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</latexit>

QED:<latexit sha1_base64="bgq6eO7GCShD4bpX2wkt8H5dwNU=">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</latexit>

Kµ =

µ
@µ jA
=
e2 µ⌫↵✏
2
4⇡<latexit sha1_base64="o8Sn4Fy3KK3UIWQDfM0szv6+drc=">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</latexit>

µa
@µ jA
=

QCD:

2

⇠E·

B

e
µ⌫↵
✏
Fµ⌫ F↵
216⇡
e2 µ⌫↵A⌫ @ ↵ A =✏A⌫ F ↵
28⇡
sha1_base64="t3YaA9saMUdF2Tmw0qypv/zX8aw=">AAACjHichVHPSxtBFP5cq8bUatSL4GVpsPYU3m6XJCqCKC09+qNRwUjY3Yx2cH+xOwnEkH9A8NxDTy2UUnrstT310n+gB/+E0qOFXnro201EPKhvmHlvvnnfm2/eOJEnE0V0MaQNPxgZHcuN5x9OPJqcKkzP7CZhK3ZFzQ29MN537ER4MhA1JZUn9qNY2L7jiT3nZCM932uLOJFh8Ep1InHo28eBPJKurRhqFBbrifT1uhN6zaTjs+s+79XdZqhuYOu9RqFIpaVq2bTKOpWIKoZppIFZsZ5ZusFIakUMbDMsfEQdTYRw0YIPgQCKYw82Eh4HMECIGDtEl7GYI5mdC/SQZ26LswRn2Iye8HrMu4MBGvA+rZlkbJdv8XjGzNSxQD/pE13SD/pMv+jfrbW6WY1US4e90+eKqDF1Nrfz916Wz17h9TXrTs0KR6hmWiVrjzIkfYXb57dP31zuLG8vdJ/Qe/rN+t/RBX3nFwTtP+6HLbH9Fnn+gKsu67cHu2bJKJfMLau4tj74ihzm8RhPud8VrOElNlHje8/xBV/xTZvULG1FW+2nakMDzixumPbiP7J1mf8=</latexit><latexit

g 2 µ⌫↵
a
✏
tr
F
F
trt
µ⌫ ↵
2
16⇡

Usually only the flavor-singlet is anomalous.
<latexit sha1_base64="O37pnCDHttF2oPXlGUEC9unAkuE=">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</latexit>µK =e2 Nf µ⌫↵✏trA⌫ (F↵216⇡2A↵ A )
3

CS currents are NOT gauge invariant … unphysical?
August 6, 2021 @ summer school online

11

CS Current
Proton Spin Crisis
Each quark has 1/2,
and three constitute 1/2

Proton Spin = 1/2

若松さんの1990年の物理学会誌の記事「陽子のスピンはどこから？」<latexit sha1_base64="0y14/bdoTTGKwIgmmV8orTPBf2A=">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</latexit>

u = 0.74 ± 0.08
d=
s=

⌃=
<latexit sha1_base64="uQ64pQRdp0n64z4YrOJEtzorgRo=">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</latexit>

0.51 ± 0.08

0.23 ± 0.08

These are old values, but
qualitatively similar todayu+d+s = 0.00 ± 0.24

Quark spins sum up to zero!?

August 6, 2021 @ summer school online

12

CS Current
Proton Spin Crisis
若松さんの1990年の物理学会誌の記事「陽子のスピンはどこから？」

Badly criticized by
Jaffe-Manohar (1990)[8] Altarelli-Ross (1988)[9] Carlitz-Collins-Mueller (1988)

August 6, 2021 @ summer school online

13

CS Current
Axial Current and Spin
Axial Current is nothing but the spin of fermions.
Rotation: xµ ! ( µ⌫ + ✏µ⌫ )x⌫<latexit sha1_base64="KEsSUQcInUWbAdHKFG9fZWMBaFc=">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</latexit><latexit sha1_base64="Oo3wWgjZtqmHJM6RW6KBY7ilo2k=">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</latexit>

Noether current: J

µ⌫

@L
=
(xµ @ ⌫ x⌫ @ µ i⌃µ⌫ )@(@ )= ¯i (xµ @ ⌫ x⌫ @ µ i⌃µ⌫ )i µµ⌫⌃ = [ ,
4<latexit sha1_base64="/I/iVE+1fdUKR3FTOB2GYty196I=">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</latexit>

J 0µ⌫ is the conserved charge (angular momentum)
S µ⌫ = ¯ 0 ⌃µ⌫ represents the spin<latexit sha1_base64="vit4fXZCtwpPhoPV0GS5wYzYwRA=">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</latexit>⌫]<latexit sha1_base64="UnxnYQeom71+VRaXRfkn+Oy4u/E=">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</latexit>

1 ijk jk
1¯ i
S = ✏ S =522<latexit sha1_base64="yyKUMPUiC17ZGL9SlxabJHaXvlc=">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</latexit>

i

1 i
= jA
2

August 6, 2021 @ summer school online14

2M2= ~p~TrF2~p).

(5.18)CS Current

Eqs. (5.14) and (5.18) have a similar interpretation: if it were not for gluon-mediated interactions, the nucleon would consist of three free, massless quarks. The
ground state mass and axial charge of a “nucleon” composed of free massless
quarks vanishes, so it is not surprising that either can be expressed (by means of
the equations of motion) in terms of gluon operators.
2 µ to evaluate K(0) inµthe parton model. A brief calculation
Supposeone
attempted
R.L.Jaffe,A.Manohar / Theg, problem
521 A
Q2(inserting (N1a5/2IT)TrFF in gluon Fock states)
yieldsOne
proton state with

Axial Current and Spin<latexit sha1_base64="gh7BwQpSHA19n/FxpPFCrYCVo8E=">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</latexit>⌃(Q )s = hN (p, s)|j |N (p, s)i

momentum p and spin s~parton(0)=Kparton(0)=—where Lig is the gluon helicity asymmetry defined in eq. (5.4). Comparing thisthe separation
proposed in refs. [12, 131 (see eq. (5.5)), two choices
Fig. 3. Coupling of result
A~to thewith
proton throughthe triangle diagram.
appear: (1)0, or (2) the parton model calculation[Jaffe-Manoharof K(0) is incomplete.
A(1990)]
possible non-partonic contribution to . has recently been discussed by Forte [531.
ere Lig is an integrated gluon distribution*,
2
Comparing eqs. (5.14) and(5.15) it is clear how one might erroneously arrive at
s
2
2
2
eq.
(5.5),
thecentralresult
of
refs.
[12,
13].
Supposeone missed
the effects that givefConclusion
Lig(Q2) f’dx[g1(x,Q2) _g1(x,Q2)j.
the s~’a mass. This would replace eq. (5.4)(5.14) by.~=<latexit sha1_base64="pbltMGo3q2R72BpcJZssvw00Rnk=">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</latexit>˜⌃(Q ) = ⌃(Q)=↵ (Q )
N
g(Q )
2⇡

2) is
Naive
value
from
Gluon
contribution???
litz et al.as [13]especiallyjet(although,seetoalsoref.Q2).
[24])
stressthatasurablea two-quark
contribution
g~(x,
The
idea
then LiF(Qis that—f(0) + K(0),the quark model?
=~(Q2) =~(Q2)- as(Q2)NLig(Q2)made in [8, 9](5.i4a)where f(0) is the residue of the if-pole(5.5)
in h(t). Next suppose one evaluated K(0) in
6, 2021@ summer
online15
the parton model as August
given byeq. (5.18).
Theschool
resultwould beh .(Q2) being quark-model like (i.e.

0.60) and approximately cancelled by

1N

CS Current
Axial Current and SpinhN (p0 , s)| ¯ µ 5 |N (p, s)i = ⌃(t)sµ + (l · s)lµ h(t)<latexit sha1_base64="OmisSJ4X4mzqZuAzbYneNxDnuaE=">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</latexit>

l = p0<latexit sha1_base64="2JQj1Kr4lPQ/dSrPYb1zOL+qht0=">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</latexit>↵shN (p , s)|Nf trF F̃ |N (p, s)i = i(l · s) (t)2⇡p,t = l2<latexit sha1_base64="aRt9sJepAl5/yUM6ps1teDlJIxA=">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</latexit>

0“Accidentally” similar form…
[Jaffe-Manohar (1990)]<latexit sha1_base64="cpL52RIPuIJTU76/3rQCyydxbAE=">AAAChHichVHLLgRBFD3ae7wGG4mFjgkhJpMa70iIsLHEGCRGJt2tjMr0K901kzCxtPEDFlYkImLLF9j4AQufIJYkNhZu93QiCG6lqk6duufWqSrdNYUvGXusUWrr6hsam5pjLa1t7R3xzq513yl5Bs8ajul4m7rmc1PYPCuFNPmm63HN0k2+oRcXg/2NMvd84dhrct/l25ZWsMWuMDRJVD7el8uIgqUNyWF1RJW55F6AZtVcUXPdgM3HEyzFwlB/gnQEEohi2YlfIocdODBQggUOG5KwCQ0+tS2kweASt40KcR4hEe5zHCJG2hJlccrQiC3SWKDVVsTatA5q+qHaoFNM6h4pVQywB3bFXtg9u2ZP7P3XWpWwRuBln2a9quVuvuO4J/P2r8qiWWLvU/WnZ4ldTIdeBXl3Qya4hVHVlw9OXjIzqwOVQXbOnsn/GXtkd3QDu/xqXKzw1VPE6APS35/7J1gfTaUnU6Mr44n5hegrmtCLfgzRe09hHktYRpbOPcI1bnCrNChJZUyZqKYqNZGmG19CmfsA/2+TNg==</latexit>⌃(t) + t h(t) = (t)<latexit sha1_base64="Wva51A+FzozRHCwKulmCPX4rRzU=">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</latexit>t ! 0<latexit sha1_base64="rV9Jmw682jL1FVXaVWZ7HhQet4M=">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</latexit>

No distinction between↵s
Nfg S and the matrix elements
2⇡of the anomaly.

Residue of “massless” h0 propagation???
Instantons make h0 massive, so this is zero!
August 6, 2021 @ summer school online

16

A side story about h0
Would-be NG boson killed by anomaly
In the large-N limit:

Quark loops are suppressed.

How can S and k be balanced in the large-N limit?
More specifically, in a theory with massless fermionsthe vacuum expectation of the anomaly is always zero(q derivatives are all zero). How can it be true?
Z<latexit sha1_base64="7sL4jibHEnmn1TTBKNqh64cg1ps=">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</latexit>

Topological susceptibility:dx hqW (x)qW (0)iCancellation between thegluon and quark loops!?

August 6, 2021 @ summer school online

17

A side story about h0
Would-be NG boson killed by anomaly
In the large-N limit:<latexit sha1_base64="mcuAyDvmgahVcp+WMxrkvjAEvAg=">AAAC63ichZHPa9RAFMdf4q92/dFtvRTqYXBpqUiX2VBaEQpFL16Ubuu2haYbJuPs7rCZJGRmF2vItQevHgTtpUIR8c/opf+Ah/4JorcKXjz0bTagtf54ITPvfed93ryZ8eNAakPpsWVfuHjp8pWR0dLVa9dvjJXHJ9Z11Eu4aPAoiJJNn2kRyFA0jDSB2IwTwZQfiA2/+3CwvtEXiZZR+NTsxGJbsXYoW5Izg5JX3nV5R3r12W7TuUOWiKt7yktdI56btN2JtNFZRtxWwnj6BPVEEZ41HcK8sOlkKUJkjjzOA3L3DKyEjsI/wIT/iirPFYZh7JUrtEpzI+edWuFUoLCVqPweXHgGEXDogQIBIRj0A2Cg8duCGlCIUduGFLUEPZmvC8ighGwPswRmMFS7OLYx2irUEONBTZ3THHcJ8E+QJDBNP9EP9IQe0Y/0M/3x11ppXmPQyw7O/pAVsTf2cnLt+38phbOBzk/qnz0baMG9vFeJvce5MjgFH/L9F69P1u6vTqcz9B39gv3v02N6iCcI+9/4QV2svoUSPkDt9+s+76w71dpC1anPV5YfFE8xAlNwG2bxvhdhGR7BCjRw36/WhDVl3bKV/cp+Y+8NU22rYG7CGbMPTgEsELth</latexit>

2
(k
)=
Q

X

ghostsPure topological susceptibilityX
Nc2 a2nNc c2n+22k
Mn mesons k 2 m2⌘This can be zero at k = 0 if mh2 ~ 1/N
p
1
1 p
µ
Nc c⌘ = h0|QW |⌘i =h0|@µ jA |⌘i =2Nf m2⌘ f⌘
2Nf
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2Nf2m⌘ = 2f⌘

PCAC<latexit sha1_base64="n9wYjlbNkZE49YUoeTHsNVHEMWM=">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</latexit><latexit sha1_base64="kkg2EO5yQM1LiEK8d8vy0a8tuGk=">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</latexit>(pure)
Q

Witten (1979)(pure)
µ ⌫
2
⇠
hQ
i
⇠
lim
q
q
hj
6 0
µ
⌫
W
Q
A jA i =q!0

Massless ghosts hidden here…

Massless (Venetizano) ghosts makeh0 massive (massless mode unobservable).

August 6, 2021 @ summer school online

18

relate the terms that appear there to local operators and to quark and gluon
distribution functions in an infinite momentum frame. Some of the issues we must
confront are:(i) What is the operator description of the spin and orbital angular momentum ofquarks and gluons?
Questions
by Jaffe-Manohar(ii)
How are theraised
matrix elementsof the angular momentum density tensor, M/*VA,
normalized?
(iii) What is the relation between the gluon spin and the anomalous current

CS Current~
528

4A~AY).

R.L. Jaffe, A. Manohar / Theg, problem(6.1)(iv) apologize
Is there atosum
rule toforwhom
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angularan infiniteWe
readers
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appearmomentum
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well
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get a contribution
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We framehave foundsomeit confusion
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in recentwork
tumg of quarks and gluons?
on1.
5 AND A GENERAL ANGULAR MOMENTUM SUM RULE
6.1. THE TENSOR M~

The current associated with translations is the stress tensor, ~which isI
also
apologize
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whom
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appearand/or wellknown.
associatedwithelementary
Lorentz transformations
is a rank-3
tensor constructed entirely
from T~[55],
M’”’5 6,~2021 @ summer—x5T’”’.
Augustschool online(6.2)19

Gauge Inv. — Integrated Rels.
Integrated relations
In some gauges, spatial surface terms are dropped.

Z ⌧2<latexit sha1_base64="NB4qZODcB4xA73VxEqpm1J7LOm8=">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</latexit>

Z⌧1⌧2

d⌧ @⌧

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d⌧ @⌧

⌧1

<latexit sha1_base64="GWoVzkLQ2JxubS/sL8DxtGxUGcY=">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</latexit>

N5 (⌧ )

ZZ

dd 1 x K 0 = ⌫(⌧2 )⌫(⌧1 )

0
dd 1 x jA= N5 (⌧2 )N5 (⌧1 )⌫(⌧ ) is a conserved charge.

These are of course all gauge invariant!
ChiralityZcharge:<latexit sha1_base64="CgoBL0CVNFSq6+PHiZWGgbXaZa0=">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</latexit>

N5 =0
dd 1 x jA

Chern-Simonscharge:Z<latexit sha1_base64="gjLJo9Dq0PMNB0eSW51huhZK+3Y=">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</latexit>⌫=

August 6, 2021 @ summer school online

dd 1 x K 0
20

Gauge Inv. — Integrated Rels.
Chern-Simons charge:<latexit sha1_base64="gjLJo9Dq0PMNB0eSW51huhZK+3Y=">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</latexit>⌫=Z

dd 1 x K 0

Gauge inv.
observable!

In the case of the U(1) theory:

2
2
e
e
ijk
K0 =
✏
Ai Fjk =
A·B
2
2
8⇡
4⇡
This itself is NOT gauge invariant, but the integral IS!
<latexit sha1_base64="Ffnapw1t0VJvHyI0zdnSRP5M5fY=">AAACz3ichVHPaxNBFH5Zf9VUbdSL4GUwVLwYXkLRIhTaCkXw0rRNW+g2y+5kUqeZ3Vl2J5F02eK1/gMeeqogIv4ZXjx68dA/QTxW7KWgbzcLVov6lp33zTfve/PNjBcqGRvEw5J17vyFi5fGLpfHr1y9NlG5fmM11v2IixbXSkfrnhsLJQPRMtIosR5GwvU9Jda83uNsfW0goljqYMUMQ7Hpu1uB7EruGqKcyvbTNrIZdt/uRi5PRLuRJtN2KCkzW4SxVDpoJ3K7l7I5R7IFJ8ngDDtVPjUqtz2tOvHQp5TMpTbvaMNOc/OpU6liDfNgZ0G9AFUoYlFX3oINHdDAoQ8+CAjAEFbgQkzfBtQBISRuExLiIkIyXxeQQpm0faoSVOES26Nxi2YbBRvQPOsZ52pOuyj6I1IymMTP+A6P8CO+xy948tdeSd4j8zKk7I20InQmXt5aPv6vyqds4Nkv1T89G+jCdO5VkvcwZ7JT8JF+sPPqaPnR0mRyF1/jV/J/gIf4gU4QDL7xN02xtA9leoD6n9d9Fqw2avUHtUZzqjo7XzzFGNyGO3CP7vshzMITWIQW7fsJvsOPElhN67m1a70YlVqlQnMTfgtr7yfrw7MI</latexit><latexit sha1_base64="xbvtfWR2AlM+f6jGoAC8J3MnOco=">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</latexit>

2eMagnetic Helicity ⌫ =4⇡ 2Zd3 x A · B

Chiral anomaly dictates a new conservation law of
the chiral charge and the magnetic helicity!
N. Yamamoto
August 6, 2021 @ summer school online

21Gauge Inv. — External A
A’s are just background fields, not necessarily dynamical?
In the case of the U(1) theory:
2
e
µ⌫↵
Kµ =
✏
A⌫ @ ↵ A =2
4⇡<latexit sha1_base64="bgq6eO7GCShD4bpX2wkt8H5dwNU=">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</latexit>

e2 µ⌫↵✏A⌫ F ↵
2
8⇡

If replaced by external “something”
gauge invariance is okay!

Gauge Inv.

QCD matter in extreme environments
If the theory is S1 compactified along the Euclidean
time,
(b)
A0 would becomegauge invariant!
(a)

4

S1

zk
3

A0 cannot be gauged awayin finite-T field theory.

R

August 6, 2021 @ summer school online
Figure 1. (a) Graphical representation of the Polyakov loop in Euclidean space-22

Gauge Inv. — External A
A’s are just background fields, not necessarily dynamical?
A0 cannot be gauged away in finite-T field theory.

d! = A0<latexit sha1_base64="zVsCv0d8Zn8645XNDkRg/GiU8Ik=">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</latexit>

B 1 (M ) = {d!|! 2 ⌦0 (M )}Z 1 (M ) = {! 2 ⌦1 (M )|d! = 0}<latexit sha1_base64="/uWLyWumbl9ml5wa1dPv1Np8eSk=">AAAClHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvoO8UZavhqKtgqxFQrpMTk56amJyrUKEAZMZl5CjH+IGacAUhVTG28gLKBngEYKGAyDKEMZQYoCMgXWM4Qw5DCkM+QzFDKkMuQypDHUAJk5zAkMhQDYTSDIYMBQwFQLJahGihWBGRlguVTGWoZuIB6S4GqUoEqEoGi2UAyHciLhormAfkgM4vBupOBtuQAcRFQpwKDqsFVg5UGnw1OGKw2eGnwB6dZ1WAzQG6pBNJJEL2pBfH8XRLB3wnqygXSJQwZCF143VzCkMZgAXZrJtDtBWARkC+SIfrLqqZ/DrYKUq1WM1hk8Bro/oUGNw0OA32QV/YleWlgatBsBi5gBBiiBzcmI8xIz9BMzyjQRNnBCRoVHAzSDEoMGsDwNmdwYPBgCGAIBdo7lWE3wxGGo0xiTDZMzkyuEKVMjFA9wgwogMkPAMkKmS8=</latexit><latexit sha1_base64="cz1OeN6kIJUUl1VOx0eIY2ToTS4=">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</latexit>

1
HdR(M ) = Z 1 (M )/B 1 (M )<latexit sha1_base64="o06wKfXeV41/VpqG0DTG/p3ynRE=">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</latexit>

1HdR(R) = 0<latexit sha1_base64="FQhCUdRbKnfom2RMNIkMRY5PiOo=">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</latexit>

1
HdR(S 1 ) ' R

This d.o.f. is the gauge
invariant A0 (zero mode)

August 6, 2021 @ summer school online

23

Gauge Inv. — External A
A’s are just background fields, not necessarily dynamical?
A0 cannot be gauged away in finite-T field theory.

In the case of the U(1) theory:
In the physical term it is nothing but a chemical potential.
QCD matterin extreme
environments
In thecase ofthe SU(N) theory:

4
(b)

S1

This corresponds
to the Polyakov loop.
(a)

ZN (center) symmetry
1-form symmetry (spatial 0-form sym.)

zk
3

R

August 6, 2021 @ summer school online

24

Figure 1. (a) Graphical representation of the Polyakov loop in Euclidean space-

Gauge Inv. — External A
A’s are just background fields, not necessarily dynamical?
2
2
e
e
µ⌫↵
µ⌫↵
Kµ =
✏
A
@
A
=
✏
A⌫ F ↵
⌫ ↵
2
2
4⇡
8⇡
<latexit sha1_base64="bgq6eO7GCShD4bpX2wkt8H5dwNU=">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</latexit>

Gauge Inv.

A0 is provided by the chemical potential.
2
e
i
Ki =
Ā
B
0
2
2⇡
<latexit sha1_base64="s73+kF15LdeL3QN3Oh5DzWaZaM8=">AAACi3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuoecdlKtgqxKQVJSZXp8YZ1VYbxRRkAmmFmKTEomrH2ngDBae4zHgBZQM9AzBQwGQYQhnKDFAQkC+wnCGGIYUhnyGZoZQhlyGVIY+hBMjOYUhkKAbCaAZDBgOGAqBYLEM1UKwIyMoEy6cy1DJwAfWWAlWlAlUkAkWzgWQ6kBcNFc0D8kFmFoN1JwNtyQHiIqBOBQZVg6sGKw0+G5wwWG3w0uAPTrOqwWaA3FIJpJMgelML4vm7JIK/E9SVC6RLGDIQuvC6uYQhjcEC7NZMoNsLwCIgXyRD9JdVTf8cbBWkWq1msMjgNdD9Cw1uGhwG+iCv7Evy0sDUoNkMXMAIMEQPbkxGmJGeoZmeUaCJsoMTNCo4GKQZlBg0gOFtzuDA4MEQwBAKtLeHYRPDboY9TLxMxkxWTDYQpUyMUD3CDCiAyRUA0t2XMA==</latexit><latexit sha1_base64="9TbZJ7oxgKRA8Olz/B6jZoccMaA=">AAACzXichVFNSxxBEH2OSTRrjKu5BLwMLoZAYOldREUQ/LjkFr9WBccsPWPv2trzwUzPwjqZXAP5AznkFCGE4M/wklNuHvwJ4lEhlySkdnYg2YhJNd31+lW96upuO1Ay0oyd9xn9d+7eGxi8Xxh6MPxwpDg6thn5ceiImuMrP9y2eSSU9ERNS63EdhAK7tpKbNmHy534VkuEkfS9Dd0OxK7Lm55sSIdrourFfcv21V7UdsklB2k9sULXXEzNedNqhNxJxMtqmlStQJK33Nj8M30pNZ/1EJbHbcVTS0tXRD2R5bReLLEyy8y8CSo5KCG3Fb/4CRb24MNBDBcCHjRhBY6Ixg4qYAiI20VCXEhIZnGBFAXSxpQlKIMTe0hrk3Y7OevRvlMzytQOnaJohqQ0McnO2Gd2xb6wE3bBvt9aK8lqdHppk7e7WhHUR94+Xv/2X5VLXmP/t+qfPWs0MJv1Kqn3IGM6t3C6+tbRu6v1ubXJ5Ak7ZpfU/wd2zk7pBl7r2vm4Ktbeo0AfUPn7uW+CzWq5Ml2urk6VFpbyrxjEOCbwlN57Bgt4jhXU6NyvuMYP/DReGLHxynjdTTX6cs0j9Jjx5hfoNLTf</latexit>r · (j A<latexit sha1_base64="AMm5xGjUe7bbziMXVJlPT72P/RQ=">AAACq3ichVFNSxxBEH1Ootlsoq7JJeBlyGLQg0utSAyBgElAhFzUdVV0ZJmZbXXcng9mehc2w/yB/IEccoogIuZf5OJZyEGSPxA8GsjFg7WzA36hVtNdr1/Vq67utgLpRIrouEd78LC371Hucf7J0/6BwcLQs6XIb4a2qNq+9MMVy4yEdDxRVY6SYiUIhelaUixbjY+d+HJLhJHje4uqHYh119z0nA3HNhVTtcKMYfmyHrVddrHhmZY0E8Ou+0ofvRzZTmqxEbr6+0Qf1y8HPiVj+judaoUilSg1/SYoZ6CIzOb8wh4M1OHDRhMuBDwoxhImIh5rKIMQMLeOmLmQkZPGBRLkWdvkLMEZJrMNXjd5t5axHu87NaNUbfMpkmfISh0j9Iv26ZQO6YD+0tmtteK0RqeXNnurqxVBbfDLi8r/e1Uue4WtC9WdPSts4E3aq8O9BynTuYXd1bc+fz2tvF0YiV/RDp1w/9/pmH7yDbzWP3t3Xix8Q54/oHz9uW+CpYlS+XVpYn6yOP0h+4ochvESo/zeU5jGLOZQ5XN/4Ai/8Ucb1yraqmZ0U7WeTPMcV0wT57dupZ8=</latexit>

K) = 0e2
jA =µB + r ⇥ C
2
2⇡

Chiral Separation Effect (CSE)
Spin expectation is induced by a finite density + B
August 6, 2021 @ summer school online

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Gauge Inv. — External A
If a vector current is wanted, the same argument leads to:e2j=µA B + r ⇥ C
2
2⇡<latexit sha1_base64="wsixjVXC5Bwzz/+DNCBckahXJHY=">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</latexit>

Chiral Magnetic Effect (CME)µA is a chiral chemical potential coupled with g5⇠ µA ¯ 0 5✓(x)<latexit sha1_base64="CCKv4x/W0JL8RBf8pgQSZTh5yro=">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</latexit><latexit sha1_base64="UB4kmOBrojEwEDHy5O6HuPs+oN8=">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</latexit>⇠ ( ¯e i 5 µA t ) 0 i@0 (e i 5 µA t )<latexit sha1_base64="5BQSQA2BO0AtvU5Dj6rX72mSYj4=">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</latexit>

D ¯D ! ei<latexit sha1_base64="ip8meJ3pZiC5MFhL94FVeWbVLs0=">AAACzHichVHLThsxFD0MlNKURwobJDYjIh6ryIlQW3VFAamsEAkNIGXSyDM4wcLz0IwTiY6yZdEf6KKrIiGE+Aw23XTJgk+oukylbqjUm8movNT2WraPj++5PrbtQMlIM3Y1YAwOPRp+PPIk83R0bHwi+2xyO/JboSMqjq/8cNfmkVDSExUttRK7QSi4ayuxYx+s9vZ32iKMpO+91YeBqLm86cmGdLgmqp5tWi7X+w5X8VrHsnkYW0EkO+ZtlgjT0r4p3sXS2hNKc9N6w12XV1/X7mb+0S/cL7BQz+ZYniVhPgSFFOSQxqafPYWFPfhw0IILAQ+asAJHRK2KAhgC4mqIiQsJyWRfoIMMaVuUJSiDE3tAY5NW1ZT1aN2rGSVqh05R1ENSmphjl+yMddkXds6+seu/1oqTGj0vhzTbfa0I6hMfprd+/lfl0qyxf6P6p2eNBl4mXiV5DxKmdwunr2+//9jdelWei+fZMftO/j+zK3ZBN/DaP5yTkih/QoY+oHD/uR+C7WK+8DxfLC3lllfSrxjBDGaxSO/9AstYxyYqdO5XdHGNX8aGoY3Y6PRTjYFUM4U7YRz9BorStD4=</latexit>[A]

D ¯0 D

U(1)A transf.

0Chiral chemical potential = t-dep. q angle!
August 6, 2021 @ summer school online

26

Gauge Inv. — External A
Equation of motion read from axion QED:
2
1
e
µ⌫↵
LCS Maxwell =Fµ⌫ F µ⌫ + ✓(x)✏Fµ⌫ F↵
2
4
16⇡<latexit sha1_base64="vTAE+Rm6gbvxs7OoF6P7sIy0Hd4=">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</latexit>

The q term would not affect the EOM if q is constant but…

e2
r · E = ⇢ + 2 (r✓) · B
2⇡<latexit sha1_base64="5l107zlDGi5MiiDHtNS/fC6C+6I=">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</latexit>

Induced charge<latexit sha1_base64="X7P8+I8Z98htuyxL9n6etby2q6s=">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</latexit>

r⇥B

e2 ⇥
@0 E = j + 2 (@0 ✓)B
2⇡r ⇥ C if static<latexit sha1_base64="CNcP7z2OR6RiB3zkbC0D9SH5bVs=">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</latexit>(r✓) ⇥ E

Induced currents

August 6, 2021 @ summer school online⇤
27

Chiral Magnetic EffectIntuitive Picture“The Chiral Magnetic Effect” by Fukushima, Kharzeev, Warringa (2008)

Right-handed particles
Momentum parallel to Spin

Left-handed particles
Momentum anti-parallel to Spin

August 6, 2021 @ summer school online

28

Rotation Induced Current
Chiral Anomaly on Geometry<latexit sha1_base64="54aoJ9J+3wyTYq3mn0NwcytZO6w=">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</latexit>µrµ j A
=

1
µ⌫⇢
↵
✏
R
R⇢ ↵
µ⌫
2
384⇡

For details, see a textbook by Parker-Toms.

Chern-Simons current
1 µ⌫⇢
K =
✏
2
96⇡
<latexit sha1_base64="E0wK61NXLFz+cw6lSzvLkNLe+AA=">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</latexit>

µ

↵
⌫

@⇢

↵

2
+
3

⇢

↵

Not gauge invariant — not written in terms of tensors only.

Is there any “something” that makes it gauge invariant?
Honestly, I do not fully understand this point yet…
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29

Rotation Induced Current
Rotation (along the z axis)
x<latexit sha1_base64="C6Otv/b+S0Gatjt19iWPLve9A+U=">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</latexit>

0y =

y

0x = !

<latexit sha1_base64="IOGQNaWRmmMtyqpbFCcpcUhKvCw=">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</latexit>!

+

Rotation angular velocity

Expand the CS current in terms of w and the first order gives:

!
0
0
K =
R
+
R
x0x
y0y
48⇡ 2
<latexit sha1_base64="AUdNRjKCbpdRIKUtKyjUdvI0lMk=">AAAC0XichVHPSxtBFH67ba2mraZ6EbwMDRaLGF6CVKkIUi9CL5oYFVwTdtdJOjj7g91N2HVZEI/tH9CDJ4VSSv+MXjwLPfgniEcFPfTQl81Kf0j1DTPzzTfve/PNjOFK4QeIp4r64OGjvsf9A7knT58NDuWfD6/5Ttszec10pONtGLrPpbB5LRCB5Buux3XLkHzd2Fns7q93uOcLx14NIpdvWXrLFk1h6gFRjbx8V99l80xreroZa47FW3oST89qrqiXE6YZoiUnWKWOjVib0+ZYiGHCJv8gIowSNkVEeEOEGRHdSKKwV8h71cgXsIhpsNuglIECZLHs5L+ABtvggAltsICDDQFhCTr41DahBAgucVsQE+cREuk+hwRypG1TFqcMndgdGlu02sxYm9bdmn6qNukUSd0jJYNx/IFf8QKP8Rue4c//1orTGl0vEc1GT8vdxtCH0erVvSqL5gDe/1bd6TmAJsymXgV5d1Omewuzp+/sfrqovqmMxy/xCM/J/yGe4ne6gd25ND+v8MoB5OgDSv8+922wVi6WXhfLK9OFhbfZV/TDGLyACXrvGViAJViGGp17AteKoqhqVY3UPXW/l6oqmWYE/gr14y8FDq6x</latexit>

z

Rxyxy

Ry xyxIf the theory is put in a finite-T and constant curvature space,this expression is simplified as
✓ 2
◆
T
R
z
jA =
!
2
12
96⇡
Direct calculation
Chiral Vortical Effect<latexit sha1_base64="mYVIGE7NTro7FHL6qdQskub7CrY=">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</latexit>

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Rotation Induced Current
Chiral Vortical Effect from Anomalous Hydrodynamics
Son-Suruwka (2009)1 µ⌫ ⇢u⌫ @ u⇢Relativistic (covariant) vorticity: ! = ✏
2 ⇣ ⌘
µ
µ
µ
µ⌫
µ ⌫
T (g + u u )@⌫+ ⇠! µCurrent: j = nuT<latexit sha1_base64="Yvk2KkNg6/f2t/CbC2+3fkHnMvs=">AAACuHichVFNa9RAGH4aP1rXj656EbwEl4qnZbIULQWh6MVj23XbQqeNk3R2O3SSCZPJQg37B/wDHjwpiIi/wLMX/4CHnrxaPFbw4sE32YBoUd+QmWee533eeWcmyrTKHWNHM96Zs+fOz85daF28dPnKfPvqtY3cFDaWg9hoY7cikUutUjlwymm5lVkpkkjLzejgYaVvjqXNlUkfu8NM7iRilKqhioUjKmz3uUnkSOzypPDv+3xoRVwGk7I38bnMcqVNuluSxtOCa6q6J7jdNxO/CInxeSasU0KHjVbRJIftDuuyOvzTIGhAB02smvYbcOzBIEaBBBIpHGENgZy+bQRgyIjbQUmcJaRqXWKCFnkLypKUIYg9oHFEq+2GTWld1cxrd0y7aPotOX0ssE/sLTthH9k7dsx+/LVWWdeoejmkOZp6ZRbOP7vR//5fV0Kzw/4v1z97dhhiqe5VUe9ZzVSniKf+8dPnJ/3l9YXyNnvFvlL/L9kR+0AnSMff4tdrcv0FWvQAwZ/XfRps9LrB3W5vbbGz8qB5ijncxC3cofu+hxU8wioGtO97fMYXHHvL3hNv5KlpqjfTeK7jt/DsTxtEq7I=</latexit>µ<latexit sha1_base64="NGiJLYdbYYPc/IZ1vPEby7m42Vc=">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</latexit>

Conclusion:<latexit sha1_base64="lj4PpBJ6qy3hRh71CRBC3wwXI4w=">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</latexit>

✓

3

2 µ n
3e+p

⇠ = C µ2◆<latexit sha1_base64="lRjtu9DPRBLloReSE6CC+dCvv1s=">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</latexit>µ@µ j =1 µ⌫↵
C✏Fµ⌫ F↵
8

Strategy: Inserting E and B and “define” the entropy
current sµ and require @µ sµ 0<latexit sha1_base64="jxninKMy5uHuaVBiehqk5yyfXRY=">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</latexit><latexit sha1_base64="erAm9ADgufPORtjnSLgcO8lFPLo=">AAACfXichVHLSiNBFD1pddSok+hsBDfBEJmFhJsgKq7E2czSRKOC0dDdlrFIv+iuBDTMfIA/MItZjSgioj/hxh9w4SeISwU3gt7uNIiK4y2q6tSpe26dqjI8SwaK6DqhdXX3fOnt608ODA59TaWHR1YCt+mbomK6luuvGXogLOmIipLKEmueL3TbsMSq0fgR7q+2hB9I11lWu57YsPW6I7elqSumaunRqqf7SupWrWo3M8FmOFbrIkO1dJbyFEXmPSjEIIs4Ft30MarYggsTTdgQcKAYW9ARcFtHAQSPuQ20mfMZyWhf4BeSrG1yluAMndkGj3Vercesw+uwZhCpTT7F4u6zMoMcXdEJ3dElndINPX5Yqx3VCL3s8mx0tMKrpfZHlx4+Vdk8K+y8qP7rWWEbs5FXyd69iAlvYXb0rb0/d0tz5Vx7gg7olv3/o2u64Bs4rXvzqCTKf5HkDyi8fe73YKWYL0zni6Wp7PxC/BV9GMM4vvN7z2AeP7GICp/7G4c4w3niSctpk1q+k6olYs03vApt5hm0vZNL</latexit>

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31

temperature and the chemical potential, so Aµ ∼ O(p )and Fµν ∼ O(p). As in first-order hydrodynamics, we
keep terms of order O(p) in the constitutive equations
for T µν and j µ (or terms of order O(p2 ) in the equationsof motion). Note that Aµ is not dynamical.

Rotation Induced Currentsµ = suµ −where ξ, ξB , D,
quiring ∂µ sµ ≥ChiralVorticalofEffectfrom Anomalous
using the followi
Inthe presence
an external
background Hydrodynamics
field thehydrodynamic ehydrodynamic
equations obtainthe form
Hydrodynamic
Equations= Conservation Laws in terms of Hydro Variables
µ
∂
ω
=−
µν
νλ
µ
µ
µ
∂µ T = F jλ , ∂µ j = CE Bµ .
(5)
"+
µ
1 µ⌫↵
µ
µ⌫
µ
∂
B
= −2ω
µ
E = F u⌫ B = ✏u ⌫ F↵
where we have defined the electric
2 and magnetic fields inthe fluid rest frame, E µ = F µν uν , B µ = 21 "µναβ uν Fαβ .
one finds that t
The right hand
sides
of
these
equationstakeintoaccount
µ⌫
µ ⌫
µ⌫
µ⌫
satisfied
T
=
(e
+
p)u
u
pg
+
⇡
the fact that external field performs work on the system,⇠ O(@)µµµ
and the anomaly.
Note
that
the
right
hand
sides are of
j = nu + ⌫∂µ Dorder O(p) or O(p2 ) in our power counting,
which
are
Dissipative terms
Ideal Hydro
within the order hydrodynamic equations.→ Entropy production∂µ D
The stress-energy tensor and the current are
2nD
August 6, 2021 @ summer school online
32"+P
T µν = (" + P )uµ uν + P g µν + τ µν ,(6)<latexit sha1_base64="OP8oEG1KycJdxqbb6t+nb88dtrY=">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</latexit><latexit sha1_base64="b8OUojuSBPDhWoiulpvB07UOUio=">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</latexit><latexit sha1_base64="PnpEnUfb8B2f3pJNTusK/Pymx78=">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</latexit><latexit sha1_base64="TwQslW6PpzVWifMF0s99f5vkVo4=">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</latexit>Rotation Induced Current
Chiral Vortical Effect from Anomalous Hydrodynamics
Hydrodynamic Equations= Conservation Laws in terms of Hydro Variables

T µ⌫ = (e + p)uµ u⌫

pg µ⌫ + ⇡ µ⌫<latexit sha1_base64="b8OUojuSBPDhWoiulpvB07UOUio=">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</latexit><latexit sha1_base64="TwQslW6PpzVWifMF0s99f5vkVo4=">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</latexit>j µ = nuµ + ⌫ µ<latexit sha1_base64="y+//kvBS1ry9UYn3+kcMAW5xcN0=">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</latexit>⌫µ =

T (g µ⌫<latexit sha1_base64="14giMijhrETy2iulbGEBhgGPT+I=">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</latexit>µs = su<latexit sha1_base64="PeW4Be8+4UorMbCAvYmd8O+g5I4=">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</latexit>µµuµ u⌫ )@⌫⇣µ⌘T

+ E µ +⇠! µ + ⇠B B µµ µ⌫ +D! µ + DB B µT22@µ s = (. . . ) + (. . . ) + . . .

⇠, ⇠B , D, DB<latexit sha1_base64="xLCkPXQmPajzEPs5LqTT2v0LMCc=">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</latexit>

All solved!

August 6, 2021 @ summer school online

33

osition

cal

ante

relate the terms that appear there to local operators and to quark and gluon
distribution functions in an infinite momentum frame. Some of the issues we must
confront are:(i) What is the operator description of the spin and orbital angular momentum ofquarks and gluons?
Questions
by Jaffe-Manohar(ii)
How are theraised
matrix elementsof the angular momentum density tensor, M/*VA,
normalized?
(iii) What is the relation between the gluon spin and the anomalous current

Coming back to…

~

4A~AY).

(6.1)

@L
µ ⌫
⌫ µ
(iv)
J Is there= a sum(xrule@ for xthe
@ nucleon’si⌃µ⌫ )angular momentum in an infinite) does it get a contribution from the orbital angular momenmomentum @(@frame andµ ⌫tum of quarks= ¯i and(xgluons?
@x⌫ @ µ i⌃µ⌫ )<latexit sha1_base64="Oo3wWgjZtqmHJM6RW6KBY7ilo2k=">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</latexit>µ⌫Similar calculations are possible for the gluon part:

1 ¯
J=+E ⇥ A i
5| {z } |2| {z }
1
2

J=

1 ¯

G

⌃

5

†

(x ⇥ r) + E(x ⇥ r)A{z} | {z }q

Lcan

August† 6, 2021 @ summer school onlinei(x ⇥ D) + x ⇥ (E ⇥ B)

g

Lcan

34

relate the terms that appear there to local operators and to quark and gluon
distribution functions in an infinite momentum frame. Some of the issues we must
confront are:(i) What is the operator description of the spin and orbital angular momentum ofquarks and gluons?
Questions
by Jaffe-Manohar(ii)
How are theraised
matrix elementsof the angular momentum density tensor, M/*VA,
normalized?
(iii) What is the relation between the gluon spin and the anomalous current

Coming back to…

~

4A~AY).

(6.1)

@L
µ ⌫
⌫ µ
(iv)
J Is there= a sum(xrule@ for xthe
@ nucleon’si⌃µ⌫ )angular momentum in an infinite) does it get a contribution from the orbital angular momenmomentum @(@frame andµ ⌫tum of quarks= ¯i and(xgluons?
@x⌫ @ µ i⌃µ⌫ )<latexit sha1_base64="Oo3wWgjZtqmHJM6RW6KBY7ilo2k=">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</latexit>µ⌫

positionSimilar calculations are possible for the gluon part:

nical

fante

Gauge Spin?

1 ¯
J=+E ⇥ A i
5| {z } |2| {z }
1
2

J=

G

⌃

1 ¯
5
2

|

i

7

Gauge Orbital?
†(x ⇥ r) + E(x ⇥ r)A{z} | {z }q

Lcan† August 6, 2021 @ summer school online(x ⇥ D) + x ⇥ (E ⇥ B){z
} | {z }

g

Lcan

35

case, First, the spin contribution toM~~(spin)=F~A”+F”’~A5(6.36)

Comingback
to…
is not directly related to the U(1) axial current, K,,. The analogy with the Diraccase would have led one to expect M~~(spin)
to be

Questions raised EIi~A(rK,,by Jaffe-Manohar
Ft’AA~+ F”’-’-A5 + ~=(6.37)

µ⌫
µ
⌫
⌫
µ
Sbut
=
A
F
A
F
gauge
the term FA”A1~ is missing from M~”(spin). Furthermore, FA~Ati is not<latexit sha1_base64="J4vmlneHrTA9h83Hq/e3hNIjbVc=">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</latexit>

separately
conservedwhichµ⌫µ so there⌫ is no⌫ superpotentialµµ⌫ can convert the spin term
✏to the form
K (6.37).
= A eti”~”K,,and
F
AM~,~(spin)
F
A
are F
not in the same representation of
the Lorentz group. The former is a vector (~,~-) while the latter contains (-i, ~)CScurrent
looksto likegauge thespin,andand spin terms in M()~”5are not~ (~,~.) in addition(-~,i.).a Finally,
orbitalseparately
gauge looks
invariant(althoughis gauge invariant up to a superpotenthe
anomaly
likea spinM(~”5conservation…???
tial). So the decomposition into spin and orbital parts is meaningful only withreference to a specific gauge.
They
are DIFFERENT by a term
corresponding to the CME!?
The distinction between M~~(spin)and ““5”K,, becomes less clear when one
constructs the rotation generators. The spin contribution tois I d3x (A X E).
The analogous integral over e’~”K,,. contains an additional term: fd3xA°B. In
A’~ 0 gauge the two expressions agree. The same is true for the analogous
generator (f d2x
1 dx M~’~)for a theory quantized on the light-cone in A + 0
Any deep physics???
gauge.
Finally, we quote the generalization of M””” to QCD. With a suitable choice of
36
a superpotential we obtain August 6, 2021 @ summer school online
5—x50”)A,,)<latexit sha1_base64="7uz9Qyl2j+5cAwpIuBTbUa/PFG4=">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</latexit>

=

—

=

Serious Problem in Spin Hydro
How to construct the hydrodynamics with J ?
Hattori-Hongo-Huang-Matsuo-Taya (2019)

J µ⌫ = xµ T ⌫<latexit sha1_base64="7BcQQR7i9fSOEyI3aaiFu7BVCFA=">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</latexit>x⌫ T µ + ⌃ µ⌫

@ J µ⌫ = 0<latexit sha1_base64="LL59YNP9eHX+Rl70c/39h7QNCz0=">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</latexit><latexit sha1_base64="C3nWCl0fHZC6aliij0OJd0S9xTY=">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</latexit>@ ⌃@ T µ=0<latexit sha1_base64="dft0PHEDeusxRSt9tGiICS2qmtE=">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</latexit>

µ⌫

=

µ⌫
2T(a)

µ⌫
T(a)
=
<latexit sha1_base64="ZkIp7mi7kzKDUVtAylQWzZnt2ew=">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</latexit>

⌫µ
T(a)

Spin tensor ~ Anti-symmetric part of the EMT<latexit sha1_base64="O5yqRpI0hUSEMHFc+aEH7pupI3U=">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</latexit><latexit sha1_base64="XwpzQ4HcRJi7ZqUp1L+Z8DIqAXE=">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</latexit>

⌃

µ⌫

=u S

µ⌫

+ ···

The dynamics of spin S<latexit sha1_base64="ieRqwCSMMEP4TchZl8aqTUXCSz4=">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</latexit>µ⌫

p = · · · + !µ⌫ S µ⌫

is constrained by @µ sµ<latexit sha1_base64="erAm9ADgufPORtjnSLgcO8lFPLo=">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</latexit>

0

Looks good… but not good in gauge theories… ?
August 6, 2021 @ summer school online

37Serious Problem in Spin Hydro
How to construct the hydrodynamics with J ?
An exercise in Peskin-Schroeder

Pseudo-gaugetransformation

August 6, 2021 @ summer school online

38Serious Problem in Spin Hydro
How to construct the hydrodynamics with J ?
In gauge theories only symmetrized EMT is gauge invariant!

@ ⌃ µ⌫ =<latexit sha1_base64="C3nWCl0fHZC6aliij0OJd0S9xTY=">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</latexit>µ⌫ is missing in a symmetric formulation?
2T(a)J µ⌫ = xµ T ⌫ x⌫ T µ + ⌃ µ⌫can be “redefined” to be a conserved angular momentum
in terms of symmetrized EMT as<latexit sha1_base64="7BcQQR7i9fSOEyI3aaiFu7BVCFA=">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</latexit>

Jb µ⌫ = xµ Tb ⌫<latexit sha1_base64="bvsblNTYztfKsbIrSs24sdnREPo=">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</latexit>x⌫ Tb µSpin apparently missing!?

I wrote a paper (Fukushima-Pu 2020) but I am notsatisfied with my own arguments yet…
August 6, 2021 @ summer school online

39

Summary
Chern-Simons current tells us a lot.
□ CS charge is a gauge invariant observable= Magnetic Helicity in QED.
□ “Something” replacing A makes it gauge invariant.

Rotation effects also induces a current.
□ Curvature / T / µ + w = (axial) current□ Anomalous hydrodynamics

Relation between CS current and Spin?
□ Spin (helicity) of gauge field well separated?
□ Spin hydrodynamics with gauge invariant EMT?
August 6, 2021 @ summer school online40