Thechiralanomaly,Berry’sphaseandchiralkine5ctheoryfromworld-linesinquantumfieldtheory

RajuVenugopalan

BrookhavenNa5onalLaboratory

TalkatCanterburyTalesWorkshop,OxfordJuly2017

BasedonNiklasMuellerandRV,arXiv:1701.03331andarXiv:1702.01233

Outlineoftalk

u  Introduc5on:topologicaleffectsinheavy-ioncollisionsu  Abini&oapproachtotheChiralMagne5cEffect:Topologicaltransi5onsintheGlasma

u World-lineformula5onofQFT&thechiralanomaly

u  Pseudo-classicalequa5onsofmo5onandBerry’sphase(coda:Fujikawa’slament)u  Outlook

Topologyinion-ioncollisions:ChiralMagne5cEffect

L or B

+External(QED)magne5cfields-1018Gauss,ofMagnetarstrength!

=

Kharzeev,McLerran,Warringa(2007) Kharzeev,Fukushima,Warringa(2008)

Overbarriertopological(sphaleron)transi5ons…analogoustoproposedmechanismforelectroweakbaryogenesis

NCS = -2 -1 0 1 2

masslessquarksinhotmedium

Topologicaltransi5onQW=nL-nR

CMEcurrentgenerated

Topologyinion-ioncollisions:ChiralMagne5cEffect

ExternalBfielddiesrapidly.Life5meofhotmader~10Fermi:effectmostsignificant,fortransi5onsatearly5mes

Consistent(caveatemptor!)withheavy-ionresultsfromRHIC&LHC

Status:Kharzeev,Liao,Voloshin,Wang,Prog.Nucl.Part.Phys.88(2016)1

CMEstudiesamajorpartofRHIC’supcomingbeamenergyscan(BESII)-possiblydefini5veresultsfromcompara5vestudyofisobarcollisions

BNLCMEtaskforcereport:V.Skokovetal.,arXiv:1608.00982

CMEincondensedmadersystems?

Q.Li, et al, Nature Physics 12, 550 (2016)

Blue:CMEmodelRed:experimentaldata

Diracsemi-metal:ZirconiumPenta-Telluride

~JCME =e2

2⇡2µ5

~B

µ5 / ~E · ~B effectofanomaly

J iCME = �ik

CME Ek

Axialchargesepara5oninexternalBfield

�ikCME / BiBk => �zz

CME / B2

TheGlasma

TheGlasmaatLOCollisionsoflumpygluon``shock”waves

DµFµ⌫,a = �

⌫+⇢

aA(x?)�(x

�) + �

⌫�⇢

aB(x?)�(x

+)

Leadingordersolu5on:Solu5onofQCDYang-Millseqns

QS(x,bT)determinedfromsatura5onmodelfitstoHERAinclusiveanddiffrac5veDISdata

h⇢aA(B)(x?)⇢aA(B)(y?)i = Q

2S,A(B)�

(2)(x? � y?)

MatchingboostinvariantYang-Millstohydrodynamics

Ini5allongitudinalpressureisnega5ve:GoestoPL=0frombelowwith5meevolu5on

Glasmaenergydensityandpressure

Krasnitz,Venugopalan(1998)Lappi(2003)

It’sthematchingoftheseresultstohydrothatgivestheIP-Glasma+MUSICmodel

TheGlasmaatNLO:plasmainstabili5es

AtLO:boostinvariantgaugefieldsAclμ,a(xT,τ)~1/g

NLO:Aμ,a(xT,τ,η)=Acl

μ,a(xT,τ)+aμ,a(η) aμ,a(η)=O(1)

Ø  Smallfluctua5onsgrowexponen5allyas~Ø  Sameorderofclassicalfieldat

epQS⌧

⌧ =1

QSln2

1

↵S

QSτ

increasing seed size

2500Ø  Resumsuchcontribu5onstoallorders

(g epQS⌧ )n

Tµ⌫resum =

Z

⌧=0+[da] Finit.[a] TLO[Acl + a]

Romatschke,VenugopalanDusling,Gelis,VenugopalanGelis,Epelbaum

GlasmaatNLO:Ini5alcondi5onsforclassical-sta5s5caldynamicsintheoverpopulatedQGP

Overpopula5onoccurs…evenstar&ngfromthe“firstprinciplesCGC”ini5alcondi5ons

Berges,Schenke,Schlich5ng,RV,NPA931(2014)348

Classical-sta5s5calreal5menumericallawcesimula5onsofanexpandinggaugetheory

Ini5alcondi5onsintheGlasma

⌧ =1

QS

⌧ =1

QSln2

1

↵S⌧ >>

1

QSln2

1

↵S

Ini5alcondi5onsintheoverpopulatedQGPChoosefortheini5alclassical-sta5s5censembleofgaugefields

Stochas5crandomvariables

Polariza5onvectorsξexpressedintermsofHankelfunc5onsinFock-SchwingergaugeAτ=0

f(p?, pz, t0) =n0

↵S⇥

✓Q�

qp2? + (⇠0pz)2

◆

Controls“prolateness”or“oblateness”ofini5almomentumdistribu5on

Occupa5on#

Coul.gauge

Temporalevolu5onintheoverpopulatedQGP

SolveHamilton’sequa5onfor3+1-DSU(2)gaugetheoryinFock-Schwingergauge

FixresidualgaugefreedomimposingColoumbgaugeateachreadout5me

Berges,Boguslavski,Schlich5ng,VenugopalanarXiv:1303.5650,1311.3005

@iAi + t�2@⌘A⌘ = 0

u Largestclassical-sta5s5calnumericalsimula5onsofexpandingYang-Millstodate:2562×4096lawces

Kine5ctheoryintheoveroccupiedregime

For1<f<1/αSadualdescrip5onisfeasibleeitherintermsofkine5ctheoryorclassical-sta5s5caldynamics… Mueller,Son(2002)

Jeon(2005)

Proper5esindependentofini5alcondi5onsSelf-similarevolu5oncharacterizedbyuniversalscalingexponents

f(pT , pz, ⌧) = (Q⌧)↵fS(⌧�pT , ⌧

�pz)

Non-thermalfixedpointinoverpopulatedQGP

BMSS:Baier,Mueller,Schiff,SonBD:BodekerKM:Kurkela,MooreBGLMV:Blaizot,Gelis,Liao,McLerran,Venugopalan

Berges,Boguslavski,Schlich5ng,Venugopalan.PRD89(2014)114007

Aremarkableuniversality

Berges,Boguslavski,Schenke,Venugopalan,PRL114(2015)061601,Editor’ssugges5on

LeadstoBose-Einsteincondensa5on

Inawideiner5alrange,scalarsandgaugefieldshaveiden5calscalingexponentsandscalingfunc5onsVerysurprisingfromasimplekine5ctheoryperspec5ve

Topologicaltransi5onsintheGlasma

Glasma

Sphalerontransi5onsinQCD

@µJµ5,f = 2mf q�5q �

g2

16⇡2F aµ⌫ F

µ⌫,a

ChiralAnomaly:

Kµ =g2

32⇡2✏µ⌫⇢�

⇣F a⌫⇢A

a� � g

3fabcA

b⌫A

c⇢A

c�

⌘Chern-Simonscurrent:

Chern-Simons#:NCS(t) =

Zd

3xK

0(t,x)

Sphaleron:spa5allylocalized,unstablefiniteenergyclassicalsolu5ons(σφαλεροs-``readytofall”) EWtheory:Klinkhamer,Manton,PRD30(1984)2212

QCD:McLerran,Shaposhnikov,Turok,Voloshin,PLB256(1991)451

RateofchangeofCS#dNCS(t)

dt

=g

2

8⇡2

Zd

3xE

ai (x)B

ai (x)

u Sphalerontransi5onrate–significantliteratureonthermalrate

�eq = lim�t�>1

h(NCS(t+ �t)�NCS(t))2ieq

V �tStateoftheartatfiniteT,Moore,Tassler,arXiv:1011.1167

Overoccupiedgaugefieldsinabox

Thermaliza5onextensivelystudiedinthiscontextemployingclassical-sta5s5calsimula5ons

Berges,Schlich5ng,Sexty,PRD86(2012)074006Schlich5ngPRD86(2012)065008York,Kurkela,Lu,Moore,PRD89(2014)074036

Overoccupiedgaugefieldsinabox

Cleansepara5onofscalesdevelopalathermalfieldtheory:Temperature(T)Electric(Debye)screening(gT)Magne5cscreening(g2T)scales Berges,Scheffler,Sexty,PRD77(2008)034504

Mace,Schlich5ng,Venugopalan,PRD93(2016),074036Berges,Mace,Schlich5ng,PRL118(2017)

Overoccupiedgaugefieldsinabox

Mace,Schlich5ng,Venugopalan,PRD93(2016),074036

Increasingcooling5me

Topologicaltransi5onsintheGlasma

Distribu5onofChern-SimonschargelocalizesaroundintegervaluesasUVmodesareremoved

“Cooled”Glueconfigura5onsintheGlasmaaretopological!

Mace,Schlich5ng,Venugopalan,PRD93(2016),074036

Topologicaltransi5onsintheGlasmaMace,Schlich5ng,Venugopalan,PRD93(2016),074036

Sphalerontransi5onratescaleswithstringtensionsquared

Topologicaltransi5onsintheGlasmaMace,Schlich5ng,Venugopalan,PRD93(2016),074036

Sphalerontransi5onratescaleswithstringtensionsquared

Improveddetermina5onofσ2andSU(3)inprogress

M.Mace

Explodingsphalerons

u  Couplesphaleronbackgroundwithfermions&externalEMfieldstosimulateabini&otheChiralMagne5cEffect!

“Explodingsphalerons”:Shuryak,Zahed,PRD67(2003)014006

u  Sphalerontransi5onrateverylargeintheGlasma-muchlargerthanequilibriumrate

Mueller,Schlich5ng,Sharma,PRL117(2016)142301Mace,Mueller,Schlich5ng,Sharma,arXiv:1612.02477

Arnold,Son,Yaffe,PRD55(1997)6264

Classical-sta5s5calsimula5ons

N.Muelleretal.PRL117,142301(2016)M.Maceetal.arXiv:1612.02477

Emergenceofchiralmagne5cwave

Kharzeev,Yee,PRD83(2011)085007

Thelimitsofclassical-sta5s5calsimula5onsSlidebyNiklasMueller

Towardsachiralkine5ctheory:

I.World-lineformula5onofQFTandthechiralanomaly

Kine5cevolu5onofthechiralmagne5ccurrentu  Quantumkine5cevolu5onofthechiralmagne5ccurrentdependson

typical5mescalesforscadering,forsphalerontransi5ons,andE&Mconduc5vityinthesystem

u  Significantworkonchiralkine5ctheorySon,Yamamoto,PRL109(2012),181602;PRD87(2013)085016Stephanov,Yin,PRL109(2012)162001Chen,Son,Stephanov,Yee,Yin,PRL113(2014)182302Chen,Son,Stephanov,PRL115(2015)021601Chen,Pu,Wang,Wang,PRL110(2013)262301Gao,Liang,Pu,Wang,Wang,PRL109(2012)232301Stone,Dwivedi,Zhou,PRD91(2015)025004Zahed,PRL109(2012)091603;Basar,Kharzeev,Zahed,PRL111(2013)161601Stephanov,Yee,Yin,PRD91(2015)125014Fukushima,PRD92(2015)054009Manuel,Torres-Rincon,PRD90(2014)074018Hidaka,Pu,Yang,arXiv:1612.04630

u Wewilldiscusshereanovelapproachbasedontheworld-lineformalisminQCD

NiklasMuellerandRV,arXiv:1701.03331and1702.01233

World-lineformalism:preliminariesReview:Corradini,Schubert,arXiv:1512.08694Also,Strassler,NPB385(1992)145

BasedonSchwinger’sproper5metrick:

Oneloopeffec5veac5onofmasslessscalarfieldcoupledtobackgroundAbelianfield

=

Z 1

0

dT

TTr exp(�TD2

)

with N =

ZDp exp(�1

2

Z T

0d⌧✏ p2) εistheEinbein:squarerootof

1Dmetric

World-lineformalism:vectorandaxialvectorfields

AisavectorgaugefieldandBisanauxiliaryaxialvectorgaugefield

Fermioneffec5veac5on:

with

Fµ⌫ = @µA⌫ � @⌫Aµ � [Aµ,A⌫ ]

S[A,B] =

Zd

4x (i/@ + /A+ �5 /B)

✓ = i/@ + /A+ �5 /B�W [A,B] = log det (✓)W [A,B] = WR + iWI

WR = �1

8

log det

⇣˜

⌃

2⌘⌘ �1

8

Tr log

⇣˜

⌃

2⌘Focusfirstontherealpart:

World-lineformalism:coherentstates

D’Hoker,Gagne,hep-ph/9508131

Berezin,Marinov,AnnalsPhys.104(1977)336

DoublingdimensionofDiracmatrices&extensionofCliffordalgebraessen5alforcoherentstatespinorrepresenta5on

Coherentstatescanbeusedtogeneratefinitedimensionalrepresenta5onsofinternalsymmetrygroups

Grassmanianpathintegralrepresenta5on

InthefermioniccoherentstateGrassmanianrepresenta5on,thetracecanberepresentedas Ohnuki,KashiwaProg.Theo.Phys.60(1978)548

D’Hoker,Gagne,hep-ph/9512080

andrewridenasthequantummechanicalpathintegral…

withthe“pointpar5cle”Lagrangian

Switchedherefrom3-Dcomplexθbasistothatof6-DMajoranafermionsψa-simplemnemonic:Γè√2ψ

WR = �1

8

Tr log

⇣˜

⌃

2⌘=

1

8

Z 1

0Tr16 exp

⇣�"

2

T ˜

⌃

2⌘

Tr16 exp

⇣�"

2

T ˜

⌃

2⌘=

Zd4z d3✓ hz,�✓| exp

⇣�"

2

T ˜

⌃

2⌘|z, ✓i

Grassmanianpathintegralrepresenta5on

InthefermioniccoherentstateGrassmanianrepresenta5on,thetracecanberepresentedas Ohnuki,KashiwaProg.Theo.Phys.60(1978)548

D’Hoker,Gagne,hep-ph/9512080

andrewridenasthequantummechanicalpathintegral…

WR = �1

8

Tr log

⇣˜

⌃

2⌘=

1

8

Z 1

0Tr16 exp

⇣�"

2

T ˜

⌃

2⌘

Tr16 exp

⇣�"

2

T ˜

⌃

2⌘=

Zd4z d3✓ hz,�✓| exp

⇣�"

2

T ˜

⌃

2⌘|z, ✓i

Thevectorcurrentcanbedefinedas(puwngBμ=0)

hjVµ (y)i = �WR

�Aµ=

1

8

Z 1

0

dT

T

NZ

PBCDx

Z

APCD j

V,cl.µ

⇣e

�R T0 d⌧LL(⌧) + e

�R T0 d⌧LR(⌧)

⌘

j

V,cl.µ =

Z T

0d⌧ [" ⌫ µ@⌫ � xµ] �

(4) (x(⌧)� y) Cancheckthat @µ jV,cl.µ = 0

Phaseofthedeterminantandthechiralanomalyu  Thephaseofthecomplexdeterminantiswellknowntobetheoriginofthechiralanomaly

K.Fujikawa,PRL42(1979)1195;PRD21(1980)2848

u  Itstreatmentinaworldlinepathintegralframeworkisalsowellknown

L.Alvarez-Gaume,E.Widen,NPB234(1984)269A.M.Polyakov,Gaugefieldsandstrings(1987),sec5on6.3

u Wewilladoptadifferentregulariza5on(duetoD’Hoker&Gagne)andapplyittoderivethequan5tyofinterest

Mueller,Venugopalan,1701.03331&1702.01233

Phaseofthedeterminantandthechiralanomaly

with

UsingatrickduetoD’Hoker&Gagne,WIcanberewrideninaformverymuchlikethatfortherealpart…

wherethetraceinser5onis

Theparameterαbreakschiralsymmetryexplicitly-sewngitto±1restoresit

WI = �1

2arg det [⌦]

PhaseofthedeterminantandthechiralanomalyTheaxialvectorcurrentcanbeexpressedas

A�ersomealgebra:i)expressingaboveasapathintegral,ii)separa5ngzero&non-zeromodes(theΓ7inser5onmakesPBCandFermionzeromodesfeasibleinthepathintegralconstruc5on)iii)andfixingFock-Schwingergaugeaboutthezeromodes,onecanshowthat

@µhJ5µ(y)i =

1

8⇡2Tr

⇣Fµ⌫F

µ⌫⌘

whichisthewellknownanomalyequa5on

AnadvantageoftheD’Hoker-Gagnéconstruc5onisthattheaxialcurrenthasthesameworld-linestructureasitsvectorcounterpart

Seeourpaper1702.01233fordetails

hj5µ(y)i =i�WI

�Bµ(y)|B=0

Towardsachiralkine5ctheory:

II.Pseudo-classicalequa5onsofmo5on

Backtotherealpart:semi-classicalworld-linesEg.,G.Dunne,C.Schubert,hep-th/0507174

Consideroursimplercaseofscalarpar5clesinabackgroundfield

Rewriteexactlyas

for

Sta5onaryphase“world-lineinstanton”offunc5onalintegral

Equa5onofmo5onofscalarpar5cleinbackgroundAbelianfield

Spinning(&colored)pseudo-classicalworld-linesBrink,DiVecchia,Howe,NPB118(1977)76Balachandran,Salomonson,Skagerstam,Winnberg,PRD15(1978)2308Barducci,Casalbuoni,Lusanna,NPB124(1977)93

ForaconsistenttreatmentoftheHamiltoniandynamics,introduceLagrangemul5pliersinac5ontoimposephysicalconstraints

S =

Z T

0d⌧

⇢pµx

µ +i

2

h µ

µ + 5 5

i�H

�

with H ="

2

�⇡2 +m2 + i µFµ⌫

⌫�+

i

2(⇡µ

µ +m 5)�

Hereεandχarethevierbeinfieldsthatimposethemassshellandhelicityconstraintsofthetheory

Q=πμψμ+mψ5isasupersymmetricchargegenera5nganN=1SUSYalgebraCanonicalmomenta:

with

whereuμisthe“anomalous”velocity

Spinning(&colored)pseudo-classicalworld-linesPseudo-classicalequa5onsofmo5onforspinningpar5clesinthe(x,p,ψ)phasespace:

OnecanalsoobtainEOMforthePauli-Lubanskivector

⌃µ = � i

2✏µ⌫⇢�P

⌫ ⇢ �

⌃µ =g

m(Fµ⌫⌃

⌫ + @⌫Fµ⌫�5) �5 = 0 1 2 3

u  Forhomogeneousfields,thisisthecovariantformoftheBargmann-Michel-Telegdiequa5on

Sµ = ⌃µ/~

mR x

µ +i

2mR

↵@

µF↵�

� + F

µ⌫x⌫ = 0

µ � 1

mRFµ⌫

⌫ = 0 5 = 0

ExtensionofframeworktocoloredGrassmaniansgivestheWongequa5onsforprecessingcolorcharges

Barducci,Casalbuoni,Lusanna,NPB124(1977)93

Non-rela5vis5climit&BerryphaseCarefullytakingnon-rela5vis5climitv/c<<1oftheworld-lineac5on

Thomasprecessionterm Larmorterm

Intheadiaba5climit B · Sm

⇡ 0spinflipsaresuppressedandpar5clespinis“slaved”toitsmo5on

Transi5onamplitudefromini5altofinalstates:

T (pf , pi.+) = hpf , +(pf )|e�iH(tf�ti)|pi, +(pi)i

exp

✓i

Zdtp · A(p)

◆

wheretheBerryconnec5onisA(p) = �ih +(p)|rp| +(p)i

Note:Alsoholdsforlargeμ

Non-rela5vis5climit&Berryphase

Transi5onamplitudefromini5altofinalstates:

T (pf , pi.+) = hpf , +(pf )|e�iH(tf�ti)|pi, +(pi)i

Note:iden5calderiva5onifmassisreplacedbylargechemicalpoten5al

u  Notefurtherthattorecoverthisdynamicsonehastotakenon-rela5vis5c,adiaba5climitsoftherealpartoftheeffec5veac5on…u  Thechiralanomalyincontrastarisesfromtheimaginaryphaseandisindependentofanykinema5climits…

Berryconnec5onandchiralkine5ctheorySon,Yamamoto,…

Canonicalexample:twocomponentspinorsa5sfyingtheWeylequa5on

hasBerryconnec5on

andcurvature

Fic55ousmagne5cfieldassociatedwitha“magne5cmonopole”

SonandYamamotoconsidertheac5on D.Xiaoetal.,PRL95(2005)137204Duvaletal.,PLB20(2006)373

with

Rela5onofBerryphaseandanomaly?

Fujikawa’slament… hep-ph/0501166

and… hep-ph/0511142

Fujikawa’sexampleMagne5cmomentinexternalBfield: H = �µ~� ·B~B(t) = B (sin ✓ cos�(t), sin ✓ sin�(t), cos ✓)

Solu5onoftheSchrödingerequa5ongives

i~@t (t) = H (t)

±(T ) = w±(T ) exp

"�i

~

Z T

0dtw†

±(t)Hw±(t)

#exp

"�i

~

Z T

0A±( ~B)

d ~B

dtdt

#

Berry connection

Foronefullperiodofthemo5on, =

Fujikawa’sexample

Intheadiaba5climit: 2⇡

µBT! 0 thephaseisnontrivial

⌦± ! 2⇡(1⌥ cos ✓)

Nonadiaba5climit:2⇡

µBT! 1 thephaseistrivial⌦± = 2⇡(1⌥ 1)

Real5meSchwinger-Kelydyshformula5oninthefirstquan5zedformalism

S.Mathur,hep-th/9306090,hep-th/9311025

TowardstheanomalousBödekertheory

u  Inhep-ph/9910299,wedevelopedareal5meworld-lineformalismthatreproducedthenon-AbelianBoltzmann-LangevinframeworkforhotQCDdevelopedbyBödekertodescribesphalerondynamics

Bödeker,PLB426(1998)351;NPB559(2000)502Li5m,Manuel,NPB562(1999)237

u  Followingtheconstruc5ondevelopedhere,theframeworkcanbeextendedtoconstructacovariantquantumkine5cdescrip5onofthetransportofchiralfermionsinthepresenceoftopologicalfluctua5ons NiklasMueller,RV,YiYin,inprogress

Foranalterna5vetreatment,seeAkamatsu,Yamamoto,PRD90(2014)125031Akamatsu,Rothkopf,Yamamoto,JHEP1603(2016)210

Spinningworld-linesinothercontexts

u  Chiralfermiontransportinastrophysicalcontexts

u  Helicityevolu5oninQCDathighenergies

Charbonneau,Zhitnitsky,JCAP108(2010)010Akamatsu,Yamamoto,PRL111(2013)052002Kaplan,Reddy,Sen,arXiv:1612.00032Dvornikov,Semikoz,arXiv:1603.07946

Bartels,Ermolaev,Ryskin,Z.Phys.C72(1996)627Kovchegov,Pitonyak,Sievert,PRL118(2017)052001

Thankyouforyouraden5on!

QuantumvsClassicalcontribu5onstothePressureBerges,Boguslavski,Schlich5ng,Venugopalan,1508.03073

Rela5onofBerryphasetoanomaly

u AreadingoftheworkofStoneetal.suggeststhatthecontentofthechiralkine5cequa5onscanbeobtainedfromthecovariantBMTequa5onStone,Dwivedi,Zhou,PRD91(2015)025004

u Inourwork,thisarisesen5relyfromtherealpartoftheeffec5veac5on…