Sphaleron transitions and the Chiral Magnetic...

Sphaleron transitions and the Chiral Magnetic Effect

Mark Mace University of Jyväskylä

Finnish Particle Physics Day November 23, 2018

Based on: MM, Schlichting, Venugopalan PRD 93 2016, MM, Mueller, Schlichting, Sharma PRD 95 2017

The Chiral Magnetic EffectTopological processes in QCD can produce imbalances of quark chirality through chiral anomaly — but how to see this? Possible in heavy-ion collisions?

Axial charge imbalance in presence of external magnetic field produces an electric (vector) current: the Chiral Magnetic Effect:Kharzeev, McLerran, Warringa NPA 803 2008; Fukushima, Kharzeev, Warringa PRD 78 2008

Red=momentum Blue=spin

2

Son & Zhitnitsky, PRD 70, 074018; Metlitski & Zhitnitsky, PRD 72, 045011Chiral Separation Effect:

Chiral Magnetic Wave: collective gapless excitation from the coupling between density waves of electric and chiral charges

jV ∼ n5B

j5 ∼ nVB

However, life-time of magnetic field in vacuum is short < 1 fm/c

B

STAR PRC 81 (2010)

Magnetic field in HICSpectators in off-central collisions create a strong magnetic field eB ~ mπ2 ~1014 T

Need detailed understanding of pre-equilibrium dynamics to make quantitive statements about CME

3

Li, Sheng, Wang PRC 94 (2016)

jV ∼ n5B

Axial charge imbalanceQCD chiral anomaly generates chiral (axial) charge

Axial current ; receives contributions from sphaleron

transitions,…

Quark mass

Klinkhamer, Manton PRD 30 1984

Sphaleron transition (real time topological transitions) leads to unit change of Chern-Simons number and induce axial charge imbalance

Pote

ntia

l Bar

rier

Vacuum

NCS

Transition overthe sphaleron

Instanton NCSPote

ntial

Bar

rier Sphaleron Transition

10 32

⇠ Eai B

ai (x, t)

V

A V

4

jV ∼ n5B

For HIC, focus on mf ≈ 0Axial charge Change in

winding #

@µjµ5 = 2mh ¯ (x)i�5 (x)i�

g2

8⇡2trFµ⌫(x)F

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�N05 = 2mf

Z

xh ¯ (x)i�5 (x)i�2�NCS

<latexit sha1_base64="voI4v/Rxq9n6hsBRdn2EMi3KHiI=">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</latexit><latexit sha1_base64="voI4v/Rxq9n6hsBRdn2EMi3KHiI=">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</latexit><latexit sha1_base64="voI4v/Rxq9n6hsBRdn2EMi3KHiI=">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</latexit><latexit sha1_base64="voI4v/Rxq9n6hsBRdn2EMi3KHiI=">AAACfnicbVFdaxQxFM2MX3Wtuir64ktwUepDtzOLxb4Ixfrgk1R028JmHe5k7+yGJpkhyUiXEPBv+gf0b5idrtS2XggczsnNuTm3bKSwLst+JumNm7du39m427u3ef/Bw/6jx0e2bg3HMa9lbU5KsCiFxrETTuJJYxBUKfG4PD1Y6cff0VhR669u2eBUwVyLSnBwkSr6PzzrHvEGZ4GyDygd0E/F7rcsvBupoqJMaFecUSZBzyVSVoLxbAHOs8aKELbOXgvK5qAUFLv0Qog8M13LX4NSthi26ejCxB98CaHoD7Jh1hW9DvI1GJB1HRb932xW81ahdlyCtZM8a9zUg3GCSww91lpsgJ/CHCcRalBop76bIdCXkZnRqjbxaEc79t8OD8rapSrjTQVuYa9qK/J/2qR11d7UC920DjU/N6paSV1NV7HTmTDInVxGANyIOCvlCzDAXVzOJRcbP7WIy+jFZPKrOVwHR6Nhng3zz28G++/XGW2Q5+QF2SI5eUv2yUdySMaEk1/JZvI0eZaS9FW6ne6cX02Tdc8TcqnSvT8/YcHH</latexit>

Theorists Overview of CMEGoal: Want to quantitatively study anomalous transport in HIC

(Son & Surowka 2009; Hirono, Hirano, Kharzeev 2014, Jiang et al 2016,…)

5

For CME, earliest times likely most important due to B

Recent progress has lead to a relatively complete qualitative picture of weak coupling thermalization

Berges, Boguslavski, Schlichting Venugopalan Phys.Rev. D89 (2014) no.11, 114007, Epelbaum, Gelis PRL 111 (2013) 232301, Kurkela, Zhu PRL 115 (2015) 18, 182301,…

Simulating early times after heavy-ion collision

Earliest time dynamics described by classical Yang-Mills

Emergence of a characteristic scale, , such that

Non-perturbatively large gluon phase space density

Highly amenable to real-time lattice simulations

Track right hand side of anomaly by measuring Chern-Simons number

Use cooling to remove short range fluctuation and isolate topological transitions

Neglect longitudinal expansion and use SU(2) for simplicity

McLerran, Venugopalan PRD49 2233 (1994)

6

Ambjorn, Krasnitz NPB 506 (1997), Moore PRD 59 (1998)

Venugopalan, Kransitz NPA 237 (1999), Berges, Scheffler, Sexty, PRD 77 034504 (2008)

Qs αS(Qs) ≪ 1

f(p ∼ Qs) ∼ 1αS

Sphalerons in the glasmaDetect integer changes in Chern-Simons number for single configuration

Chern-Simons diffusion of many configurations shows transitions between different topological sectors occurs

-8-6-4-2 0 2 4 6 8

10 12 14

10 15 20 25 30 35 40 45 50 55Time: Qst

Gradient Flow NCSCooled NCS (Qs

2τc=1/4) Cooled NCS (Qs

2τc=250)

MM, Schlichting, Venugopalan PRD 93 2016

Can isolate transitions from background Significant number can occur at early times

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Qst=10Qsδt=10

Pro

ba

bil

ity

: P(Δ

NC

S)

ΔNCS

Qs2τc=250

Qs2τc=108

Qs2τc=36

Qs2τc=12

7

ΔNCS

Non-equilibrium sphaleron transition rate

Strongly time dependent, largest are early times — dynamics controlled by magnetic screening

Contributions from field strength fluctuations

Non-markovian, not random walk like thermal equilibrium

10-8

10-7

10-6

10-5

10 100

(Qst) -4/3

Γsp

hne

q / Q

s4

Time: Qs t

Qs2τc=1

Qs2τc=12

Qs2τc=36

Qs2τc=108

Qs2τc=216

Qs2τc=324

Increased cooling

MM, Schlichting, Venugopalan PRD 93 2016

8

Time

Qs4

Qst=1 QstQuant=αS-7/4

αS Qs

7/3 4

αS T45

Γ sph

Non-equilibrium,classical regime

Equilibration,Quantum

regimeThermal

equilibrium

Dominant amount of axial charge (should be)

generated at pre-equilibrium stage

Arnold, Son, Yaffee; Moore et al.

Axial Charge from Sphaleron Transition

�NCS =g2

8⇡

Zd4xEa ·Ba

�J05 = �2�NCS + 2mf

Zd4xh i�5 i

eB = 0, m ⌧ r�1sph

-0.1 0

0.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4t/tsph

0

0.5

1

1.5

2

Ja0+2ΔNCS

Ja0

-2 ΔNCSJa

0: Wilson NLO Ja

0: Overlap M=1.5

MM, Mueller, Schlichting, Sharma PRD 95 2017

Good agreement found between discretization

Consider isolated sphaleron of size rsph, timescale tsph

Unit change in Chern-Simons number induces axial charge imbalance

9

To study full CME dynamics, need to extend non-equilibrium gauge field studies to include fermions

Cannot use classical approximation for fermions

Solve operator Dirac equation: (i /D �m) = j<latexit sha1_base64="Zk9Qseawpse9PpB4adGCGiye6hE=">AAACIHicbZDLSsNAFIYn9VbrLerSTWgR6sKSiKAboagLlxXsBZpQJpNJO3ZmEmYmQgnZ+xw+gFt9BHfiUl/A13CaZmGrBwY+/v+cOTO/H1MilW1/GqWl5ZXVtfJ6ZWNza3vH3N3ryCgRCLdRRCPR86HElHDcVkRR3IsFhsynuOuPr6Z+9wELSSJ+pyYx9hgcchISBJWWBma1TlxJoRzhIL3OjtmRO4IqdWNJsosc77OBWbMbdl7WX3AKqIGiWgPz2w0ilDDMFdJXy75jx8pLoVAEUZxV3ETiGKIxHOK+Rg4Zll6a/yWzDrUSWGEk9OHKytXfEylkUk6YrzsZVCO56E3F/7x+osJzLyU8ThTmaLYoTKilImsajBUQgZGiEw0QCaLfaqERFBApHd/cliKvrKKTcRZz+Audk4ZjN5zb01rzssioDA5AFdSBA85AE9yAFmgDBB7BM3gBr8aT8Wa8Gx+z1pJRzOyDuTK+fgA/06QZ</latexit><latexit sha1_base64="Zk9Qseawpse9PpB4adGCGiye6hE=">AAACIHicbZDLSsNAFIYn9VbrLerSTWgR6sKSiKAboagLlxXsBZpQJpNJO3ZmEmYmQgnZ+xw+gFt9BHfiUl/A13CaZmGrBwY+/v+cOTO/H1MilW1/GqWl5ZXVtfJ6ZWNza3vH3N3ryCgRCLdRRCPR86HElHDcVkRR3IsFhsynuOuPr6Z+9wELSSJ+pyYx9hgcchISBJWWBma1TlxJoRzhIL3OjtmRO4IqdWNJsosc77OBWbMbdl7WX3AKqIGiWgPz2w0ilDDMFdJXy75jx8pLoVAEUZxV3ETiGKIxHOK+Rg4Zll6a/yWzDrUSWGEk9OHKytXfEylkUk6YrzsZVCO56E3F/7x+osJzLyU8ThTmaLYoTKilImsajBUQgZGiEw0QCaLfaqERFBApHd/cliKvrKKTcRZz+Audk4ZjN5zb01rzssioDA5AFdSBA85AE9yAFmgDBB7BM3gBr8aT8Wa8Gx+z1pJRzOyDuTK+fgA/06QZ</latexit><latexit sha1_base64="Zk9Qseawpse9PpB4adGCGiye6hE=">AAACIHicbZDLSsNAFIYn9VbrLerSTWgR6sKSiKAboagLlxXsBZpQJpNJO3ZmEmYmQgnZ+xw+gFt9BHfiUl/A13CaZmGrBwY+/v+cOTO/H1MilW1/GqWl5ZXVtfJ6ZWNza3vH3N3ryCgRCLdRRCPR86HElHDcVkRR3IsFhsynuOuPr6Z+9wELSSJ+pyYx9hgcchISBJWWBma1TlxJoRzhIL3OjtmRO4IqdWNJsosc77OBWbMbdl7WX3AKqIGiWgPz2w0ilDDMFdJXy75jx8pLoVAEUZxV3ETiGKIxHOK+Rg4Zll6a/yWzDrUSWGEk9OHKytXfEylkUk6YrzsZVCO56E3F/7x+osJzLyU8ThTmaLYoTKilImsajBUQgZGiEw0QCaLfaqERFBApHd/cliKvrKKTcRZz+Audk4ZjN5zb01rzssioDA5AFdSBA85AE9yAFmgDBB7BM3gBr8aT8Wa8Gx+z1pJRzOyDuTK+fgA/06QZ</latexit><latexit sha1_base64="Zk9Qseawpse9PpB4adGCGiye6hE=">AAACIHicbZDLSsNAFIYn9VbrLerSTWgR6sKSiKAboagLlxXsBZpQJpNJO3ZmEmYmQgnZ+xw+gFt9BHfiUl/A13CaZmGrBwY+/v+cOTO/H1MilW1/GqWl5ZXVtfJ6ZWNza3vH3N3ryCgRCLdRRCPR86HElHDcVkRR3IsFhsynuOuPr6Z+9wELSSJ+pyYx9hgcchISBJWWBma1TlxJoRzhIL3OjtmRO4IqdWNJsosc77OBWbMbdl7WX3AKqIGiWgPz2w0ilDDMFdJXy75jx8pLoVAEUZxV3ETiGKIxHOK+Rg4Zll6a/yWzDrUSWGEk9OHKytXfEylkUk6YrzsZVCO56E3F/7x+osJzLyU8ThTmaLYoTKilImsajBUQgZGiEw0QCaLfaqERFBApHd/cliKvrKKTcRZz+Audk4ZjN5zb01rzssioDA5AFdSBA85AE9yAFmgDBB7BM3gBr8aT8Wa8Gx+z1pJRzOyDuTK+fgA/06QZ</latexit>

J05

Anomalous transport in real time

B

Sphaleron transition in background B field

10

OutlookSphaleron transition rate enhanced during the early stages of HIC

Development of real-time lattice techniques to study pre-equilibrium dynamics of fermions

Successful microscopic description of anomalous transport phenomena in simplistic setup — framework set up for full back-reaction QED/QCD

Ongoing and planned studies of quark production and axial charge production for variety of gauge field configurations

Calculation of spectral functions, transport coefficients, etc.

First calculations of axial charge production based on weakly coupled pre-equilibrium dynamics

11

Fermion production in HIC — needed for full description of HIC thermalization

Non-perturbative study of QED Schwinger pair production — variety of space- and time-dependent electric fields

Study chirality transfer between gauge fields and fermions — chiral plasma instability MM, Mueller, Schlichting, Sharma, in prep

MM, Mueller, Schlichting, Sharma, in prep

Thanks!

12

BACKUP

Early stages of HICs

J+ J�E

⌘

B⌘

colliding nuclei Glasma flux tubes over-occupiedplasma

min-jets + soft bath equilibrium

time

classical-statistical lattice gauge theory eff. kinetic theory hydro

strong fields quasi particles 1-2 fm/c

Fig. S. Schlichting

Initial nuclei described as Color Glass

Condensates

Recent progress has lead to a relatively complete qualitative picture of weak coupling thermalization

Berges, Boguslavski, Schlichting Venugopalan Phys.Rev. D89 (2014) no.11, 114007, Epelbaum, Gelis PRL 111 (2013) 232301, Kurkela, Zhu PRL 115 (2015) 18, 182301 14

Real-time QFTs

• Longstanding goal of particle and nuclear physics is to access the real time dynamics of QCD

• Ex. Thermalization and transport properties of hot and dense QCD, parton distributions in nucleons,…

• Euclidean formulation works well for some static observables*

• For real-time (non-equilibrium) dynamics, formulate path-integral on Schwinger-Keldysh contour

• A sign problem prevents general computations, avoiding this is a grand challenge

* with well known caveats (like finite baryon chemical potential)

Schwinger, J. Math. Phys. 2, 407 (1961), Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964)

e.g. Berges, Stamatescu PRL 95 (2005) 202003, Alexandru et al, PRL117, 081602 (2016)

15

Space-time evolution of HIC

Classical-Statistical Approximation

• Typical situation: weak coupling, large coherent field or highly occupied gauge fields → correspondence principle

• Systematic expansion of the 'quantum' fields around the 'classical' fields

‘Real-time’ partition function

ZC =

Z[dA]

Z[d d ]⇢(t0)exp(iSG + iSF )

<latexit sha1_base64="2TodgZXEoObpAriX+JXR7kusw00=">AAACUXicbZBNTxsxEIYnS8tHKBDgyMVqhBSEFO0iJLggpSDRHqloEkR2tfJ6J8SK90P2bEW02h/G7+DEiSPtT+itzsehQEey/eidGY3njXIlDbnuU81Z+vBxeWV1rb7+aWNzq7G90zNZoQV2RaYyfRNxg0qm2CVJCm9yjTyJFPaj8cU03/+J2sgs/UGTHIOE36VyKAUnK4WN69vQTziNBFflRXXmy5TYIP4SsBlZ9COuSz83sornT8CYr0dZi0L3gPmE91TifV615HX49dBelwdho+m23Vmw9+AtoAmLuAobL36ciSLBlITixgw8N6eg5JqkUFjV/cJgzsWY3+HAYsoTNEE5W75i+1aJ2TDT9tgvz9R/O0qeGDNJIls5XdS8zU3F/+UGBQ1Pg1KmeUGYivmgYaEYZWzqJIulRkFqYoELLe1fmRhxzQVZv19NMXapEcZV3TrjvfXhPfSO2p7b9r4fNzvnC49WYQ8+Qws8OIEOfIMr6IKAB3iGX/C79lj744DjzEud2qJnF16Fs/4XydSzpg==</latexit><latexit sha1_base64="2TodgZXEoObpAriX+JXR7kusw00=">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</latexit><latexit sha1_base64="2TodgZXEoObpAriX+JXR7kusw00=">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</latexit><latexit sha1_base64="2TodgZXEoObpAriX+JXR7kusw00=">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</latexit>

ZclC =

Z[dA]

Zd⇧0⇢W (A0,⇧0)�[@F [A]� j]

<latexit sha1_base64="sbbV0/U0OU3EyoeeNOuZsBEt+sE=">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</latexit><latexit sha1_base64="sbbV0/U0OU3EyoeeNOuZsBEt+sE=">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</latexit><latexit sha1_base64="sbbV0/U0OU3EyoeeNOuZsBEt+sE=">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</latexit><latexit sha1_base64="sbbV0/U0OU3EyoeeNOuZsBEt+sE=">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</latexit>

A+µ,n = A+

µ,n +1

2A+

µ,n<latexit sha1_base64="g9LzmtwPEEvJt7MR4jpq1etqagI=">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</latexit><latexit sha1_base64="g9LzmtwPEEvJt7MR4jpq1etqagI=">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</latexit><latexit sha1_base64="g9LzmtwPEEvJt7MR4jpq1etqagI=">AAACOnicbZDLSsNAFIYn9VbrrerSTbAIQqUkRdCFQqsblxXsBZpaJpOTduhkEmYmQgl5GZ/DB3CrO7duRNz6AE4voG09MPDz/+dw5nxuxKhUlvVmZJaWV1bXsuu5jc2t7Z387l5DhrEgUCchC0XLxRIY5VBXVDFoRQJw4DJouoPrUd58ACFpyO/UMIJOgHuc+pRgpa1u/qJ6X+wmThCf8PTScbFIqumvU3R8gUlip0k5dRRlHszE3XzBKlnjMheFPRUFNK1aN//heCGJA+CKMCxl27Yi1UmwUJQwSHNOLCHCZIB70NaS4wBkJxlfmZpH2vFMPxT6cWWO3b8TCQ6kHAau7gyw6sv5bGT+l7Vj5Z93EsqjWAEnk0V+zEwVmiNkpkcFEMWGWmAiqP6rSfpYg1Ea7MwWqY/qg5fmNBl7nsOiaJRLtlWyb08Llaspoyw6QIfoGNnoDFXQDaqhOiLoET2jF/RqPBnvxqfxNWnNGNOZfTRTxvcPooauaQ==</latexit><latexit sha1_base64="g9LzmtwPEEvJt7MR4jpq1etqagI=">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</latexit>

A�µ,n = A�

µ,n � 1

2A�

µ,n<latexit sha1_base64="F5/briopCh/yAsjremg77ED89hE=">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</latexit><latexit sha1_base64="F5/briopCh/yAsjremg77ED89hE=">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</latexit><latexit sha1_base64="F5/briopCh/yAsjremg77ED89hE=">AAACOnicbZDLSsNAFIYn9VbrrerSTbAILmxJiqALhVY3LivYCzS1TCYn7dDJJMxMhBLyMj6HD+BWd27diLj1AZxeQNt6YODn/8/hzPnciFGpLOvNyCwtr6yuZddzG5tb2zv53b2GDGNBoE5CFoqWiyUwyqGuqGLQigTgwGXQdAfXo7z5AELSkN+pYQSdAPc49SnBSlvd/EX1vthNnCA+4eml42KRVNNfp+j4ApPETpNy6ijKPJiJu/mCVbLGZS4KeyoKaFq1bv7D8UISB8AVYVjKtm1FqpNgoShhkOacWEKEyQD3oK0lxwHITjK+MjWPtOOZfij048ocu38nEhxIOQxc3Rlg1Zfz2cj8L2vHyj/vJJRHsQJOJov8mJkqNEfITI8KIIoNtcBEUP1Xk/SxBqM02JktUh/VBy/NaTL2PIdF0SiXbKtk354WKldTRll0gA7RMbLRGaqgG1RDdUTQI3pGL+jVeDLejU/ja9KaMaYz+2imjO8fsFqucQ==</latexit><latexit sha1_base64="F5/briopCh/yAsjremg77ED89hE=">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</latexit>

}

Fermion field integration

Wigner distribution over quantum ICs Classical EoMKeep first term in A<latexit sha1_base64="Sz5HXDUTi7FiyGmaAz27KJBGlZg=">AAACBXicbVDLSsNAFJ3UV62vqks3g0VwVRIRdFl147KCbZU2lMnkph06MwkzE6GErP0At/oJ7sSt3+EX+BtO2yxs64ELh3Pu5d57goQzbVz32ymtrK6tb5Q3K1vbO7t71f2Dto5TRaFFYx6rh4Bo4ExCyzDD4SFRQETAoROMbiZ+5wmUZrG8N+MEfEEGkkWMEmOlx55hPITsKu9Xa27dnQIvE68gNVSg2a/+9MKYpgKkoZxo3fXcxPgZUYZRDnmll2pICB2RAXQtlUSA9rPpwTk+sUqIo1jZkgZP1b8TGRFaj0VgOwUxQ73oTcT/vG5qoks/YzJJDUg6WxSlHJsYT77HIVNADR9bQqhi9lZMh0QRamxGc1u0fWoIYV6xyXiLOSyT9lndc+ve3XmtcV1kVEZH6BidIg9doAa6RU3UQhQJ9IJe0Zvz7Lw7H87nrLXkFDOHaA7O1y8F6JlU</latexit><latexit sha1_base64="Sz5HXDUTi7FiyGmaAz27KJBGlZg=">AAACBXicbVDLSsNAFJ3UV62vqks3g0VwVRIRdFl147KCbZU2lMnkph06MwkzE6GErP0At/oJ7sSt3+EX+BtO2yxs64ELh3Pu5d57goQzbVz32ymtrK6tb5Q3K1vbO7t71f2Dto5TRaFFYx6rh4Bo4ExCyzDD4SFRQETAoROMbiZ+5wmUZrG8N+MEfEEGkkWMEmOlx55hPITsKu9Xa27dnQIvE68gNVSg2a/+9MKYpgKkoZxo3fXcxPgZUYZRDnmll2pICB2RAXQtlUSA9rPpwTk+sUqIo1jZkgZP1b8TGRFaj0VgOwUxQ73oTcT/vG5qoks/YzJJDUg6WxSlHJsYT77HIVNADR9bQqhi9lZMh0QRamxGc1u0fWoIYV6xyXiLOSyT9lndc+ve3XmtcV1kVEZH6BidIg9doAa6RU3UQhQJ9IJe0Zvz7Lw7H87nrLXkFDOHaA7O1y8F6JlU</latexit><latexit sha1_base64="Sz5HXDUTi7FiyGmaAz27KJBGlZg=">AAACBXicbVDLSsNAFJ3UV62vqks3g0VwVRIRdFl147KCbZU2lMnkph06MwkzE6GErP0At/oJ7sSt3+EX+BtO2yxs64ELh3Pu5d57goQzbVz32ymtrK6tb5Q3K1vbO7t71f2Dto5TRaFFYx6rh4Bo4ExCyzDD4SFRQETAoROMbiZ+5wmUZrG8N+MEfEEGkkWMEmOlx55hPITsKu9Xa27dnQIvE68gNVSg2a/+9MKYpgKkoZxo3fXcxPgZUYZRDnmll2pICB2RAXQtlUSA9rPpwTk+sUqIo1jZkgZP1b8TGRFaj0VgOwUxQ73oTcT/vG5qoks/YzJJDUg6WxSlHJsYT77HIVNADR9bQqhi9lZMh0QRamxGc1u0fWoIYV6xyXiLOSyT9lndc+ve3XmtcV1kVEZH6BidIg9doAa6RU3UQhQJ9IJe0Zvz7Lw7H87nrLXkFDOHaA7O1y8F6JlU</latexit><latexit sha1_base64="Sz5HXDUTi7FiyGmaAz27KJBGlZg=">AAACBXicbVDLSsNAFJ3UV62vqks3g0VwVRIRdFl147KCbZU2lMnkph06MwkzE6GErP0At/oJ7sSt3+EX+BtO2yxs64ELh3Pu5d57goQzbVz32ymtrK6tb5Q3K1vbO7t71f2Dto5TRaFFYx6rh4Bo4ExCyzDD4SFRQETAoROMbiZ+5wmUZrG8N+MEfEEGkkWMEmOlx55hPITsKu9Xa27dnQIvE68gNVSg2a/+9MKYpgKkoZxo3fXcxPgZUYZRDnmll2pICB2RAXQtlUSA9rPpwTk+sUqIo1jZkgZP1b8TGRFaj0VgOwUxQ73oTcT/vG5qoks/YzJJDUg6WxSlHJsYT77HIVNADR9bQqhi9lZMh0QRamxGc1u0fWoIYV6xyXiLOSyT9lndc+ve3XmtcV1kVEZH6BidIg9doAa6RU3UQhQJ9IJe0Zvz7Lw7H87nrLXkFDOHaA7O1y8F6JlU</latexit>

� h�2iQ2

⇠ 1<latexit sha1_base64="AxMG8IVUOY57P2BFPXrub/lMzr4=">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</latexit><latexit sha1_base64="AxMG8IVUOY57P2BFPXrub/lMzr4=">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</latexit><latexit sha1_base64="AxMG8IVUOY57P2BFPXrub/lMzr4=">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</latexit><latexit sha1_base64="AxMG8IVUOY57P2BFPXrub/lMzr4=">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</latexit>

Coupling strength

field2 expectation value (vacuum/

thermal equilibrium/nonequilibrium)

characteristic energy/momentum2

Classical background

Quantum fluctuation

ZC =

Z[dA]

Z⇢G(A)exp(Tr log�C [A]�1 + iSG)

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Observables:

Kasper et al PRD 90 (2014) no.2, 025016

16hO[A]i =

Z[dA]

Zd⇧0⇢W (A0,⇧0)O[A]�[@F [A]� j]

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Where can we use this?

• In general, over-occupied and highly correlated systems

• Heavy ion collisions - initial fields strength A~1/g

• Pre-heating dynamics of the early universe

• Ultra-cold atoms far from equilibrium

• Can model quantum simulator systems

e.g. Berges, Boguslavski, Chatrchyan, Jaeckel PRD 96 (2017) no.7, 076020

eg. McLerran, Venugopalan PRD 49 2233 (1994), Venugopalan, Kransitz NPA 237 (1999) ,Berges, Boguslavski, Schlichting Venugopalan Phys.Rev. D89 (2014) no.11, 114007, MM et al…

c.f. J. Berges arXiv:1503.02907

e.g. Kasper et al Phys.Lett. B760 (2016) 742-746

Many of the methods discussed in this talk can be applied to a large variety of systems

e.g. Kasper et al New J.Phys. 19 (2017) no.2, 023030

17

High energy heavy ions• Heavy-ion collision experiments currently conducted at RHIC and the LHC

• Used to study many-body QCD in terms of quarks and gluons

• Formation of Quark Gluon Plasma (QGP)

• Possible to study QCD phase diagram, chiral symmetry restoration, transport properties of QCD, nuclear EoS,…

Difficult quantum many-body problem with phase transition, impossible to model entire evolution from first principles Can study initial times using high energy EFT of QCD

18

Photo taken 4/10/18 flying to UMD

Initial conditions of HICs

Problem can then be studied using effective theory of high energy QCD - Color Glass Condensate

Emergent semi-hard scale, Qs, becomes dominant scale of the problem

The initial nuclear wave function in high energy collisions dominated by gluons

Iancu and Venugopalan, QGP 249-3363; Iancu, Jalilian-Marian, Venugopalan, Ann. Rev. Nucl. Part. Sci. 60, 463 (2010)

↵S(Q2s) ⌧ 1

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Weakly coupled, but highly correlated system

Initially gluons highly over-occupied:f(p < Qs) ⇠ 1/↵S � 1

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19

Real-time simulations• Gauge field dynamics from Kogut-Susskind Hamiltonian

• From plasma instabilities, initial flux-tubes break up, over-occupied distribution up to scale Qs after timescale 1/Qs

H =a3

2

X

j,x

Eaj (x)E

aj (x) +

2

g2a

X

⇤ReTr

h1� U⇤

i

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Berges et al NPA 931 (2014) 348-353

f(p) = ✓(Qs � |p|)<latexit sha1_base64="pDswx1l7/0rFGOOIzcwNrv9ME8c=">AAACG3icbVDLSgNBEJz1GeMr6lGExSAkB8OuCHoRgl48JmAekA1hdtKbDJl9MNMrhE1Ofocf4FU/wZt49eAX+BtOkj2YxIKGoqqb7i43ElyhZX0bK6tr6xubma3s9s7u3n7u4LCuwlgyqLFQhLLpUgWCB1BDjgKakQTquwIa7uBu4jceQSoeBg84jKDt017APc4oaqmTO/EKUfHGwT4gLVQ76nzk+BT7rpdE41Gxk8tbJWsKc5nYKcmTFJVO7sfphiz2IUAmqFIt24qwnVCJnAkYZ51YQUTZgPagpWlAfVDtZPrG2DzTStf0QqkrQHOq/p1IqK/U0Hd15+RGtehNxP+8VozedTvhQRQjBGy2yIuFiaE5ycTscgkMxVATyiTXt5qsTyVlqJOb26L0U33ojrM6GXsxh2VSvyjZVsmuXubLt2lGGXJMTkmB2OSKlMk9qZAaYeSJvJBX8mY8G+/Gh/E5a10x0pkjMgfj6xcTc6FL</latexit><latexit sha1_base64="pDswx1l7/0rFGOOIzcwNrv9ME8c=">AAACG3icbVDLSgNBEJz1GeMr6lGExSAkB8OuCHoRgl48JmAekA1hdtKbDJl9MNMrhE1Ofocf4FU/wZt49eAX+BtOkj2YxIKGoqqb7i43ElyhZX0bK6tr6xubma3s9s7u3n7u4LCuwlgyqLFQhLLpUgWCB1BDjgKakQTquwIa7uBu4jceQSoeBg84jKDt017APc4oaqmTO/EKUfHGwT4gLVQ76nzk+BT7rpdE41Gxk8tbJWsKc5nYKcmTFJVO7sfphiz2IUAmqFIt24qwnVCJnAkYZ51YQUTZgPagpWlAfVDtZPrG2DzTStf0QqkrQHOq/p1IqK/U0Hd15+RGtehNxP+8VozedTvhQRQjBGy2yIuFiaE5ycTscgkMxVATyiTXt5qsTyVlqJOb26L0U33ojrM6GXsxh2VSvyjZVsmuXubLt2lGGXJMTkmB2OSKlMk9qZAaYeSJvJBX8mY8G+/Gh/E5a10x0pkjMgfj6xcTc6FL</latexit><latexit sha1_base64="pDswx1l7/0rFGOOIzcwNrv9ME8c=">AAACG3icbVDLSgNBEJz1GeMr6lGExSAkB8OuCHoRgl48JmAekA1hdtKbDJl9MNMrhE1Ofocf4FU/wZt49eAX+BtOkj2YxIKGoqqb7i43ElyhZX0bK6tr6xubma3s9s7u3n7u4LCuwlgyqLFQhLLpUgWCB1BDjgKakQTquwIa7uBu4jceQSoeBg84jKDt017APc4oaqmTO/EKUfHGwT4gLVQ76nzk+BT7rpdE41Gxk8tbJWsKc5nYKcmTFJVO7sfphiz2IUAmqFIt24qwnVCJnAkYZ51YQUTZgPagpWlAfVDtZPrG2DzTStf0QqkrQHOq/p1IqK/U0Hd15+RGtehNxP+8VozedTvhQRQjBGy2yIuFiaE5ycTscgkMxVATyiTXt5qsTyVlqJOb26L0U33ojrM6GXsxh2VSvyjZVsmuXubLt2lGGXJMTkmB2OSKlMk9qZAaYeSJvJBX8mY8G+/Gh/E5a10x0pkjMgfj6xcTc6FL</latexit><latexit sha1_base64="pDswx1l7/0rFGOOIzcwNrv9ME8c=">AAACG3icbVDLSgNBEJz1GeMr6lGExSAkB8OuCHoRgl48JmAekA1hdtKbDJl9MNMrhE1Ofocf4FU/wZt49eAX+BtOkj2YxIKGoqqb7i43ElyhZX0bK6tr6xubma3s9s7u3n7u4LCuwlgyqLFQhLLpUgWCB1BDjgKakQTquwIa7uBu4jceQSoeBg84jKDt017APc4oaqmTO/EKUfHGwT4gLVQ76nzk+BT7rpdE41Gxk8tbJWsKc5nYKcmTFJVO7sfphiz2IUAmqFIt24qwnVCJnAkYZ51YQUTZgPagpWlAfVDtZPrG2DzTStf0QqkrQHOq/p1IqK/U0Hd15+RGtehNxP+8VozedTvhQRQjBGy2yIuFiaE5ycTscgkMxVATyiTXt5qsTyVlqJOb26L0U33ojrM6GXsxh2VSvyjZVsmuXubLt2lGGXJMTkmB2OSKlMk9qZAaYeSJvJBX8mY8G+/Gh/E5a10x0pkjMgfj6xcTc6FL</latexit>

Consider an over-occupied plasma

20

Over-occupied QCD plasma• Loss of memory of initial

conditions

• System eneters a turbulent cascade

• Direct energy cascade, inverse particle number cascade

• Kinetic scenario can be identified from simulations

• Emergent scales from initial scale Q

Berges, Boguslavski, Schlichting Venugopalan Phys.Rev. D89 (2014) no.11, 114007,

21

Other effects and discovery• Chiral Separation Effect:

• Chiral Magnetic Wave: collective gapless excitation from the coupling between density waves of electric and chiral chargesKharzeev & Yee, PRD 83, 085007; Newman JHEP 0601 2006

Son & Zhitnitsky, PRD 70, 074018; Metlitski & Zhitnitsky, PRD 72, 045011ja ⇠ nV B

• Predicted and obseverd in Dirac and Weyl semi-metals

• Effects under intense search at RHIC (right now!) and LHC

• Tantalizing hints from RHIC, but many experimental and theory uncertainties

First found in ZrTe5: Kharzeev et al. Nature Physics 2016

22STAR, B. I. Abelev et al., Phys. Rev. Lett. 103, 251601 (2009), 0909.1739; ALICE, B. Abelev et al., Phys. Rev. Lett. 110, 012301 (2013), STAR, L. Adamczyk et al., Phys. Rev. Lett. 113, 052302 (2014), 1404.1433

The Chiral Magnetic Effect• Chiral Magnetic Effect: jV ⇠ n5B

Kharzeev, McLerran, Warringa NPA 803 2008; Fukushima, Kharzeev, Warringa PRD 78 2008

New kind of parity odd conductivity, non-dissipative

jV =e2

2⇡2µ5B

P-odd T-odd

P-even T-odd

P-odd T-even

Coefficient fixed from the anomaly• Anomalously produced axial charge — connection between

topology of QCD vacuum

• Tells us about confinement and chiral symmetry transition: deconfinement and chiral symmetry restoration needed

Red=momentum Blue=spin

Macroscopic electric current from quark chirality imbalance, n5, in external B field

23

Open question: Where does n5 come from? How much is created?Real-time information needed!

Universal dynamics

• Initial particle production leads to universal scaling

• Reaches non-thermal fixed point, self-similar evolution

Berges, MM, Schlichting PRL 118 (2017) no.19, 192005

Berges, Boguslavski, Schlichting, Venugopalan PRL114 (2015)

f(t,p) = s↵/�f(s�1/�t, sp)<latexit sha1_base64="1bJzPtVFbUxb6uj48Aj1jlUBf6k=">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</latexit><latexit sha1_base64="1bJzPtVFbUxb6uj48Aj1jlUBf6k=">AAACPHicbVA9axtBEN2TnUSRnVhxyjSLhUEGRbkzgaSxEUmT0oHoA3SymFvNSYv3PtidC4jj/o1/h3+AW6dJH3Bh0qbOSr5CH36w8Pa9GWbmBamShlz3t1PZ2X32/EX1ZW1v/9Xrg/qbw55JMi2wKxKV6EEABpWMsUuSFA5SjRAFCvvB1deF3/+J2sgk/kHzFEcRTGMZSgFkpXH9PGxSy4+AZkGYp8XJmbnMfVDpDD74ARIUPGxa6b1Xfqll+Er5uN5w2+4SfJt4JWmwEhfj+r0/SUQWYUxCgTFDz01plIMmKRQWNT8zmIK4gikOLY0hQjPKl3cW/NgqEx4m2r6Y+FJd7cghMmYeBbZysaLZ9BbiU94wo/DzKJdxmhHG4nFQmClOCV+ExidSoyA1twSElnZXLmagQZCNdm2KsUfNcFLUbDLeZg7bpHfa9ty29/1jo/OlzKjK3rEj1mQe+8Q67Bu7YF0m2DW7ZXfsl3Pj/HEenL+PpRWn7HnL1uD8+w9BTq6a</latexit><latexit sha1_base64="1bJzPtVFbUxb6uj48Aj1jlUBf6k=">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</latexit><latexit sha1_base64="1bJzPtVFbUxb6uj48Aj1jlUBf6k=">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</latexit>

s�1/�t = 1<latexit sha1_base64="6LGfsVNMB+Lo+YVKtUGRIUdNJMY=">AAACDXicbVBLSgNBEO2Jvxg/GXXpZjAIbowzIuhGCLpxGcF8IBlDT09N0qTnQ3eNEIY5gwdwq0dwJ249gyfwGnY+C5P4oODxXhVV9bxEcIW2/W0UVlbX1jeKm6Wt7Z3dsrm331RxKhk0WCxi2faoAsEjaCBHAe1EAg09AS1veDv2W08gFY+jBxwl4Ia0H/GAM4pa6pll9ZidOmddD5DmeO30zIpdtSewlokzIxUyQ71n/nT9mKUhRMgEVarj2Am6GZXImYC81E0VJJQNaR86mkY0BOVmk8Nz61grvhXEUleE1kT9O5HRUKlR6OnOkOJALXpj8T+vk2Jw5WY8SlKEiE0XBamwMLbGKVg+l8BQjDShTHJ9q8UGVFKGOqu5LUo/NQA/L+lknMUclknzvOrYVef+olK7mWVUJIfkiJwQh1ySGrkjddIgjKTkhbySN+PZeDc+jM9pa8GYzRyQORhfv+iam0o=</latexit><latexit sha1_base64="6LGfsVNMB+Lo+YVKtUGRIUdNJMY=">AAACDXicbVBLSgNBEO2Jvxg/GXXpZjAIbowzIuhGCLpxGcF8IBlDT09N0qTnQ3eNEIY5gwdwq0dwJ249gyfwGnY+C5P4oODxXhVV9bxEcIW2/W0UVlbX1jeKm6Wt7Z3dsrm331RxKhk0WCxi2faoAsEjaCBHAe1EAg09AS1veDv2W08gFY+jBxwl4Ia0H/GAM4pa6pll9ZidOmddD5DmeO30zIpdtSewlokzIxUyQ71n/nT9mKUhRMgEVarj2Am6GZXImYC81E0VJJQNaR86mkY0BOVmk8Nz61grvhXEUleE1kT9O5HRUKlR6OnOkOJALXpj8T+vk2Jw5WY8SlKEiE0XBamwMLbGKVg+l8BQjDShTHJ9q8UGVFKGOqu5LUo/NQA/L+lknMUclknzvOrYVef+olK7mWVUJIfkiJwQh1ySGrkjddIgjKTkhbySN+PZeDc+jM9pa8GYzRyQORhfv+iam0o=</latexit><latexit sha1_base64="6LGfsVNMB+Lo+YVKtUGRIUdNJMY=">AAACDXicbVBLSgNBEO2Jvxg/GXXpZjAIbowzIuhGCLpxGcF8IBlDT09N0qTnQ3eNEIY5gwdwq0dwJ249gyfwGnY+C5P4oODxXhVV9bxEcIW2/W0UVlbX1jeKm6Wt7Z3dsrm331RxKhk0WCxi2faoAsEjaCBHAe1EAg09AS1veDv2W08gFY+jBxwl4Ia0H/GAM4pa6pll9ZidOmddD5DmeO30zIpdtSewlokzIxUyQ71n/nT9mKUhRMgEVarj2Am6GZXImYC81E0VJJQNaR86mkY0BOVmk8Nz61grvhXEUleE1kT9O5HRUKlR6OnOkOJALXpj8T+vk2Jw5WY8SlKEiE0XBamwMLbGKVg+l8BQjDShTHJ9q8UGVFKGOqu5LUo/NQA/L+lknMUclknzvOrYVef+olK7mWVUJIfkiJwQh1ySGrkjddIgjKTkhbySN+PZeDc+jM9pa8GYzRyQORhfv+iam0o=</latexit><latexit sha1_base64="6LGfsVNMB+Lo+YVKtUGRIUdNJMY=">AAACDXicbVBLSgNBEO2Jvxg/GXXpZjAIbowzIuhGCLpxGcF8IBlDT09N0qTnQ3eNEIY5gwdwq0dwJ249gyfwGnY+C5P4oODxXhVV9bxEcIW2/W0UVlbX1jeKm6Wt7Z3dsrm331RxKhk0WCxi2faoAsEjaCBHAe1EAg09AS1veDv2W08gFY+jBxwl4Ia0H/GAM4pa6pll9ZidOmddD5DmeO30zIpdtSewlokzIxUyQ71n/nT9mKUhRMgEVarj2Am6GZXImYC81E0VJJQNaR86mkY0BOVmk8Nz61grvhXEUleE1kT9O5HRUKlR6OnOkOJALXpj8T+vk2Jw5WY8SlKEiE0XBamwMLbGKVg+l8BQjDShTHJ9q8UGVFKGOqu5LUo/NQA/L+lknMUclknzvOrYVef+olK7mWVUJIfkiJwQh1ySGrkjddIgjKTkhbySN+PZeDc+jM9pa8GYzRyQORhfv+iam0o=</latexit>

f(t,p) = t↵fs(t�p)

<latexit sha1_base64="Rx9bPd+DKIdsxxRV3iqBO5GPGoM=">AAACLnicbVDLSgNBEJz1bXxFPXoZDEIECbsi6EUQvXiMYBIhiaF30psMzj6Y6RXCkt/wO/wAr/oJggfxqJ/hJNlDfBQM1FR1093lJ0oact03Z2Z2bn5hcWm5sLK6tr5R3NyqmzjVAmsiVrG+8cGgkhHWSJLCm0QjhL7Chn93MfIb96iNjKNrGiTYDqEXyUAKICt1im5QpoNWCNT3gywZ7p/SbQtU0gcedEzZfnwk4FMFnWLJrbhj8L/Ey0mJ5ah2ip+tbizSECMSCoxpem5C7Qw0SaFwWGilBhMQd9DDpqURhGja2fiyId+zSpcHsbYvIj5WpzsyCI0ZhL6tHK1ofnsj8T+vmVJw0s5klKSEkZgMClLFKeajmHhXahSkBpaA0NLuykUfNAiyYf6YYuxRfewOCzYZ73cOf0n9sOK5Fe/qqHR2nme0xHbYLiszjx2zM3bJqqzGBHtgT+yZvTiPzqvz7nxMSmecvGeb/YDz9Q1P1qkr</latexit><latexit sha1_base64="Rx9bPd+DKIdsxxRV3iqBO5GPGoM=">AAACLnicbVDLSgNBEJz1bXxFPXoZDEIECbsi6EUQvXiMYBIhiaF30psMzj6Y6RXCkt/wO/wAr/oJggfxqJ/hJNlDfBQM1FR1093lJ0oact03Z2Z2bn5hcWm5sLK6tr5R3NyqmzjVAmsiVrG+8cGgkhHWSJLCm0QjhL7Chn93MfIb96iNjKNrGiTYDqEXyUAKICt1im5QpoNWCNT3gywZ7p/SbQtU0gcedEzZfnwk4FMFnWLJrbhj8L/Ey0mJ5ah2ip+tbizSECMSCoxpem5C7Qw0SaFwWGilBhMQd9DDpqURhGja2fiyId+zSpcHsbYvIj5WpzsyCI0ZhL6tHK1ofnsj8T+vmVJw0s5klKSEkZgMClLFKeajmHhXahSkBpaA0NLuykUfNAiyYf6YYuxRfewOCzYZ73cOf0n9sOK5Fe/qqHR2nme0xHbYLiszjx2zM3bJqqzGBHtgT+yZvTiPzqvz7nxMSmecvGeb/YDz9Q1P1qkr</latexit><latexit sha1_base64="Rx9bPd+DKIdsxxRV3iqBO5GPGoM=">AAACLnicbVDLSgNBEJz1bXxFPXoZDEIECbsi6EUQvXiMYBIhiaF30psMzj6Y6RXCkt/wO/wAr/oJggfxqJ/hJNlDfBQM1FR1093lJ0oact03Z2Z2bn5hcWm5sLK6tr5R3NyqmzjVAmsiVrG+8cGgkhHWSJLCm0QjhL7Chn93MfIb96iNjKNrGiTYDqEXyUAKICt1im5QpoNWCNT3gywZ7p/SbQtU0gcedEzZfnwk4FMFnWLJrbhj8L/Ey0mJ5ah2ip+tbizSECMSCoxpem5C7Qw0SaFwWGilBhMQd9DDpqURhGja2fiyId+zSpcHsbYvIj5WpzsyCI0ZhL6tHK1ofnsj8T+vmVJw0s5klKSEkZgMClLFKeajmHhXahSkBpaA0NLuykUfNAiyYf6YYuxRfewOCzYZ73cOf0n9sOK5Fe/qqHR2nme0xHbYLiszjx2zM3bJqqzGBHtgT+yZvTiPzqvz7nxMSmecvGeb/YDz9Q1P1qkr</latexit><latexit sha1_base64="Rx9bPd+DKIdsxxRV3iqBO5GPGoM=">AAACLnicbVDLSgNBEJz1bXxFPXoZDEIECbsi6EUQvXiMYBIhiaF30psMzj6Y6RXCkt/wO/wAr/oJggfxqJ/hJNlDfBQM1FR1093lJ0oact03Z2Z2bn5hcWm5sLK6tr5R3NyqmzjVAmsiVrG+8cGgkhHWSJLCm0QjhL7Chn93MfIb96iNjKNrGiTYDqEXyUAKICt1im5QpoNWCNT3gywZ7p/SbQtU0gcedEzZfnwk4FMFnWLJrbhj8L/Ey0mJ5ah2ip+tbizSECMSCoxpem5C7Qw0SaFwWGilBhMQd9DDpqURhGja2fiyId+zSpcHsbYvIj5WpzsyCI0ZhL6tHK1ofnsj8T+vmVJw0s5klKSEkZgMClLFKeajmHhXahSkBpaA0NLuykUfNAiyYf6YYuxRfewOCzYZ73cOf0n9sOK5Fe/qqHR2nme0xHbYLiszjx2zM3bJqqzGBHtgT+yZvTiPzqvz7nxMSmecvGeb/YDz9Q1P1qkr</latexit>

• Self similarity observed for distribution function seen between scalars and gauge theory

0

0.05

0.1

0.15

0.2

0 1 2 3

Mo

men

t: (p

/Q)3 g

2 f(p,

t)

Momentum: p/Q

tQNc2=450

tQNc2=900

tQNc2=1800

tQNc2=3600

0 1 2 3 0

0.05

0.1

0.15

0.2

0.25

Scale

d: (

t/t0

Nc2 )-β

(p/Q

)3 g2 f(

p,t)

SU(3)SU(2)

Qt0=100

Scaled: (t/t0Nc2)β p/Q

24Berges et al PRL114 (2015)

Universal dynamics

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

5

0 20 40 60 80 100 120 140 160 180 200

ζ=3/5 | t0Q=100

SU(2) SU(3)

Wilso

n L

oo

p: -

1/C

F lo

g(〈W

(t,A)〉)

Scaled Area: (t/t0)-ζ A Q2/Nc

tQ=200tQ=400tQ=600tQ=800

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

5

0 20 40 60 80 100 120 140 160 180 200

ζ=3/5 | t0Q=100

SU(2) SU(3)

Wilso

n L

oo

p: -

1/C

F lo

g(〈W

(t,A)〉)

Scaled Area: (t/t0)-ζ A Q2/Nc

Berges, MM, Schlichting PRL 118 (2017) no.19, 192005

1

Ha

rd S

ca

le:

10-2

10-1

102 103

Λ(t)

2 / Q

2m

D(t)

2 / (N

c Q2 )

(tQ)2/7

(tQ)-2/7

SU(2) SU(3)

1

De

by

e S

ca

le:

Time: t Q Nc2

• In gauge theories, universal, self-similar scaling for a large range of scales, including IR, which controls topological transitions

0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80 100 120 140 160 180 200

SU(2) SU(3)Wil

so

n L

oo

p: -

1/C

F lo

g(〈W

(t,A)〉)

Area: A Q2/Nc

tQ=200tQ=400tQ=600tQ=800

25

W =1

NcTrPeig

Hc dxµAµ(x)

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Dynamical separation of scales

MM, Schlichting, Venugopalan PRD 074036 (2016)

Hard Scale

Electric Screening

Magnetic Screening

meqD ⇠ gT

⇤eq ⇠ T

⇤eqs ⇠

p� ⇠ g2T

Initially one scale , dynamically generate scales

QsIn equilibrium, hierarchy of scales

�neqsph(t) = 2⇥ 10�2�2(t)Identify that sphaleron transition rate:

Qua

ntum

True approach to equilibrium needs quantum treatment

26

Sphaleron transition in action

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

8 10 12 14 16

NC

S

Time: t

UncooledCooled τc=4

27

@Aai (x)

@⌧= � @H

@Aai (x)

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Real-time Lattice QCDTo study full CME dynamics, need to extend non-equilibrium gauge field studies to include fermions

Cannot resort to classical approximation for fermions because of Pauli principle

Expand initial fermion field in operator basis, evolve wave function by solving Dirac equation

Measure operator expectation values, i.e

Fermions in HIC are nearly chiral, want chiral formulations

Work in quenched limited (no fermion backreaction) for first study

SU(Nc)+U(1) x(t) =1pV

X

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Kasper et al PRD 90 (2014) no.2, 025016, MM, Mueller, Schlichting, Sharma PRD 95 2017

i�0@t x = (�i /DsW +m) x ,

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A0=0 3+1D

28

Fermions on the lattice

Overlap Fermions:

Exact Chiral symmetry. Separate chiral and continuum limit. Can be massless. Extremely numerically expensive. First Minkowski study in 3+1D.

Narayanan & Neuberger PRL71 3251, Neuberger PLB417 141

Karsten & Smit NPB 183 (1983)

Simulate with both as a cross check

Wilson (tree level-improved):

Add term to action to remove doublers. Explicitly breaks chiral symmetry, only recovered in continuum limit. Has own anomaly relation

In the classical-statistical limit mass renormalized perturbatively, Wilson fermions arbitrary close to massless limit

�i /Dsov = M

1+

�0HW (M)pHW (M)2

!

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29

Magnetic field dependence

MM, Mueller, Schlichting, Sharma PRD 95 (2017)

finite magnetic field: → important deviations

from ‘ideal' picture of CME

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

Vect

or c

harg

e se

para

tion:

Δ

J0 V

Magnetic field: qBrsph2

maxt/tsph=1.5

-0.6-0.4-0.2

0 0.2 0.4

-5 -4 -3 -2 -1 0 1 2 3 4 5z/rsph

t = 2.67 tsph

qB = 0.8 rsph-2

qB = 1.6 rsph-2

-0.6-0.4-0.2

0 0.2 0.4

Vector density Axial density

t = 1.67 tsph

-0.6-0.4-0.2

0 0.2 0.4 jv

0(z,t)

t = 0.67 tsph

0 0.2 0.4 0.6 0.8

1

-5 -4 -3 -2 -1 0 1 2 3 4 5

qB = 3.5 rsph-2

qB = 7 rsph-2

0 0.2 0.4 0.6 0.8

1 0

0.2 0.4 0.6 0.8

1 ja0(z,t)

asymptotic limit

“simple” estimates work

30

Quark mass dependence

MM, Mueller, Schlichting, Sharma PRD 95 (2017)

0

0.2

0.4

0.6

0.8

1

0 0.5 1

Cha

rge

sepa

ratio

n

ΔJV0

Quark mass: mrsph

maxt/tsph=1.5

0 0.5 1

ΔJA0

finite quark mass effects → anomalous transport

suppressed for heavy quarks

31

Chiral magnetic effect

Non-zero magnetic field -> vector current is generated jzVBz

With addition of B field, sphaleron transition induces local imbalance

of axial charge density

Vector current leads to separation of electric charges along the B-field direction

j0VjzV

j05

xz

y

B

32(MM, Mueller, Schlichting, Sharma PRD 95 2017)

Vector charge j0VVector current jzVAxial charge j05

(Slide adapted from S. Schlichting CERN 2016)

Non-zero magnetic field -> vector current is generated jzVBz

Sphaleron transition induces local imbalance of axial charge density j05

Vector current leads to separation of electric charges along the B-field direction

j0VjzV

Chiral magnetic effect

33(MM, Mueller, Schlichting, Sharma PRD 95 2017)


xz

y

B

(Slide adapted from S. Schlichting CERN 2016)

Chiral magnetic wave


Vector charge imbalance generates an axial current so that axial charge also flows along the B-field direction

j0V jz5

Shock-wave of vector charge and axial charge propagating along B-field direction

34

xz

y

B

(MM, Mueller, Schlichting, Sharma PRD 95 2017)(Slide adapted from S. Schlichting CERN 2016)



j0V jz5



35

xz

y

B




j0V jz5



36

xz

y

B


LI ET AL, NATURE PHYSICS, DOI: 10.1038/NPHYS3648 STAR, B. I. ABELEV ET AL., PHYS. REV. LETT. 103, 251601 (2009), 0909.1739; ALICE, B. ABELEV ET AL., PHYS. REV. LETT. 110, 012301 (2013), STAR, L. ADAMCZYK ET AL., PHYS. REV. LETT. 113, 052302 (2014), 1404.1433 S. PRATT, S. SCHLICHTING, AND S. GAVIN, PHYS. REV. C84, 024909 (2011), 1011.6053.; Y. HATTA, A. MONNAI, AND B.-W. XIAO, (2015), 1507.04690.

CHIRAL MAGNETIC EFFECT IN REALITY▸ In heavy ion collisions

▸ STAR, ALICE published results in line with CME (and other anomalous transport phenomena) predictions

▸ Alternative explanations exist

▸ Local charge conservation

▸ Viscous hydrodynamics

▸ Recent discovery in condensed matter systems

Image from H. Ke’s RHIC/AGS 2015 users meeting talk

37

Sphalerons in the glasma

• Non-equilibrium, over occupied gluons

• Initially only one scale,

• Classical-Statistical framework: sample over ensemble of stochastic initial conditions

Qs

38

(MM, Schlichting, Venugopalan PRD 074036)

Dynamical scales at a glance• Hard scale

• Average typical momentum of hard modes

• Electric screening scale

• Appeal to perturbative formula

• Magnetic screening scale

• Measure spatial string tension

• Large area fall limit of spatial Wilson loop

⇤eqs ⇠

p� ⇠ g2T

⇤eq ⇠ T

meqD ⇠ gT

Qs

Qs

Qs

hW (A, t)i ⇡ exp(��A)

Kurkela and Moore, JHEP 1112, 044 (2011); Schlichting PRD86, 065008 (2012)

Dumitru, Lappi, and Nara, PLB734, 7 (2014)

Kurkela and Moore, PRD86, 056008 (2012)m2

D = 4g2Nc

Zd3p

(2⇡)3f(p)

p

⇤(t)2 =2

3

Rd3p p3 f(p)Rd3p p f(p)

Non-equilibrium sphaleron transition rate

-1

0

1

2

3

4

5

6

7

40 60 80 100 120 140 160 180 200 220

x 10-3

Γne

qsp

h / σ

2

Time: Qs t

Qs2τc=36

• In equilibrium, sphaleron rate controlled by magnetic modes

• From dynamical separation of scales, non-equilibrium sphaleron rate controlled by modes of order magnetic screening

• Same scaling exponent expected for SU(2) and SU(3), greater coefficient for SU(3)

(MM, Schlichting, Venugopalan PRD 93 2016)

�neqsph(t) = 2⇥ 10�2�2(t)

�2(t) ⇡ Q2s(Qst)

�2/3

40

(Berges, MM, Schlichting arXiv:1703.00697)

Transport

• Mass, B dependence on charge separation

0

0.5

1

1.5

0 0.5 1 1.5 2Δ

J vz / Δ

J a0 (t)t/tsph

2

4

6

8

10

ΔJ az /

ΔJ v0 (t)

qB = 0.8 rsph-2

qB = 1.6 rsph-2

qB = 3.5 rsph-2

qB = 7 rsph-2

• B dep. for ratios of charge separation

Nontrivial behavior away from asymptotic limit

Vector charge separation vanishes for large m

�Jzv (t)

�J0a(t)

�Jza (t)

�J0v (t)

MM, Mueller, Schlichting, Sharma PRD 95 2017Finite “relaxation time” should be taken into account

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

Vect

or c

harg

e se

para

tion Δ

J V0

qBr2sph

maxt/tsph=1.5

0 0.5 1mrsph

Application: Pair production in strong field

QED

42

Schwinger pair production• Above a critical field strength, F=me2/e=1018 V/m,

classical electrodynamics breaks down

• Produce fermion pairs from vacuum due to non-linearities in QED

• Many analytical/numerical results, however including full backreaction of fermions difficult

• Large coherent field highly amenable to classical-statistical approximation

e.g. Cooper, Mottola PRD 40 (1989), Kluger, Eisenberg, Svetitsky, Cooper, Mottola PRD 45 (1992), …, Dunne et al…

Strong E and B fields in HIC also above limit

J. S. Schwinger, Phys. Rev. 82, 664 (1951).

Kasper et al PRD 90 (2014) no.2, 025016

May be accessible in next generation of high intensity user experiments (ELI)

43

Schwinger pair production• Without back-reaction able to

reproduce continuum pair production rate

• Novel dynamics begin when including fermion back-reaction

MM, Schlichting, Sharma, in progress

0.015

0.02

0.025

0.03

0.035

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Schw

inge

r pai

r pro

duct

ion

rate

: d/d

t dN

/d3 x

[m4 ]

Lattice spacing: maT

Wilson (LO): VTm2=36cont. extrapolation: s0 + s1(maT) + s2 (maT)2

Wilson (LO): VTm2=81cont. extrapolation: s0 + s1(maT) + s2 (maT)2

Wilson (NLO): VTm2=36cont. extrapolation: s0 + s0 (maT)2

continuum result

Narozhny et al Phys. Lett. A 330, 1 (2004); JETP Lett. 80, 382 (2004); JETP 129, 14 (2006).

dN

dtd3x=

e2E2s

4⇡2✏⌘ coth

⇣⇡⌘✏

⌘exp

⇣�⇡

✏

⌘

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-3

-2

-1

0

1

2

3

0 20 40 60 80 100 120 140

Elec

tric

field

: eE/

me2

Evolution time: me t

r-Avg Wilson (NLO) NT=16 NL=40 maT= 0.375 maL=0.05

e2Nf=1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 20 40 60 80 100 120 140

e+ e- pai

r den

sity

: dN

e+ e- /d3 x

[me3 ]

Evolution time: me t

r-Avg Wilson (NLO) NT=16 NL=40 maT= 0.375 maL=0.05

e2Nf=1 rW=+1e2Nf=1 rW= -1

e2Nf=1No BR

limit

44

Real-time dynamics

45

MM, S. SCHLICHTING, R. VENUGOPALAN ARXIV:1601.07342 [HEP-PH]

COOLING: A CARTOON46

real-time evolution

cooling

calib

ratio

n

NCS(t2) - NCS(t1) NCS(t3) - NCS(t2)

NC

S(t 1

) - N

CS(

t 1)

Vac

NC

S(t 3

) - N

CS(

t 3)

Vac

τ=0

τ=τc

τ=∞ vacuum

t2t1 t3

@Aai (x)

@⌧= � @H

@Aai (x)

TOPOLOGY ON THE LATTICE▸

▸ On the lattice,

47

dNCS

dt' g2

8⇡2

X

x2lattice,i

(Eai (x, t+ �t/2) + Ea

i (x, t� �t/2))

22X

8⇤tr(

i⌧aU⇤(x)4

)

Ez(x, t+ �t/2)� Ez(x, t� �t/2)

2Bz

dNCS(t)

dt=

Zd3x

g2

8⇡2Ea

i Bai

INDEPENDENT TEST

▸ Can compute the change in Chern-Simons number with two completely independent methods which agree well

48

*Shifted curves for

clarity


ROBUSTNESS OF OUR RESULTS 49

0

5x10-7

1x10-6

1.5x10-6

2x10-6

2.5x10-6

3x10-6

0 10 20 30 40 50 60 70 80

Qst=50

<(N

CS(

t+δt

)-NC

S(t))

2 >/(Q

s3 V)

Qsδt

N=64, Qsa=1, Qs2τc=12

N=128, Qsa=0.5, Qs2τc=12

N=96, Qsa=1, Qs2τc=12

N=192, Qsa=0.5, Qs2τc=12

0

5x10-7

1x10-6

1.5x10-6

2x10-6

2.5x10-6

0 10 20 30 40 50 60 70 80

<(N

CS(

t+δt

)-NC

S(t))

2 >/(Q

s3 V)

Qsδt

N=96, Qs2τc=12

N=128, Qs2τc=12

N=192, Qs2τc=12

Volume independence (for two cooling lengths)

Lattice spacing independence

0 2x10-8 4x10-8 6x10-8 8x10-8 1x10-7

1.2x10-7 1.4x10-7 1.6x10-7 1.8x10-7

2x10-7

0 10 20 30 40 50 60 70 80

<(N

CS(

t+δt

)-NC

S(t))

2 >/(Q

s3 V)

Qsδt

N=64, Qs2τc=324

N=96, Qs2τc=324


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