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…
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