Separating fact from fantasy*: The chiral anomaly and the ...shovkovy.faculty.asu.edu/colloquium/slides/Colloquium_Slides... · Europe and Asia. In support of this new direction:

Separating fact from fantasy*: The chiral anomaly and the proton spin puzzle

Raju Venugopalan(BNL)

Arizona State Online Theory Colloquium, August 19, 2020

Separating fact from fantasy*: The chiral anomaly and the proton spin puzzle

R. L. Jaffe A. Manohar

30th anniversary of their seminal paper which strongly inspired the work reported here:

The G(1) Problem: Fact and Fantasy on the Spin of the Proton.Nucl. Phys., B337:509–546, 1990

Work* in collaboration with Andrey Tarasov (The OSU and CFNS)

* Our first paper will be on arXiv tonight (6 pm ASU time) and two others are in preparation

Talk outlinev Polarized DIS – preliminaries

v Iso-singlet axial vector current and topology of QCD vacuum: the role of the triangle graph

v Fun with world-lines

v World-line effective action: i) Finding a triangle in a box in Bjorken and Regge asymptoticsii) Hidden triangles dress infrared poles; the WZW termiii) Small xBj effective action for polarized DIS:

Extended PCAC and the 𝜂′ as an emergent axion-like field iv) QCD evolution of the small x effective action

v Conjecture: How spin diffuses from polarized large x partons to small x via sphalerons transitions

v General considerations/outlook

On the DOE web page: January 9, 2020

Secretary Brouillette also approved Critical Decision-0, “Approve Mission Need,” for the EIC on December 19, 2019

WASHINGTON, D.C. – Today, the U.S. Department of Energy (DOE) announced the selection of Brookhaven National Laboratory in Upton, NY, as the site for a planned major new nuclear physics research facility.

The Electron Ion Collider (EIC), to be designed and constructed over ten years at an estimated cost between $1.6 and $2.6 billion, will smash electrons into protons and heavier atomic nuclei in an effort to penetrate the mysteries of the “strong force” that binds the atomic nucleus together.

“The EIC promises to keep America in the forefront of nuclear physics research and particle accelerator technology, critical components of overall U.S. leadership in science,” said U.S. Secretary of Energy Dan Brouillette. “This facility will deepen our understanding of nature and is expected to be the source of insights ultimately leading to new technology and innovation.”“

What’s the Electron-Ion Collider ?

The deeply inelastic scattering (DIS) femtoscope

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#$%

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θ

θ

µµMeasure of resolution power

Measure of inelasticity

Bjorken variable: Measure of momentum fraction of struck quark

Inclusive DIS measurements: e+↑p/A à e’+XMeasure only the scattered lepton in the detector: structure functions (pdfs: F2,FL,g1,g2,…)

Semi-inclusive DIS (SIDIS) measurements: e+↑p/A à e’+h(p,K,p,jet)+XMeasure electrons in coincidence with identified hadrons/jets (Transverse momentum dependent dists. -TMDs)

Exclusive measurements (DVCS,…): e+↑p/A à e’+ photon/hadron + p/AMeasure electron, photon (or e.g., vector meson), intact proton/nucleus (Generalized parton dists. -GPDs)

Diffractive measurements: e+↑p/A à e’+ hadrons/jets + rapidity gap (coherent diff.→ 𝐢𝐧𝐭𝐚𝐜𝐭 𝐩/𝐀)As for exclusive, but with rapidity veto on particle production (Diff. structure functions, F2

D, FLD )

Also, measure nuclear fragments, multiparticle correlations, …

Landscape of scattering in the strong interaction

Aschenauer et al., arXiv:1708.01527Rep.Prog. Phys. 82, 024301 (2019)

Many open questions: three-D structure, spin and orbital dynamics, many-body correlations, small x

Higher twists

The Electron-Ion Collider

v Electron storage ring with frequent injection of fresh polarized electron bunches

v Hadron storage ring with strong cooling or frequent injection of hadron bunches

Polarized protons up to 275 GeV; Nuclei up to ~ Z/A*275 GeV/nØ Existing RHIC complex: Storage (Yellow), injectors (source,

booster, AGS)Ø Need few modificationsØ RHIC beam parameters fairly close to those required for

EIC@BNL

Electrons up to 18 GeVØ Storage ring, provides the range sqrt(s) = 20-140 GeV. Beam

current limited by RF power of 10 MWØ Electron beam with variable spin pattern (s) accelerated in on-

energy, spin transparent injector (Rapid-Cycling-Synchrotron) with 1-2 Hz cycle frequency

Ø Polarized e-source and a 400 MeV s-band injector LINAC in the existing tunnel

Design optimized to reach 1034 cm-2sec-1

The Electron-Ion Collider

Ø First polarized electron-polarized proton collider

Ø First electron-nucleus collider

Ø Luminosities up to ~ 1000 times that of HERA

Ø Fine resolution inside the proton down to ~10-18 m

Larry McLerran, Raju Venugopalan et al. at BNL led the workshop on Nuclei

in eRHIC with some discussion of

polarized protons by A. D. & Richard

Milner

While not the highest of energies

hoped for (in comparison with

HERA), it was the first time, that polarized proton, and nuclear beam

proponents worked together.

1510/21/2018 Abhay Deshpande at INT 18-03 Week 4: History of EIC Project

NSAC 2007 Long Range Plan�An Electron-Ion Collider (EIC) with polarized beams has been embraced by the U.S. nuclear science community as embodying the vision for reaching the next QCD frontier. EIC would provide unique capabilities for the study of QCD well beyond those available at existing facilities worldwide and complementary to those planned for the next generation of accelerators in Europe and Asia. In support of this new direction:

We recommend the allocation of resources to develop accelerator and detector technology necessary to lay the foundation for a polarized Electron Ion Collider. The EIC would explore the new QCD frontier of strong color fields in nuclei and precisely image the gluons in the proton.�


The Yale group organized a first workshop

fully focused on polarized eRHIC:

A.D., Vernon Hughes,

+ George Igo (UCLA)

+ A. De Roeck (DESY/CERN)

Encouraged by G. Garvey, P. Paul and others,

identify eRHIC as a particularly promising

avenue

(Roser et al. showed polarization at RHIC

was possible)


By 2001-2003

BNL & RIKEN-BNL Research

Center became the focus of EIC

activities

AD, R. Venugopalan, W.

Vogelsang et al. locally

supported by L. McLerran, T.D.

Lee, P. Paul (Acting D. Director

for Science & Technology), and

G. Garvey (on Sabbatical)

18

Symbol is born!(BNL Arts department)


Preparation for the 2002 Long Range Plan

White Paper prepared/edited by:

A. Deshpande, R. Milner & R. Venugopalan

Institutions:

BNL, Budker Institute, CERN, U. of Colorado (Boulder), FNAL, UIUC,

Indiana U., LBNL, Los Alamos, MIT, INP Poland, U. of Paris VI, Penn

State, Regensburg, RIKEN-BNL, Saclay, Triumf, Yale

Actively Supported by:

G. Garvey (Los Alamos) & Peter Paul

(Acting Director BNL Dep. Director) & T. D. Lee (RBRC/Columbia)


NSAC 2002 Long Range Plan: First Recognition

20

The Electron-Ion Collider is a newaccelerator concept that has been proposed to extend our understanding of the structure of matter in terms of itsquark and gluon constituents. Two classes of machine design for the EIChave been considered: a ring-ring optionwhere both electron and ion beams circulate in storage rings, and a ring-linacoption where a linear electron beam incident on a stored ion beam.

There is a strong consensus among nuclearscientists to pursue R&D over the next three years to address a number of EIC design issues.In parallel, the scientific case for the EIC will besignificantly refined.


34

Institution List for EIC Working Group Institutes 2007 LRP

ANL, BARC/india, BNL, Buenos Aires, UCLA, CERN, U. Colorado, Columbia, DESY,Glasgow, Hampton U., UIUC, Iowa State, Jlab, U. Kyoto, LBNL, Los Alamos, U. Mass (A), MIT, MPI Munich, U. of Michigan, NMSU, ODU, Penn State, RIKEN, RIKEN-BNL, SINS Poland,Stony Brook, Tel Aviv, TJNAF


INT program 2010

Organizers:

D. Boer, M. Diehl, R. Milner, R. Venugopalan, W.

Vogelsang

Broad community input and participation. A

document containing 500+ pages and 160+ writers

summarized the entire gamut of EIC physics.

Identified the most compelling measurements to make

the case for the EIC: Golden, Silver, Bronz. (not

appreciated by all but) very convincing to the outsiders

, the broader community of our sincere effort. –

Important understated lesson


EIC White Paper for LRP 2015

38

In 2012 R. McKeown and S. Vigdorappointed a group:A. Accardi et al. 1212.1701.v3Ed: AD, Jianwei Qiu & Zein-Eddine Meziani

Pre-LRP QCD Town Meeting Temple University:Hot & Cold QCD Working Groups unanimously declared EIC to be the most desired future facility for US Nuclear Science – September 2014

1212.1701.v3A. Accardi et al


NSAC 2015 Long Range Plan

40

science.energy.gov/np/nsac/


1999/20002015

2007 LRP

2002 LRP

2018

A long and winding road…1st eRHIC workshop, BNL Dec. 3-4, 1999

2010 INT reportRedux: 2018

NAS panel

2012 EIC white paper 2018 BNL Report

(Garvey,Ludlam,McLerran,Strikman,Venugopalan)

Polarized DIS at small x – reaching into terra incognita with EIC

The proton’s spin puzzle: a many-body picture

Fixed target DIS experiments showed that quarks (ΔΣ) carry only about 30% of the proton’s spin

“Spin crisis”: failure of the quark model (“Ellis-Jaffe sum rule”) picture of relativistic “constituent” quarks

D. De Florian, R. Sassot, M. Stratmann, W. Vogelsang, PRL 113 (2014)

Evidence for gluon spin (∆𝐺) from RHIC but large uncertainties from small x

Resolving the proton’s spin puzzle: the g1 structure function

x=5.2×10-5 (+53)

8.2×10-5 (+45)

1.3×10-4 (+38)

2.1×10-4 (+33)

3.3×10-4 (+28)

5.2×10-4 (+24)

8.2×10-4 (+21)

1.3×10-3 (+19)

2.1×10-3 (+17)3.3×10-3 (+15.5)5.2×10-3 (+14)

8.2×10-3 (+13)1.3×10-2 (+12)

2.1×10-2 (+11)3.3×10-2 (+10)

5.2×10-2 (+9)8.2×10-2 (+8)

1.3×10-1 (+7)2.1×10-1 (+6)

3.3×10-1 (+5)5.2×10-1 (+4)

g1(x,Q2) + const(x)

Q2 [GeV2]

DSSV 201490% C.L. ∆g

EIC 15×100 GeVEIC 15×250 GeVEIC 20×250 GeV

10

20

30

40

50

1 10 102

103

world datauntil eRHIC

-

dg1

d log(Q2 )~ −Δg(x,Q2 )→ gluon contr.

noweRHIC

dramatic reduction of uncertainties:

g1 extracted from longitudinal spin asymmetry

�⌃(Q2) /Z 1

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Resolving the proton’s spin puzzle: the g1 structure function

Aschenauer et al., arXiv:1708.01527Rep. Prog. Phys. 82, 024301 (2019)

Iso-singlet axial vector current and topology of the QCD vacuumReviews:S. D. Bass, RMP, hep-ph/0411005G. M. Shore, hep-ph/0701171

g1 structure function: formal definitions

Hadron tensor 𝑊"# =

with

Most generally, where 2𝑊"# is the antisymmetric part of 𝑊"#

𝑆" = $%&!

𝑃" and 𝜆 = ±1/2

Generalized parton model (“leading twist”):

Where the quark helicity pdf is defined to be

Iso-singlet axial vector current

In general, the first moment of g1:

Z 1

0g1(x,Q

2) =1

18

�3F +D + 2⌃(Q2)

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Combination of triplet axial vector current (gives gA) measured in 𝛽 𝑑𝑒𝑐𝑎𝑦 and octet axial vector current measured in hyperon decays

with the iso-singlet quark helicity given by

In the parton model picture, this mixes, under evolution, with other isospin blind moment Δ𝐺 = ∫'(𝑑𝑥 Δ𝑔(𝑥, 𝑄2)

Splitting functions known to high loop order Moch, Rogal, Vermaseren, Vogt Talk by Vogelsang at POETIC IX, LBNL; DeFlorian, Vogelsang (2019)

Iso-singlet axial vector current and the chiral anomaly

UA(1) violation from the chiral anomaly: @µJ

µ5 = 2nf@µK

µ +

nfX

i=1

2imiq̄i�5qi<latexit sha1_base64="BtFJlH/1OeKkIMBQQn/3YSEEjy0=">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</latexit><latexit sha1_base64="BtFJlH/1OeKkIMBQQn/3YSEEjy0=">AAACTXicbVHPSxwxFM5sW3+sra726OXRRRAKy4woehFEL6W9KHRV2NkNb7KZNWySGZOMsAzzD3op9OZ/4cWDRcTMuof640HCx/d9j/fyJcmlsC4Mb4LGh4+f5uYXFptLn78sr7RW105tVhjGuyyTmTlP0HIpNO864SQ/zw1HlUh+loyPav3sihsrMv3bTXLeVzjSIhUMnadoaxjnaJxASWNVwE+6MyhrUME+bIGmKbzQfw3q+3tsC0VLsR9Vg9J7Km8VoKiAMk7QwGXlYTxCpZDuwCUVtNUOO+G04C2IZqBNZnVMW3/jYcYKxbVjEq3tRWHu+mW9CZO8asaF5TmyMY54z0ONitt+OU2jgg3PDCHNjD/awZT9v6NEZe1EJd6p0F3Y11pNvqf1Cpfu9Uuh88JxzZ4HpYUEl0EdLQyF4czJiQfIjPC7ArtAg8z5D2j6EKLXT34LTrc6UdiJTrbbB4ezOBbIOvlGNklEdskB+UGOSZcwck1uyT35F/wJ7oKH4PHZ2ghmPV/Ji2rMPwHrhrKM</latexit><latexit sha1_base64="BtFJlH/1OeKkIMBQQn/3YSEEjy0=">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</latexit><latexit sha1_base64="BtFJlH/1OeKkIMBQQn/3YSEEjy0=">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</latexit>

where the Chern-Simons current

For massless quarks, conserve Jµ5 � 2nfK

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∝

So then the “real” ΔΣ is

offering a possible explanation of empirical small ΔΣ (in addition to flavorSU(3) violation)…

ca., 1988Efremov, TeryaevAltarelli, Ross Carlitz, Collins, Mueller


UA(1) violation from the anomaly: @µJµ5 = 2nf@µK

µ +

nfX

i=1

2imiq̄i�5qi<latexit sha1_base64="BtFJlH/1OeKkIMBQQn/3YSEEjy0=">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</latexit><latexit sha1_base64="BtFJlH/1OeKkIMBQQn/3YSEEjy0=">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</latexit><latexit sha1_base64="BtFJlH/1OeKkIMBQQn/3YSEEjy0=">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</latexit><latexit sha1_base64="BtFJlH/1OeKkIMBQQn/3YSEEjy0=">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</latexit>

where the Chern-Simons current

But, identification of CS charge with ΔG is intrinsically ambiguous… 𝑡ℎ𝑒 𝑙𝑎𝑡𝑡𝑒𝑟 𝑖𝑠 𝑔𝑎𝑢𝑔𝑒 𝑖𝑛𝑣𝑎𝑟𝑖𝑎𝑛𝑡, 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚𝑒𝑟 𝑖𝑠 𝑛𝑜𝑡

Jaffe, Manohar (1990)

”Large gauge transformation”- deep consequence of topology

R. Jaffe: identification of 𝐾" with ΔGa source of much confusion in the literature (Varenna lectures, 2007)


However, identifying CS − charge with ∆G “works” remarkably well …(Vogelsang, POETIC IX)

∝ ∆𝐺

NNLO splitting function computations confirm this At LO, Altarelli, Lampe (1990)NNLO: Vogt, Moch, Rogal, Vermaseren, arXiv:0807.1238

This suggests that 𝛥𝛴 𝑜𝑛𝑙𝑦 𝑚𝑖𝑥𝑒𝑠 𝑤𝑖𝑡ℎ 𝑖𝑡𝑠𝑒𝑙𝑓; 𝑙𝑖𝑘𝑒𝑤𝑖𝑠𝑒 ∆𝐺 𝑎𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑, 𝑜𝑛𝑙𝑦 𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝛥𝛴with the same splitting function

We will discuss later the deeper physics behind these expressions and their relation to the topology of the QCD vacuum

Alternative picture: topological charge screening of spinShore, Veneziano, PLB (1990); NPB (1992)Narison, Shore, Veneziano, hep-ph/9812333

Anomalous chiral Ward identities and extended PCAC from UA(1) breaking

Example: Witten-Veneziano formula m2⌘0 =

2nf

f2⇡

�YM(0) +O((nf

Nc)2)

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where the topological susceptibility �YM(p2) = i

Zdx eiP ·xh0|T (Q(x)Q(0)|0i

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with Q(x) =↵S

8⇡Tr

⇣Fµ⌫ F̃

µ⌫⌘

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In this picture,

In chiral limit 𝜒 0 → 0, ΔΣ “controlled” by the slope χ) at p2=0 – estimated to be small by Veneziano et al.

What about dynamics – can we reconcile this picture with the perturbative one ?

OZI-allowed OZI-suppressed

⌃(Q2) =1

3mN�CS

1 (↵S)⇣gQNN�(0) + g⌘0NN

p�0(0)

⌘

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Perturbative & nonperturbative interplay: The triangle graph

The key role of the UA(1) anomaly is seen from the structure of the triangle graph In the off-forward (𝑙" → 0, 𝑡 = 𝑙$ → 0) matrix element of < 𝑃),S| 𝐽"*|P, S >

hP 0, S|Jµ5 |P, Si = GA(t)Sµ + l · S lµGP (t)

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The computation of this has an infrared pole

proportional to +"

+#𝐹 �𝐹

! l · S lµ

l2(t) +

✓Sµ � l · S lµ

l2

◆�(t)

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For that to hold, one must have (0) = �(0) / FF̃<latexit sha1_base64="D2rMamwTVSP4PsyzpYcyxjkZKb8=">AAACFHicbVDNS8MwHE39nPOr6tFLcAgTYbQi6EUYCsPjBPcBaxlpmm5haROSVBhlf4QX/xUvHhTx6sGb/41Z14NuPgi8vN/vkbwXCEaVdpxva2l5ZXVtvbRR3tza3tm19/bbiqcSkxbmjMtugBRhNCEtTTUjXSEJigNGOsHoZjrvPBCpKE/u9VgQP0aDhEYUI22kvn3qjZAQqOqcwCvoMWMM84snJBeawwbMPE1ZSGBj0rcrTs3JAReJW5AKKNDs219eyHEak0RjhpTquY7QfoakppiRSdlLFREIj9CA9AxNUEyUn+WhJvDYKCGMuDQn0TBXfzsyFCs1jgOzGSM9VPOzqfjfrJfq6NLPaCJSTRI8eyhKGTRppw3BkEqCNRsbgrCk5q8QD5FEWJsey6YEdz7yImmf1Vyn5t6dV+rXRR0lcAiOQBW44ALUwS1oghbA4BE8g1fwZj1ZL9a79TFbXbIKzwH4A+vzBzu6nHA=</latexit><latexit 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the topological charge density

The perturbative/nonperturbative interplay gives a highly non-trivial result:

hP 0, S|Jµ5 |P, Si = GA(t)Sµ + l · S lµGP (t)

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The infrared pole in GA(t) must be canceled by a pole in GP(t) – the corresponding Wess-Zumino-Witten term for the 𝜂)

Jaffe. Manohar (1990)

Perturbative & nonperturbative interplay: The triangle graph

with the (manifestly) gauge invariant result for the forward matrix element

Suprisingly (since known to Jaffe+Manohar & Veneziano et al) not addressed in a lot of the “pQCD” literatureIt is however deeply profound since it is intimately tied to the mechanism whereby the 𝜂) gets its mass

This result, generalization to g1(x,Q2), and the interplay with non-perturbative physics can be explored efficiently in a worldline framework

Thinking properly about anomalies with worldlines

Review: Schubert, Phys. Repts. (2001)N. Mueller, RV: 1701.03331.1702.01233,1901.10492Tarasov, RV: 1903.11624 and in preparation

The worldline formulation of QFT is equivalent to the string amplitude formalism of Bern and Kosower,as shown by Strassler

Bern,Kosower, NPB 379 (1992) 145; Bern, TASI lectures, hep-ph/9304249Strassler, NPB 385 (1992) 145

World-line formalism: vector and axial vector fields

with

Heat kernel representation of χ: 𝑠𝑎𝑛𝑑𝑤𝑖𝑐ℎ between 0 + 1 − Dboson x, p and fermion Grassmann coherent states

Wigner-Weyl formalism: i=0,1#𝑎!±

Grassmann Majorana fermions:

Berezin, Marinov (1976);Brink et al; Barducci et al; Balachandran et al (1976-77); D’Hoker, Gagne (1996)

S[A,A5] =

Zd4x ̄ (i/@ + /A+ �5/A5)

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One loop effective action �[A,A5] = log det (�)<latexit sha1_base64="8W42ntvBYImkv3XEmtKr0TXtchw=">AAACH3icbZDJSgNBEIZ73I1b1KOXxiAohDAjbhfB5aDHCEaFzBB6OjVJk+6ZobtGCEPexIuv4sWDIuLNt7GzHDTxh4aPv6qorj9MpTDout/O1PTM7Nz8wmJhaXllda24vnFnkkxzqPFEJvohZAakiKGGAiU8pBqYCiXch53Lfv3+EbQRSXyL3RQCxVqxiARnaK1G8ci/Ykqx+nn5vHEY0FOa+1pRmbR6fnnITcCeLyHCXZ+3ha9Fq417jWLJrbgD0UnwRlAiI1UbxS+/mfBMQYxcMmPqnptikDONgkvoFfzMQMp4h7WgbjFmCkyQD+7r0R3rNGmUaPtipAP390TOlDFdFdpOxbBtxmt9879aPcPoJMhFnGYIMR8uijJJMaH9sGhTaOAouxYY18L+lfI204yjjbRgQ/DGT56Eu/2K51a8m4PS2cUojgWyRbbJLvHIMTkj16RKaoSTJ/JC3si78+y8Oh/O57B1yhnNbJI/cr5/AOFcoaw=</latexit><latexit sha1_base64="8W42ntvBYImkv3XEmtKr0TXtchw=">AAACH3icbZDJSgNBEIZ73I1b1KOXxiAohDAjbhfB5aDHCEaFzBB6OjVJk+6ZobtGCEPexIuv4sWDIuLNt7GzHDTxh4aPv6qorj9MpTDout/O1PTM7Nz8wmJhaXllda24vnFnkkxzqPFEJvohZAakiKGGAiU8pBqYCiXch53Lfv3+EbQRSXyL3RQCxVqxiARnaK1G8ci/Ykqx+nn5vHEY0FOa+1pRmbR6fnnITcCeLyHCXZ+3ha9Fq417jWLJrbgD0UnwRlAiI1UbxS+/mfBMQYxcMmPqnptikDONgkvoFfzMQMp4h7WgbjFmCkyQD+7r0R3rNGmUaPtipAP390TOlDFdFdpOxbBtxmt9879aPcPoJMhFnGYIMR8uijJJMaH9sGhTaOAouxYY18L+lfI204yjjbRgQ/DGT56Eu/2K51a8m4PS2cUojgWyRbbJLvHIMTkj16RKaoSTJ/JC3si78+y8Oh/O57B1yhnNbJI/cr5/AOFcoaw=</latexit><latexit sha1_base64="8W42ntvBYImkv3XEmtKr0TXtchw=">AAACH3icbZDJSgNBEIZ73I1b1KOXxiAohDAjbhfB5aDHCEaFzBB6OjVJk+6ZobtGCEPexIuv4sWDIuLNt7GzHDTxh4aPv6qorj9MpTDout/O1PTM7Nz8wmJhaXllda24vnFnkkxzqPFEJvohZAakiKGGAiU8pBqYCiXch53Lfv3+EbQRSXyL3RQCxVqxiARnaK1G8ci/Ykqx+nn5vHEY0FOa+1pRmbR6fnnITcCeLyHCXZ+3ha9Fq417jWLJrbgD0UnwRlAiI1UbxS+/mfBMQYxcMmPqnptikDONgkvoFfzMQMp4h7WgbjFmCkyQD+7r0R3rNGmUaPtipAP390TOlDFdFdpOxbBtxmt9879aPcPoJMhFnGYIMR8uijJJMaH9sGhTaOAouxYY18L+lfI204yjjbRgQ/DGT56Eu/2K51a8m4PS2cUojgWyRbbJLvHIMTkj16RKaoSTJ/JC3si78+y8Oh/O57B1yhnNbJI/cr5/AOFcoaw=</latexit><latexit sha1_base64="8W42ntvBYImkv3XEmtKr0TXtchw=">AAACH3icbZDJSgNBEIZ73I1b1KOXxiAohDAjbhfB5aDHCEaFzBB6OjVJk+6ZobtGCEPexIuv4sWDIuLNt7GzHDTxh4aPv6qorj9MpTDout/O1PTM7Nz8wmJhaXllda24vnFnkkxzqPFEJvohZAakiKGGAiU8pBqYCiXch53Lfv3+EbQRSXyL3RQCxVqxiARnaK1G8ci/Ykqx+nn5vHEY0FOa+1pRmbR6fnnITcCeLyHCXZ+3ha9Fq417jWLJrbgD0UnwRlAiI1UbxS+/mfBMQYxcMmPqnptikDONgkvoFfzMQMp4h7WgbjFmCkyQD+7r0R3rNGmUaPtipAP390TOlDFdFdpOxbBtxmt9879aPcPoJMhFnGYIMR8uijJJMaH9sGhTaOAouxYY18L+lfI204yjjbRgQ/DGT56Eu/2K51a8m4PS2cUojgWyRbbJLvHIMTkj16RKaoSTJ/JC3si78+y8Oh/O57B1yhnNbJI/cr5/AOFcoaw=</latexit>

� = i/@ + /A+ �5 /A5<latexit sha1_base64="Y878R7LLd36+c7vgYxWO37Bjekk=">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</latexit><latexit 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sha1_base64="Y878R7LLd36+c7vgYxWO37Bjekk=">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</latexit><latexit sha1_base64="Y878R7LLd36+c7vgYxWO37Bjekk=">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</latexit>

World-line formalism: vector and axial vector fields

with

S[A,A5] =

Zd4x ̄ (i/@ + /A+ �5/A5)

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One loop effective action �[A,A5] = log det (�)<latexit sha1_base64="8W42ntvBYImkv3XEmtKr0TXtchw=">AAACH3icbZDJSgNBEIZ73I1b1KOXxiAohDAjbhfB5aDHCEaFzBB6OjVJk+6ZobtGCEPexIuv4sWDIuLNt7GzHDTxh4aPv6qorj9MpTDout/O1PTM7Nz8wmJhaXllda24vnFnkkxzqPFEJvohZAakiKGGAiU8pBqYCiXch53Lfv3+EbQRSXyL3RQCxVqxiARnaK1G8ci/Ykqx+nn5vHEY0FOa+1pRmbR6fnnITcCeLyHCXZ+3ha9Fq417jWLJrbgD0UnwRlAiI1UbxS+/mfBMQYxcMmPqnptikDONgkvoFfzMQMp4h7WgbjFmCkyQD+7r0R3rNGmUaPtipAP390TOlDFdFdpOxbBtxmt9879aPcPoJMhFnGYIMR8uijJJMaH9sGhTaOAouxYY18L+lfI204yjjbRgQ/DGT56Eu/2K51a8m4PS2cUojgWyRbbJLvHIMTkj16RKaoSTJ/JC3si78+y8Oh/O57B1yhnNbJI/cr5/AOFcoaw=</latexit><latexit sha1_base64="8W42ntvBYImkv3XEmtKr0TXtchw=">AAACH3icbZDJSgNBEIZ73I1b1KOXxiAohDAjbhfB5aDHCEaFzBB6OjVJk+6ZobtGCEPexIuv4sWDIuLNt7GzHDTxh4aPv6qorj9MpTDout/O1PTM7Nz8wmJhaXllda24vnFnkkxzqPFEJvohZAakiKGGAiU8pBqYCiXch53Lfv3+EbQRSXyL3RQCxVqxiARnaK1G8ci/Ykqx+nn5vHEY0FOa+1pRmbR6fnnITcCeLyHCXZ+3ha9Fq417jWLJrbgD0UnwRlAiI1UbxS+/mfBMQYxcMmPqnptikDONgkvoFfzMQMp4h7WgbjFmCkyQD+7r0R3rNGmUaPtipAP390TOlDFdFdpOxbBtxmt9879aPcPoJMhFnGYIMR8uijJJMaH9sGhTaOAouxYY18L+lfI204yjjbRgQ/DGT56Eu/2K51a8m4PS2cUojgWyRbbJLvHIMTkj16RKaoSTJ/JC3si78+y8Oh/O57B1yhnNbJI/cr5/AOFcoaw=</latexit><latexit sha1_base64="8W42ntvBYImkv3XEmtKr0TXtchw=">AAACH3icbZDJSgNBEIZ73I1b1KOXxiAohDAjbhfB5aDHCEaFzBB6OjVJk+6ZobtGCEPexIuv4sWDIuLNt7GzHDTxh4aPv6qorj9MpTDout/O1PTM7Nz8wmJhaXllda24vnFnkkxzqPFEJvohZAakiKGGAiU8pBqYCiXch53Lfv3+EbQRSXyL3RQCxVqxiARnaK1G8ci/Ykqx+nn5vHEY0FOa+1pRmbR6fnnITcCeLyHCXZ+3ha9Fq417jWLJrbgD0UnwRlAiI1UbxS+/mfBMQYxcMmPqnptikDONgkvoFfzMQMp4h7WgbjFmCkyQD+7r0R3rNGmUaPtipAP390TOlDFdFdpOxbBtxmt9879aPcPoJMhFnGYIMR8uijJJMaH9sGhTaOAouxYY18L+lfI204yjjbRgQ/DGT56Eu/2K51a8m4PS2cUojgWyRbbJLvHIMTkj16RKaoSTJ/JC3si78+y8Oh/O57B1yhnNbJI/cr5/AOFcoaw=</latexit><latexit sha1_base64="8W42ntvBYImkv3XEmtKr0TXtchw=">AAACH3icbZDJSgNBEIZ73I1b1KOXxiAohDAjbhfB5aDHCEaFzBB6OjVJk+6ZobtGCEPexIuv4sWDIuLNt7GzHDTxh4aPv6qorj9MpTDout/O1PTM7Nz8wmJhaXllda24vnFnkkxzqPFEJvohZAakiKGGAiU8pBqYCiXch53Lfv3+EbQRSXyL3RQCxVqxiARnaK1G8ci/Ykqx+nn5vHEY0FOa+1pRmbR6fnnITcCeLyHCXZ+3ha9Fq417jWLJrbgD0UnwRlAiI1UbxS+/mfBMQYxcMmPqnptikDONgkvoFfzMQMp4h7WgbjFmCkyQD+7r0R3rNGmUaPtipAP390TOlDFdFdpOxbBtxmt9879aPcPoJMhFnGYIMR8uijJJMaH9sGhTaOAouxYY18L+lfI204yjjbRgQ/DGT56Eu/2K51a8m4PS2cUojgWyRbbJLvHIMTkj16RKaoSTJ/JC3si78+y8Oh/O57B1yhnNbJI/cr5/AOFcoaw=</latexit>

� = i/@ + /A+ �5 /A5<latexit sha1_base64="Y878R7LLd36+c7vgYxWO37Bjekk=">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</latexit><latexit sha1_base64="Y878R7LLd36+c7vgYxWO37Bjekk=">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</latexit><latexit 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�[A,A5] = �R + i�I<latexit sha1_base64="UJLP+m0MQiozAzKzu6B9D9d5AjY=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6UgKCURRTdCqwt1V8U+IA1hMp20Q2eSMDMRSugXuPFX3LhQxK1rd/6N0zaCth4YOPece7lzjx8zKpVlfRm5ufmFxaX8cmFldW19w9zcasgoEZjUccQi0fKRJIyGpK6oYqQVC4K4z0jT71+M/OY9EZJG4Z0axMTlqBvSgGKktOSZpfYl4hw51YOqd+zCMzipvVu4D+lPce2ZRatsjQFniZ2RIshQ88zPdifCCSehwgxJ6dhWrNwUCUUxI8NCO5EkRriPusTRNEScSDcdnzOEJa10YBAJ/UIFx+rviRRxKQfc150cqZ6c9kbif56TqODUTWkYJ4qEeLIoSBhUERxlAztUEKzYQBOEBdV/hbiHBMJKJ1jQIdjTJ8+SxmHZtsr2zVGxcp7FkQc7YBfsARucgAq4AjVQBxg8gCfwAl6NR+PZeDPeJ605I5vZBn9gfHwDH2aZng==</latexit><latexit sha1_base64="UJLP+m0MQiozAzKzu6B9D9d5AjY=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6UgKCURRTdCqwt1V8U+IA1hMp20Q2eSMDMRSugXuPFX3LhQxK1rd/6N0zaCth4YOPece7lzjx8zKpVlfRm5ufmFxaX8cmFldW19w9zcasgoEZjUccQi0fKRJIyGpK6oYqQVC4K4z0jT71+M/OY9EZJG4Z0axMTlqBvSgGKktOSZpfYl4hw51YOqd+zCMzipvVu4D+lPce2ZRatsjQFniZ2RIshQ88zPdifCCSehwgxJ6dhWrNwUCUUxI8NCO5EkRriPusTRNEScSDcdnzOEJa10YBAJ/UIFx+rviRRxKQfc150cqZ6c9kbif56TqODUTWkYJ4qEeLIoSBhUERxlAztUEKzYQBOEBdV/hbiHBMJKJ1jQIdjTJ8+SxmHZtsr2zVGxcp7FkQc7YBfsARucgAq4AjVQBxg8gCfwAl6NR+PZeDPeJ605I5vZBn9gfHwDH2aZng==</latexit><latexit sha1_base64="UJLP+m0MQiozAzKzu6B9D9d5AjY=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6UgKCURRTdCqwt1V8U+IA1hMp20Q2eSMDMRSugXuPFX3LhQxK1rd/6N0zaCth4YOPece7lzjx8zKpVlfRm5ufmFxaX8cmFldW19w9zcasgoEZjUccQi0fKRJIyGpK6oYqQVC4K4z0jT71+M/OY9EZJG4Z0axMTlqBvSgGKktOSZpfYl4hw51YOqd+zCMzipvVu4D+lPce2ZRatsjQFniZ2RIshQ88zPdifCCSehwgxJ6dhWrNwUCUUxI8NCO5EkRriPusTRNEScSDcdnzOEJa10YBAJ/UIFx+rviRRxKQfc150cqZ6c9kbif56TqODUTWkYJ4qEeLIoSBhUERxlAztUEKzYQBOEBdV/hbiHBMJKJ1jQIdjTJ8+SxmHZtsr2zVGxcp7FkQc7YBfsARucgAq4AjVQBxg8gCfwAl6NR+PZeDPeJ605I5vZBn9gfHwDH2aZng==</latexit><latexit sha1_base64="UJLP+m0MQiozAzKzu6B9D9d5AjY=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6UgKCURRTdCqwt1V8U+IA1hMp20Q2eSMDMRSugXuPFX3LhQxK1rd/6N0zaCth4YOPece7lzjx8zKpVlfRm5ufmFxaX8cmFldW19w9zcasgoEZjUccQi0fKRJIyGpK6oYqQVC4K4z0jT71+M/OY9EZJG4Z0axMTlqBvSgGKktOSZpfYl4hw51YOqd+zCMzipvVu4D+lPce2ZRatsjQFniZ2RIshQ88zPdifCCSehwgxJ6dhWrNwUCUUxI8NCO5EkRriPusTRNEScSDcdnzOEJa10YBAJ/UIFx+rviRRxKQfc150cqZ6c9kbif56TqODUTWkYJ4qEeLIoSBhUERxlAztUEKzYQBOEBdV/hbiHBMJKJ1jQIdjTJ8+SxmHZtsr2zVGxcp7FkQc7YBfsARucgAq4AjVQBxg8gCfwAl6NR+PZeDPeJ605I5vZBn9gfHwDH2aZng==</latexit>

Phase of the determinant..An elegant way to represent the chiral anomaly which Fundamentally arises from non-invariance of pathintegral measure under a chiral rotation

Remarkably, this phase can be re-expressed in a form that is nearly identical to ΓR !

Useful mnemonic: Odd powers of γ5 contribute to ΓI and Even powers to ΓR

D’Hoker, Gagne, hep-ph/9512080See also Mondragon, Nellen, Schmidt, Schubert,hep-th/9502125

The triangle anomaly in the worldline formalism

Axial vector couplingsSpin precessionWilson line

Famous infrared pole of anomaly

Point particle Bose and Grassmann path integralsTarasov, RV

Polarized DIS in the worldline formalism

Polarization tensor from world-line Hamiltonian

Gauge (gluon) background

The box diagram for polarized DIS (g1(x,Q2))

Polarization tensor(antisymmetric piece)

Box diagram

We can compute the box explicitly in both the Bjorken limit of QCD (Q$ → ∞, s → ∞, x = �ixed)and the Regge limit (x → 0, s → ∞, Q2 = �ixed). The latter result is new

DIS with worldlines,Tarasov, RV (2019)

Finding triangles in boxes in Bjorken and Regge asymptotics

Bjorken Regge

Remarkably, g1(xB,Q2) has the same structure in both limits, dominated by the triangle anomaly !

+ non-pole ,#$%&-#

+ non-pole $!

"

%" << 1

Hence g1 is topological in both asymptotic limits of QCD…its relation to Δg x, Q2 is unclear

The r.h.s in the previous expressions is infrared divergent – how is this cured to give a finite result?

The role of pseudoscalar fields in resolving the UA(1) problem

We **did not** previously write down the most general form of the **imaginary part** of the worldline effective action :

with

+i " µ 5Dµ⇧+ ↵ " 5 6[⇧,�]<latexit sha1_base64="aWA4V4rOzo7P5H93h/lCn6dRSd8=">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</latexit><latexit sha1_base64="aWA4V4rOzo7P5H93h/lCn6dRSd8=">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</latexit><latexit sha1_base64="aWA4V4rOzo7P5H93h/lCn6dRSd8=">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</latexit><latexit sha1_base64="aWA4V4rOzo7P5H93h/lCn6dRSd8=">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</latexit>

and

D’Hoker, Gagne,hep-th/9508131

J (0) / 5 6{⇧,�}<latexit sha1_base64="1GfooPKBH6dzQucz/3Chjd8Wo9k=">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</latexit><latexit sha1_base64="1GfooPKBH6dzQucz/3Chjd8Wo9k=">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</latexit><latexit sha1_base64="1GfooPKBH6dzQucz/3Chjd8Wo9k=">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</latexit><latexit sha1_base64="1GfooPKBH6dzQucz/3Chjd8Wo9k=">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</latexit>

where 𝛷 𝑖𝑠 𝑡ℎ𝑒 𝑐ℎ𝑖𝑟𝑎𝑙 𝑐𝑜𝑛𝑑𝑒𝑛𝑠𝑎𝑡𝑒

Expanding out the worldline Lagrangian, the first nontrivial contribution to WI is the Wess-Zumino-Witten term !

The role of pseudoscalar fields in resolving the UA(1) problem

The WZW term has a contribution ∝ 𝜂) 𝐹 �𝐹

The zero mode part of it exactly cancels the +"

+#“perturbative” contribution to the anomaly

(Leutwyler (1996); Herrara-Sikody et al (1997); Leutwyler-Kaiser (2000) )

→ 𝑃)|1

𝑖𝑙 ∗ 𝑆⍵ |𝑃 where the topological charge density𝑔1 𝑥, 𝑄2

Tarasov, RV, in preparation

!(X) =↵s

8⇡Tr

⇣Fµ⌫ F̃

µ⌫⌘

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Computing the worldline matrix element

Berezin,Marinov (1976)Tarasov,RV

Wilson line Spin precession

Shockwave limit

Spin precession and smearing of width of shock wave become important to obtain helicity flip contributions

But the key effect is 𝛾* which induces a zero mode in WI that couples to the topological charge

See alsoBakker,Leader,Trueman,Hep0ph/0406139

Worldline quark trajectory: polarized case

Only this term survives in the density matrix – giving F �𝐹

𝜓* changes anti-periodic b.c.’s to periodic b.c’s for Majorana modesAnomaly term arises from the surviving zero mode

Rq ⌘ ig2Z

d4x ✓(x)TrcFµ⌫ F̃µ⌫

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+ig2Z

d4x�5(x)TrcFµ⌫ F̃µ⌫

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sha1_base64="SRy0A92SyWyMe/8MpM/RTz8u7Cc=">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</latexit>

from the WZW term in the worldline effective action…

Emergent ”axion” effective action for polarized DIS at small xTarasov, RV: in preparation

Conjecture: “Axion-like” effective action for polarized DIS

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Narison, Shore and Veneziano (see hep-ph/0701171) argue that the result is dominated by the derivative of the topological susceptibility (𝜒′), which they compute (using QCD sum rules) to be in agreement with the HERMES and COMPASS data

≈ 0 �0(p2 = 0)<latexit sha1_base64="SzyUX3dTlhckCtQLX5HojnrJhcc=">AAAB/HicdVDLSgMxFM3UV62vapdugkWomyEzDm1dCEU3LivYVuiLTJq2oZnMkGSEMtRfceNCEbd+iDv/xkxbQUUPXDiccy/33uNHnCmN0IeVWVldW9/Ibua2tnd29/L7B00VxpLQBgl5KG99rChngjY005zeRpLiwOe05U8uU791R6ViobjR04h2AzwSbMgI1kbq5wsdMma9TiRZQGEp6rnn6KSfLyL7rFp2vTJENkIVx3VS4la8Uw86RklRBEvU+/n3ziAkcUCFJhwr1XZQpLsJlpoRTme5TqxohMkEj2jbUIEDqrrJ/PgZPDbKAA5DaUpoOFe/TyQ4UGoa+KYzwHqsfnup+JfXjvWw2k2YiGJNBVksGsYc6hCmScABk5RoPjUEE8nMrZCMscREm7xyJoSvT+H/pOnaDrKda69Yu1jGkQWH4AiUgAMqoAauQB00AAFT8ACewLN1bz1aL9brojVjLWcK4Aest09ioZPw</latexit><latexit sha1_base64="SzyUX3dTlhckCtQLX5HojnrJhcc=">AAAB/HicdVDLSgMxFM3UV62vapdugkWomyEzDm1dCEU3LivYVuiLTJq2oZnMkGSEMtRfceNCEbd+iDv/xkxbQUUPXDiccy/33uNHnCmN0IeVWVldW9/Ibua2tnd29/L7B00VxpLQBgl5KG99rChngjY005zeRpLiwOe05U8uU791R6ViobjR04h2AzwSbMgI1kbq5wsdMma9TiRZQGEp6rnn6KSfLyL7rFp2vTJENkIVx3VS4la8Uw86RklRBEvU+/n3ziAkcUCFJhwr1XZQpLsJlpoRTme5TqxohMkEj2jbUIEDqrrJ/PgZPDbKAA5DaUpoOFe/TyQ4UGoa+KYzwHqsfnup+JfXjvWw2k2YiGJNBVksGsYc6hCmScABk5RoPjUEE8nMrZCMscREm7xyJoSvT+H/pOnaDrKda69Yu1jGkQWH4AiUgAMqoAauQB00AAFT8ACewLN1bz1aL9brojVjLWcK4Aest09ioZPw</latexit><latexit sha1_base64="SzyUX3dTlhckCtQLX5HojnrJhcc=">AAAB/HicdVDLSgMxFM3UV62vapdugkWomyEzDm1dCEU3LivYVuiLTJq2oZnMkGSEMtRfceNCEbd+iDv/xkxbQUUPXDiccy/33uNHnCmN0IeVWVldW9/Ibua2tnd29/L7B00VxpLQBgl5KG99rChngjY005zeRpLiwOe05U8uU791R6ViobjR04h2AzwSbMgI1kbq5wsdMma9TiRZQGEp6rnn6KSfLyL7rFp2vTJENkIVx3VS4la8Uw86RklRBEvU+/n3ziAkcUCFJhwr1XZQpLsJlpoRTme5TqxohMkEj2jbUIEDqrrJ/PgZPDbKAA5DaUpoOFe/TyQ4UGoa+KYzwHqsfnup+JfXjvWw2k2YiGJNBVksGsYc6hCmScABk5RoPjUEE8nMrZCMscREm7xyJoSvT+H/pOnaDrKda69Yu1jGkQWH4AiUgAMqoAauQB00AAFT8ACewLN1bz1aL9brojVjLWcK4Aest09ioZPw</latexit><latexit sha1_base64="SzyUX3dTlhckCtQLX5HojnrJhcc=">AAAB/HicdVDLSgMxFM3UV62vapdugkWomyEzDm1dCEU3LivYVuiLTJq2oZnMkGSEMtRfceNCEbd+iDv/xkxbQUUPXDiccy/33uNHnCmN0IeVWVldW9/Ibua2tnd29/L7B00VxpLQBgl5KG99rChngjY005zeRpLiwOe05U8uU791R6ViobjR04h2AzwSbMgI1kbq5wsdMma9TiRZQGEp6rnn6KSfLyL7rFp2vTJENkIVx3VS4la8Uw86RklRBEvU+/n3ziAkcUCFJhwr1XZQpLsJlpoRTme5TqxohMkEj2jbUIEDqrrJ/PgZPDbKAA5DaUpoOFe/TyQ4UGoa+KYzwHqsfnup+JfXjvWw2k2YiGJNBVksGsYc6hCmScABk5RoPjUEE8nMrZCMscREm7xyJoSvT+H/pOnaDrKda69Yu1jGkQWH4AiUgAMqoAauQB00AAFT8ACewLN1bz1aL9brojVjLWcK4Aest09ioZPw</latexit>

Conjecture: “Axion-like” effective action for Regge limit

If we now assume (at small x), the background field couples to a large # of quark and gluon world-line trajectories, one can construct the effective action for an ensemble of spinning, colored partons at a given x<< 1:

⇥ exp

Zd4X

� !2

2�YM�r

Nf

2! +

1

2F 2 @2

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�YM =

Zd4Xh!(X)!(0)iA

<latexit sha1_base64="wwvnmeCZfuteWw7dgnhFss5O9z4=">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</latexit><latexit sha1_base64="wwvnmeCZfuteWw7dgnhFss5O9z4=">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</latexit><latexit sha1_base64="wwvnmeCZfuteWw7dgnhFss5O9z4=">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</latexit><latexit sha1_base64="wwvnmeCZfuteWw7dgnhFss5O9z4=">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</latexit>

Topological susceptibility:

𝜔 𝑋 𝑖𝑠 𝑡ℎ𝑒 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙 𝑐ℎ𝑎𝑟𝑔𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦, 𝐹 ψ 𝑖𝑠 𝑡ℎ𝑒 𝜂) field and F is the 𝜂) decay constant

g1(x,Q2) /

Z[D⇢]WP

Y [⇢]

Z[D ]W̃P,S

P [ ]<latexit sha1_base64="4DHoNjHtjWCKcgi+QedGfB/J/kA=">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</latexit><latexit sha1_base64="4DHoNjHtjWCKcgi+QedGfB/J/kA=">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</latexit><latexit sha1_base64="4DHoNjHtjWCKcgi+QedGfB/J/kA=">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</latexit><latexit sha1_base64="4DHoNjHtjWCKcgi+QedGfB/J/kA=">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</latexit>

Veneziano, Mod. Phys. Lett. (1989)Hatsuda, PLB (1990)

Conjecture: “Axion-like” effective action for Regge limit

⇥ exp

Zd4X

� !2

2�YM�r

Nf

2! +

1

2F 2 @2

!!

<latexit sha1_base64="JK08Ha86Um0Hlmp9KiKG8JvNLSU=">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</latexit><latexit sha1_base64="JK08Ha86Um0Hlmp9KiKG8JvNLSU=">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</latexit><latexit sha1_base64="JK08Ha86Um0Hlmp9KiKG8JvNLSU=">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</latexit><latexit sha1_base64="JK08Ha86Um0Hlmp9KiKG8JvNLSU=">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</latexit>

The term

can be rewritten as the “Veneziano effective action” of a glueball and 𝜂) field

exp

✓�Z

d4X

✓� G2

2�YM+

1

2⌘0�@2 +m2

⌘0�⌘0◆◆

<latexit sha1_base64="C0krsE7mDhOp4J/W6Hnaupw8GRA=">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</latexit><latexit sha1_base64="C0krsE7mDhOp4J/W6Hnaupw8GRA=">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</latexit><latexit sha1_base64="C0krsE7mDhOp4J/W6Hnaupw8GRA=">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</latexit><latexit sha1_base64="C0krsE7mDhOp4J/W6Hnaupw8GRA=">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</latexit>

with 𝑚.' given by the Witten-Veneziano formula m2⌘0 =

2Nf �YM

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or equivalently as exp

✓Zd4X

�F 2 @2 + ( + ✓)! + �YM ( + ✓)2

�◆

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ChPT formulation for the nonet,Herrera-Siklody et al, hep-ph/9610549

Spin diffusion to small x

A lot of impressive work on perturbative computations of g1 at small x…

Kirschner Lipatov (1980’s); Bartels, Ermolaev, Ryskin (1990’s); Kovchegov, Pitonyak, Sievert (2015-),Kovchegov,Cougoulic (2019); Boussarie, Hatta, Yuan (2019)

Can we reconcile this work with the dynamics of the chiral anomaly at small x ?

QCD evolution to small x

After smearing of the target in 𝑥/, the effective action reproduces itself (in a double log approximation)

– renormalizing the “axion effective action” at small x, one should recover the triangle structure in the “box diagram”

Fascinating interplay between gluon saturation (controlled by the saturation scale QS) and spin diffusion (controlled by the topological charge density 𝜔)

Spin diffusion via sphaleron transitionsIn a hot plasma (McLerran,Mottola,Shaposhnikov, 1990), diffusion of topological charge

Out of equilibrium, for overoccupied glue, the scale is set by the spatial string tension and gluon saturation scale QS Mace, Schlichting, RV: 1601.07342

The analogy of time here is the rapidity Y = Ln(1/x)

So the diffusion to small x is controlled by the rate of sphaleron transitions

Spin diffusion via sphaleron transitions in topologically disordered media

Helicity �lip for massless quarks given by nL − nR = 2 N0 νwhere Γ123456789: ∝ ν2

Spin diffusion via sphaleron transitions in topologically disordered media

Helicity �lip for massless quarks given by nL − nR = 2 N0 νwhere Γ123456789

: ∝ ν2

Outlook: these ideas can be tested at the EIC !Conformal Anomaly

Chiral Anomaly

Color Glass Condensate

Confined many-body parton dynamics

Precision probes of the strong interplay between perturbative many-body parton dynamics and non-perturbative structure (“the ether”) of the QCD vacuum

Outlook

Semi-classical approximations

Worldline QCD+QED Hamiltonian:

Powerful semi-classical intuition from corresponding Hamilton equations:

Ø Bargmann-Michel-Telegdi equations for spinning particles in gauge backgrounds,Ø Wong equations for colored particles precessing in gauge field backgrounds Ø Papapetrou-Dixon equations for spinning massive particles in gravity

Useful in semi-classical (“Wigner function”) approaches in theories with internal symmetries

Mueller, Venugopalan, arXiv:1901.10492, and references thereinApplications to chiral kinetic theory:Mueller, Venugopalan, arXiv:1701.03331, arXiv: 1702.01233

Also, “world-line instanton” techniques can be used to compute pair-production (Schwinger mechanism)

Dunne,Schubert, hep-th/0507174

Worldline inspired ”axion” effective action for polarized DISTarasov, RV: in preparation

Spin-dependent contributions at high energies are ”sub-eikonal”- in addition to shockwave Coulomb/Glauber field A+ , there is a Δ = 1/P+ → 0 suppressed contribution Ai

World-line formalism: preliminariesReview: Corradini, Schubert, arXiv:1512.08694 Also, Strassler, NPB385 (1992) 145Based on Schwinger’s proper time trick:

One loop effective action of massless scalar field coupled to background Abelian field

=

Z 1

0

dT

TTr exp(�TD2)

with N =

ZDp exp(�1

2

Z T

0d⌧✏ p2)

ε is the Einbein: square root of 1D metric

Computing structure functions: world-line formalism“Schwinger-Keldysh” approach to computing

real-time correlations of quantum fields

Break up problem into two parts:

Initial conditions for gluon fields

Yang-Mills action

withInitial conditions for “valence” quarks

Dirac action

Zf [A] can be rexpressed as the effective action corresponding to a quantum mechanical “world-line” Hamiltonian describing spinning and colored Grassmann point particles (qubits) in gluon fields

Documents

Separating fact from fantasy*: The chiral anomaly and the ...shovkovy.faculty.asu.edu/colloquium/slides/Colloquium_Slides... · Europe and Asia. In support of this new direction: