JHEP05(2021)209
Published o SISSA by Sp inge
Recei ed:Decembe 24, 2020
Re ised:Ma ch 8, 2021
Accep ed:May 2, 2021
Published:May 24, 2021
Chi al o sional e ec s in anomalous luids in he mal
equilib ium
Juan L. Mañes,aManuel Valleaand Miguel Á. Vázquez-Mozob
aDepa amen o de Física, Uni e sidad del País Vasco UPV/EHU,
Apa ado 644, 48080 Bilbao, Spain
bDepa amen o de Física Fundamen al, Uni e sidad de Salamanca,
Plaza de la Me ced s/n, 37008 Salamanca, Spain
E-mail: [email p o ec ed],[email p o ec ed],
[email p o ec ed]
Abs ac : Using he simila i y be ween space ime o sion and axial gauge couplings, we
s udy o sional con ibu ions o he equilib ium pa i ion unc ion in a s a iona y back-
g ound. In he case o a cha ged luid minimally coupled o o sion, we spo he exis ence
o linea o sional magne ic and o ical e ec s, while he axial- ec o cu en and he
spin ene gy po en ial do no ecei e co ec ions in he o sion a linea o de . The co a i-
an ene gy-momen um enso , on he o he hand, does con ain e ms linea in he o sion
enso . The case o a wo- la o had onic supe luid is also analyzed, and he o sional
con ibu ions o he cons i u i e ela ions compu ed. Ou esul s show he exis ence o a
o sional elec ic chi al e ec media ed by he cha ged pions.
Keywo ds: Anomalies in Field and S ing Theo ies, The mal Field Theo y
A Xi eP in : 2012.08449
Open Access,c
The Au ho s.
A icle unded by SCOAP3.h ps://doi.o g/10.1007/JHEP05(2021)209
JHEP05(2021)209
Con en s
1 In oduc ion 1
2 Equilib ium pa i ion unc ion and co a ian cu en s wi h backg ound
o sion 3
3 The spin ene gy po en ial and he ene gy-momen um enso 10
4 To sional chi al e ec s in a wo- la o had onic supe luid 14
5 Closing ema ks 18
A Basics o space ime o sion 19
B A summa y o exp essions om e . [20]21
1 In oduc ion
The coupling o hyd odynamic sys ems o ex e nal sou ces h ough anomalous cu en s
gi es ise o a a ie y o chi al anspo e ec s [1–8]. These ha e been shown o be
ele an o he unde s anding o di e se physical phenomena, anging om condensed
ma e o cosmology. Due o he opological na u e o anomalies, he e ec i e ac ion
o he long wa eleng h modes can be compu ed using di e en ial geome y echniques,
om which he pa i y- iola ing e ms in he luid cons i u i e ela ions can be de i ed [9–
20]. By placing he sys em on a cu ed backg ound s a iona y me ic, and pe o ming
dimensional educ ion on o he compac i ied Euclidean ime, i is possible o inco po a e
physical e ec s such as o ici y and accele a ion, which a e sou ced espec i ely by he
backg ound Kaluza-Klein (KK) gauge ield and he g adien o he ime componen o he
me ic. Phenomena linked o he exis ence o mixed gauge-g a i a ional anomalies [21]
ha e been subjec o di ec de ec ion in he labo a o y [22].
Despi e i s absence in s anda d gene al ela i i y, o sion has been he ocus o a en ion
in physics o almos a cen u y, since his geome ical no ion was i s in oduced in he
classical wo ks o Élie Ca an [23–25]. An ob ious mo i a ion o hese in es iga ions has
been he possibili y o ou space ime ha ing a small albei non anishing o sion, gi ing ise
o new physics (see [26,27] o a e iew o di e en physical scena ios).
Toge he wi h he p ospec s o spo ing undamen al mic oscopic o sion in high-ene gy
physics, o sional geome ies p o ide a p ac ical way o implemen ing physical e ec s in
condensed ma e physics. Focusing on he physics a la ge dis ances, he ec o s de ining
he links a each node o he ion la ice build up an e ec i e d eibein whose geome y models
la ice i egula i ies. Fo example, hei cu a u e and o sion espec i ely implemen
la ice disloca ions and disclina ions [28]. In sys ems wi h linea dispe sion ela ions, such
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JHEP05(2021)209
as he case o Weyl semime als, his geome y p o ides a s a ic nondynamical e ec i e
backg ound on which e mions p opaga e [29].
To sion is also known o ha e an in e play wi h chi al anomalies in quan um ield
heo y [30–33]. The axial anomaly ecei es a con ibu ion gi en by he so-called Nieh-Yan
e m [34,35], which comes om a bubble diag am wi h wo axial- ec o cu en inse ions
and is quad a ically di e gen . As a consequence, he coe icien o he Nieh-Yan e m
depends o he squa e o he cu o , o any o he ele an UV scale o he heo y [36]. The
ole o his e m in condensed ma e physics has been explo ed in a numbe o wo ks (see,
o example, [28,29,37–39]).
In he con ex o chi al luids, he in e es in o sion a ose in connec ion wi h he s udy
o Hall anspo [40–43] (see [44] o a comp ehensi e e iew). The ela ion be ween he
Hall iscosi y and he mean o bi al spin pe pa icle sugges ed a connec ion wi h he spin
cu en , which is sou ced by he backg ound o sion. In he ela i is ic se up, his link
was u he s udied in e . [45] by a i s p inciples calcula ion o he e ec i e ac ion and
cons i u i e ela ions o a e mion gas on a (2 + 1)-dimensional space ime wi h o sion.
The issue o o sional anspo e ec s in ou dimensions has also ecei ed some a -
en ion la ely [46]. Besides he sec o associa ed o he Nieh-Yan anomaly,1 he pa i ion
unc ion con ains o he con ibu ions which a e induced by he iangle diag ams associ-
a ed wi h he e ec i e axial- ec o ield encoding he an isymme ic pa o he o sion.
In e . [48], anspo phenomena induced by o sion we e s udied, bo h a ze o and ini e
empe a u e. The exis ence o a chi al magne ic and elec ic e ec s was ound, esul ing
om e mions minimally coupled o he an isymme ic pa o he o sion enso , which can
be ecas in e ms o i s dual ec o ield. This ex e nal sou ce couples o he e mionic sin-
gle axial- ec o cu en , which is a ec ed by a ’ Hoo anomaly. To sional con ibu ions
o spin anspo we e also ecen ly s udied in [49].
The e ec s associa ed wi h he ’ Hoo anomaly o he gauge ield dual o he an-
isymme ic pa o he o sion can be eadily compu ed using he s anda d di e en ial
geome y me hods employed in he analysis o anomalous luids. In he p esen wo k, we
apply he echniques de eloped in [19,20] o ca y ou a s udy o he linea e ec s o o -
sion in hyd odynamics, wi h and wi hou Nambu-Golds one bosons. Fo a cha ged luid
coupled o an ex e nal elec omagne ic ield, we e i y he exis ence o o sional magne ic
and o ical e ec s. The axial- ec o cu en , on he o he hand, does no con ain any co -
ec ions linea in he o sion. This is also he case o he co a ian spin ene gy po en ial,
which is w i en in e ms o he co a ian axial- ec o cu en . The componen s o he
co a ian ene gy-momen um enso can be also exp essed in e ms o he co a ian axial-
ec o cu en s, bu in his case he coe icien s depend linea ly on he o sion enso . Once
w i en in e ms o he o sion, hey gi e ise o new o sion-induced con ibu ions o he
cons i u i e ela ions om whe e he co esponding anspo coe icien s can be ob ained.
A e analyzing he Abelian case, we ocus ou a en ion on he case o a wo- la o
had onic supe luid in he p esence o o sion, in he phase in which chi al symme y
1I has ecen ly been p oposed, howe e , ha he e is no genuine o sional chi al dissipa ionless anspo
in ha sec o [47].
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JHEP05(2021)209
U(2)L×U(2)Ris spon aneously b oken o i s ec o subg oups. We compu e he co ec ions
o he co a ian cu en s and anspo coe icien s linea in he o sion enso and ind he
exis ence o a o sional chi al elec ic e ec media ed by he wo cha ged pions. The chi al
sepa a ion e ec s ound in [20], on he o he hand, do no ecei e any con ibu ions linea
in he o sion.
The p esen a icle is o ganized as ollows. Sec ion 2is de o ed o he analysis o he
equilib ium pa i ion unc ion o a cha ged plasma in he p esence o o sion, including
he compu a ion o he co a ian cu en s. This models is u he elabo a ed in sec ion 3
wi h he calcula ion o linea o sional con ibu ions o he spin ene gy po en ial and he
ene gy-momen um enso . In sec ion 4, a e a b ie discussion o he linea coupling o
Nambu-Golds one bosons o o sion in he Abelian case, we compu e he linea o sional
co ec ions o he cons i u i e ela ions o a wo- la o had onic supe luid. Finally, ou
indings a e summa ized in sec ion 5. To make ou p esen a ion mo e sel -con ained, we
e iew in appendix Asome basic ac s abou geome ic o sion, while in appendix Bwe
lis some exp essions o e . [20] ele an o ou discussion.
2 Equilib ium pa i ion unc ion and co a ian cu en s wi h backg ound
o sion
We begin wi h he discussion o he dynamics o massless Di ac e mions p opaga ing on
a space ime wi h o sion.2The ac ion o a massless Di ac spino minimally coupled o
g a i y can be w i en as [26,50,51]
S=1
2Zd4x(de e)ψ−→
∇/ ψ −ψ←−
∇/ ψ,(2.1)
whe e he le and igh co a ian de i a i es inside he in eg al a e de ined espec i ely by
−→
∇/ ψ =γAeµ
A∂µψ+1
4γAγ[BγC]ωBCAψ,
ψ←−
∇/=eµ
A∂µψγA−1
4ψγ[BγC]γAωBCA,(2.2)
and he Di ac ma ices e i y he Minkowskian Cli o d algeb a {γA, γB}= 2ηAB1. W i ing
he ull spin connec ion in e ms o he auxilia y o sionless Le i-Ci i a connec ion and he
con o sion enso as ωA
BC =ωA
BC +κA
BC, we ge he ollowing exp ession o he le
co a ian de i a i e in e ms o i s Le i-Ci i a coun e pa
−→
∇/ ψ =−→
∇/ ψ +1
4γCγ[AγB]κABCψ
=−→
∇/ ψ +1
4γBηAC −γAηBC +iABCDγDγ5κABC ψ, (2.3)
whe e we indica e by a ba all geome ic quan i ies e e ed o he Le i-Ci i a connec ion
and ha e used he gamma ma ices iden i y
γAγ[BγC]=γCηAB −γBηAC +iABCDγDγ5.(2.4)
2The basics o geome ic o sion, as well as he no a ion used in he ollowing, a e summa ized in
appendix A.
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JHEP05(2021)209
A simila calcula ion o he igh co a ian de i a i e in eq. (2.2) gi es
ψ←−
∇/=ψ←−
∇/−1
4ψγ[AγB]γCκACB
=ψ←−
∇/−1
4ψγAηBC −γBηAC +iABCDγDγ5κABC .(2.5)
Plugging hese esul s in o he ac ion (2.1), we a i e a he exp ession
S=1
2Zd4x(de e)ψ−→
∇/ ψ +1
4ψγBηAC −γAηBC +iABCDγDγ5ψκABC
−ψ←−
∇/ ψ +1
4ψγAηBC −γBηAC +iABCDγDγ5ψκABC .(2.6)
The impo an poin he e is ha he e m p opo ional o (γBηAC −γAηBC )κABC , which
con ains he symme ic componen s o he con o sion in he wo las indices, cancels ou .
This means ha e mions only couple o i s an isymme ic piece, κA[BC], which as shown
in appendix A[see eq. (A.9)] is gi en by he componen s o he o sion enso
S=1
2Zd4x(de e)ψ−→
∇/ ψ −ψ←−
∇/ ψ +i
4ψγDγ5ψ BC
DA TABC .(2.7)
This o m o he ac ion sugges s he in oduc ion o he e ec i e ec o ield
SA=−1
8CD
AB TBCD,(2.8)
o w i e
S=1
2Zd4x(de e)ψ−→
∇/ ψ −ψ←−
∇/ ψ −2iψγAγ5ψSA.(2.9)
Thus, he whole e ec o backg ound o sion on he dynamics o he e mion is codi-
ied h ough i s axial- ec o coupling o an ex e nal e ec i e gauge ield, which, ollowing
e . [48], we call sc ew o sion. The minimal coupling o g a i y selec s only one among all
possible dimension- ou ope a o s coupling Di ac e mions o o sion [52].
Be o e p oceeding any u he , some cla i ica ion on he ac ion (2.1) is in o de . A
ace alue, he heo y i desc ibes seems o be equi alen o ha o a Di ac e mion axially
coupled o an ex e nal gauge ield. The c ucial di e ence, howe e , is ha his ex e nal
gauge ield Sis a “composi e” exp essed as he Hodge dual o he an isymme ic pa o
he o sion componen s, which is he “ undamen al” ex e nal sou ce. This is impo an ,
because once exp essed in a coo dina e basis he componen s o his gauge ield depend no
only on he o sion, bu on he me ic enso as well. As a esul , he ene gy-momen um
enso and he spin ene gy po en ial o he heo y include con ibu ions ha would be
absen in he heo y o a “ undamen al” gauge ield axially coupled o a Di ac e mion
(see sec ion 3). These new e ms a e associa ed wi h no el anspo coe icien s in he
cons i u i e ela ions o he co esponding ene gy-momen um and spin co a ian cu en s.
In addi ion o he couplings shown in eq. (2.1), he au ho s o e . [48] conside ed an
addi ional coupling o he Di ac e mion o an ex e nal Abelian ec o gauge ield. This
so-called edge o sion ec o ield is p opo ional o he o sion ec o TB
BA and mixes o
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JHEP05(2021)209
many p ac ical pu poses wi h he elec omagne ic ield. In wha ollows we s ick o he
minimal coupling p esc ip ion (2.1) and only conside he coupling o he sc ew o sion, in
he unde s anding ha in all ou esul s he edge o sion would be eabso bed by a shi
in he physical elec omagne ic ield.
Rema ks on gauge in a iance. We ha e seen how he ac ion o a Di ac e mion mini-
mally coupled o g a i y only depends on he ully an isymme ic componen s o he o sion
enso TABC ≡T[ABC], which can be used o de ine he h ee- o m3
T=1
3!TABCeAeBeC.(2.10)
I s ou independen componen s a e encoded in he sc ew- o sion acco ding o eq. (2.8),
which can be w i en using he Hodge s a ope a o as
Sµ=−1
8αβ
µν Tναβ =⇒ S =3
4?T.(2.11)
The ield s eng h o he Abelian sc ew o sion, FS=dS, can be w i en hen as
FS=3
4d ? T=−3
4? δT,(2.12)
whe e δ≡ − ? d? deno es he codi e en ial ac ing on a h ee- o m. In componen s, his
equa ion eads
Sµν =−3
8∇σTσαβαβµν.(2.13)
Looking a eq. (2.11) abo e, we see ha he gauge a ia ion o he sc ew o sion ec o
ield Sby an exac one- o m, S → S +dα, co esponds o he ollowing ans o ma ion o
he o sion h ee- o m T
T −→ T +δβ, (2.14)
wi h β∼?α a ou - o m. The nihilpo ency o he codi e en ial, δ2= 0, gua an ees he
gauge in a iance o he sc ew o sion ield s eng h (2.12).
The equilib ium pa i ion unc ion. In he con ex o hyd odynamics, he coupling o
he backg ound o sion o he mic oscopic e mionic deg ees o eedom gi es he p esc ip-
ion o he cons uc ion o he e ec i e unc ional desc ibing he long- ange exci a ions o
a luid [9,10]. He e we a e going o employ he di e en ial geome y me hods in oduced
in [19,20] o build he equilib ium pa i ion unc ion o luids wi h o sion. In he ol-
lowing, we s udy he case o a luid coupled o an ex e nal Abelian ec o sou ce in he
p esence o backg ound o sion on a gene ic s a ic backg ound geome y. A e his, he
mo e gene al case o a wo la o had onic (supe ) luid will be analyzed in sec ion 4.
Besides i s coupling o o sion h ough S, we also assume ha he mic oscopic e mionic
deg ees o eedom a e coupled o an ex e nal ec o Abelian gauge ield V, which emains
3Unlike in e s. [19,20], he e no −iis ac o ed ou o he componen s o di e en ial o ms.
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JHEP05(2021)209
anomaly- ee and will be e en ually associa ed o he elec omagne ic ield. In ou dimen-
sions, he (nonlocal) anomalous pa o he e ec i e ac ion can be compu ed in e ms o
he Che n-Simons o m (see, o example, [19]). Keeping only e ms linea in he o sion,
we ha e
e
ω0
5(S,FV,FS)=6SF2
V,(2.15)
whe e FV=dVis he ec o ield s eng h. This gi es he Ba deen o m o he anomaly,
which explici ly p ese es ec o gauge ans o ma ions. The p ope ly no malized Che n-
Simons nonlocal e ec i e ac ion encoding he linea e ec s o o sion is hen gi en by
Γ[V,S]CS =1
4π2Z
D5
SF2
V,(2.16)
whe e D5is a i e-dimensional mani old whose bounda y is iden i ied wi h he Euclidean
ou -dimensional physical space ime. To compu e he equilib ium pa i ion unc ion om
he Che n-Simons e ec i e ac ion, we ake he me ic o he ou -dimensional space ime
∂D5 o be he gene ic s a ic line elemen
ds2=−e2σ(x)hdx0+ai(x)dxii2+gij(x)dxidxj,(2.17)
and ake all ields o be independen o x0. We implemen dimensional educ ion on o he
compa i ied Euclidean ime by se ing D5=S1×D4, whe e he leng h o he S1equals
he in e se o he equilib ium empe a u e T0. Vec o ields a e hen w i en in e ms
o componen s ha emain in a ian unde KK ans o ma ions [20], ac ing acco ding o
x0→x0+φand ai→ai−∂iφ,
V ≡ Vµdxµ=V−e−σV0u,
S ≡ Sµdxµ=S−e−σS0u, (2.18)
whe e we ha e in oduced he ou - eloci y one- o m ugi en by
u=−eσdx0+aidxi≡ −eσdx0+a,(2.19)
and he KK-in a ian spa ial one- o ms a e de ined by
V=Vi−V0aidxi≡Vidxi,
S=Si−S0aidxi≡Sidxi.(2.20)
A simila elec ic-magne ic decomposi ion can be w i en o he ec o ield s eng h
FV≡B+uE
=dV−de−σuV0+ue−σdV0(2.21)
=FV+V0da+ue−σdV0,
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JHEP05(2021)209
whe e FV≡dVand we ha e used ha d(e−σu) = −da. The elec ic Eand magne ic
Bcomponen s o he ield s eng h will be la e iden i ied wi h he elec ic and magne ic
ields [c . (2.36)]. The equi alen exp ession o he ield s eng h associa ed o he sc ew
o sion eads
FS≡BS+uES
=FS+S0da+ue−σdS0,(2.22)
wi h FS=dS.
Finally, we implemen he dimensional educ ion on he h ee- o m (2.10) as well by
decomposing i as
T=TB+uTE.(2.23)
In a coo dina e basis, he elec ic and magne ic componen s a e espec i ely gi en by
TE=1
3!e−σh2gi`T`0j+e2σT0
ij −2aiT0
0j+e2σa`T`ij −2aiT`0jidxjdxk,
TB=1
3!gj`T`kn −2akT`0ndxjdxkdxn.(2.24)
Being a ou - o m, he gauge unc ion βin eq. (2.14) does no ha e any magne ic compo-
nen , β=uβE. As a consequence, only he elec ic pa o T ans o ms unde (2.14)
TE−→ TE+δ⊥βE,
TB−→ TB,(2.25)
whe e δ⊥=∗d∗, wi h ∗ he h ee-dimensional Hodge dual (no o be con used wi h i s
ou -dimensional cou e pa deno ed by ?). Since unde ou -dimensional Hodge duali y
he elec ic and magne ic componen s in e change
?T=−∗TE−u∗TB,(2.26)
we ind om eqs. (2.11) and (2.18)
S0=3
4eσ∗TB,
S=−3
4∗TE.(2.27)
We see ha he gauge in a iance o TBimplies he same p ope y o S0, whe eas S
unde goes he s anda d gauge ans o ma ion gene a ed by he ze o- o m ∗βE. Using in
addi ion eq. (2.24), we can w i e he sc ew o sion ield in e ms o he componen s o he
o sion enso as
S0=1
8eσijkgi`T`jk −2ajT`0k,(2.28)
S=−1
8e−σijkgmih2g`jT`0k+e2σa`T`jk −2ajT`0k+e2σT0
jk −2ajT0
0kidxm.
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JHEP05(2021)209
This dependence o he sc ew o sion gauge ield on he me ic and o sion componen s
is wha dis inguishes ou heo y om ha o a Di ac e mion coupled o an ex e nal
gauge ield h ough he axial- ec o cu en . As al eady poin ed ou , his has impo an
consequences o he cons i u i e ela ions o he ene gy-momen um and spin cu en s.
Ha ing a i ed a his pa ame iza ion o he e ec i e sc ew o sion, we p oceed o
compu e he e ms in he e ec i e ac ion induced by he ’ Hoo anomaly a ec ing he
gauge in a iance (2.25) and coming om iangle diag ams.4As shown in [11,19], he
dimensionally- educed e ec i e ac ion spli s in o a local anomalous and a nonlocal in a ian
piece, espec i ely gi en by [19]
W[V0,S0,V,S, da]anom =1
4π2T0Z
S32V0FV+daV2
0S,(2.29)
and
W[V0,S0,V,S, da]in =1
4π2T0Z
D4
hS0F2
V+ 2V0FVFS(2.30)
+daV0V0FS+ 2S0FV+ (da)2S0V2
0i.
In he second exp ession, we ha e in oduced he componen s o he ield s eng h associ-
a ed wi h S
FS=dS≡1
2Sijdxidxj.(2.31)
The co a ian cu en s can be now compu ed om he in a ian pa o he pa i ion
unc ion [11,19,20]. We begin wi h he one associa ed wi h he ec o cu en
hJVico =T0
δ
δFV
W[V0,S0,V,S]in
=1
2π2hS0FV+daV0+V0FSi,(2.32)
whe eas o he one co esponding o he o sional axial- ec o gauge ield, he esul is
hJSico =T0
δ
δFS
W[V0,S0,V,S]in
=1
2π2V0FV+1
2daV2
0.(2.33)
In bo h cases, he ze o componen s anish, hJV0ico =hJS0ico = 0. We obse e ha ,
unlike he ec o cu en , hJSico does no pick any linea dependence on he o sion.
These co a ian cu en s will be e y ele an in he ollowing.
In addi ion, he co esponding Ba deen-Zumino (BZ) cu en s a e gi en by [19]
hJViBZ =1
2π2SFV,
hJSiBZ =1
6π2SFS.(2.34)
4As s a ed abo e, in his wo k we do no conside he sec o associa ed wi h he Nieh-Yan anomaly.
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JHEP05(2021)209
end o he calcula ion, he linea couplings o he single Nambu-Golds one boson αin he
local WZW ac ion akes a pa icula ly simple o m [19]
W[α, . . .]WZW ≡W[. . . , A+S, . . .]anom −W[. . . , A+S+dα, . . .]anomA=0
=. . . −1
6π2T0Z
S3A0FS+S0dadα, (4.1)
whe e he ellipsis in he second line indica es he o sion-independen coupling o he
Nambu-Golds one boson o he backg ound ields V0,V, and A0. We see ha he Nambu-
Golds one boson couples linea ly o BS, he magne ic componen o he sc ew o sion ield
s eng h de ined in eq. (2.22). Exp essed in componen s, he ele an e m in he WZW
ac ion eads
W[α, . . .]WZW =. . . −Z
S3
d3x√gµ5
6π2TBi
S∂iα, (4.2)
whe e Bi
Swas de ined in eq. (2.37) and we also in oduced he local empe a u e T=e−σT0
and he chi al chemical po en ial µ5=e−σA0.
A e his b ie discussion o he Abelian case, we u n ou a en ion o he analysis o
o sional chi al e ec s in he wo- la o had onic supe luid s udied in e . [20], whe e we
e e he eade o de ails. This luid couples o a ec o and an axial- ec o ex e nal gauge
ields, espec i ely deno ed by Vand A, ans o ming in he la o g oup U(2)L×U(2)R.
The alues o hese backg ound ields a e es ic ed o he Ca an subalgeb a gene a ed by
0=1
21, 3=1
2σ3.(4.3)
We assume he sys em unde goes spon aneous symme y b eaking o i s ec o subg oups,
U(2)L×U(2)R→U(1)V×SU(2)V, gene a ing in he p ocess a iple o Nambu-Golds one
bosons π0, π±encoded in he ma ix
U= exp
i√2
π
1
√2π0π+
π−−1
√2π0
.(4.4)
The i s issue o add ess is how o inco po a e he o sional ec o ields in o ou
analysis. The le Di ac ope a o , including he ec o , axial, and sc ew- o sion ields,
akes he o m
−→
D/ ψ =−→
∇/−iV/−iA/ γ5−iS/1γ5ψ
=n−→
∇/−iV/0 0+V/3 3−ihA/0+ 2 S/ 0+A/3 3iγ5oψ, (4.5)
and simila ly o he igh ope a o . The s uc u e o hese e ms shows ha , o ake
in o accoun he e ec o o sion in he analysis o e . [20], i is enough o implemen he
eplacemen
A0µ−→ A0µ+ 2Sµ,(4.6)
while lea ing all emaining ields unchanged.
– 15 –
JHEP05(2021)209
Wi h all his in mind, we can compu e he linea o sional co ec ions o he co a ian
gauge cu en s a leading o de in he de i a i e expansion. To a oid cumbe some exp es-
sions, he e we only gi e he e ms in he cu en s depending linea ly on he o sion, ha
we deno e by h∆Jµ
aV ico and h∆Jµ
aAico . These should be added o he exp essions ound
in e . [20] o he co esponding cu en s. The esul s a e
h∆Jµ
0Vico =−Nc
8π2µναβSνV0αβ,
h∆Jµ
3Vico =−Nc
24π2µναβhH+ 3SνV3αβ −∂αSνTβi,
h∆Jµ
0Aico =−Nc
24π2µναβSνA0αβ + 2A0ν∂αSβ,(4.7)
h∆Jµ
3Aico =Nc
24π2µναβ∂νSαIβ,
whe e we ha e used he enso s uc u es in oduced in [20]
H≡T hU−1QU −QQi,
Iµ≡T hRµ+LµQi,(4.8)
Tµ≡T hQRµ−Lµi+ 2V3µT hU−1QU −QQi,
wi h Rµ=iU−1∂µUand Lµ=i∂µUU−1, while he cha ge ma ix is gi en by
Q=1
3 0+ 3.(4.9)
The i s hing o be no iced he e is ha he o sional couplings o he pions a e es ic ed
o he h ee- la o componen s o he ec o and axial- ec o co a ian cu en s.
In o de o compu e he longi udinal and ans e se componen s o hese cu en s, and
w i e he cons i u i e ela ions o he had onic supe luid, we need o in oduce a numbe
o scala and ec o s uc u es, in addi ion o hose used in e . [20].7They ep esen he
linea coupling o he Nambu-Golds one bosons o o sion. To he i e scala ones, we add
S6=µναβuµ∂νSαIβ,
S7=µναβuµ∂νSαTβ,(4.10)
while o he ec o s uc u es we ex end he no a ion Pµ
1,a and Pµ
3,a o include a=S, and
add a new e m Pµ
5
Pµ
1,S =µναβuνIα∂βµS
T,
Pµ
3,S =µναβuνTα∂βµS
T,(4.11)
Pµ
5=µναβuνSα∂βH.
7Fo he eade ’s con enience, hese a e summa ized in appendix B.
– 16 –
JHEP05(2021)209
He e we ha e used he sc ew o sion chemical po en ial (2.39), as well as he local empe -
a u e T=e−σT0.
The longi udinal and ans e se componen s o he co a ian ec o and axial- ec o
cu en s a e now w i en in e ms o hese quan i ies. The new o sional nondissipa i e
chi al anspo coe icien s can be ead om he esul ing exp essions. We s a wi h he
0- la o ec o cu en
uµh∆Jµ
0Vico =Nc
4π2SµBµ
0,
Pµ
σh∆Jσ
0Vico =Nc
4π2µSBµ
0+TµναβuνSα∂βµ0
T,(4.12)
whe e we ha e in oduced he magne ic ield
Bµ
a=1
2µναβuνVaαβ a= 0,3.(4.13)
The longi udinal and ans e se componen s o he 3- la o ec o cu en , on he o he
hand, ead
uµh∆Jµ
3Vico =Nc
24π2h2H+ 3SµBµ
3−S7i,
Pµ
σh∆Jσ
3Vico =Nc
24π22H+ 3µSBµ
3+H+ 6TµναβuνSα∂βµ3
T
−µSPµ
4+TPµ
3,S +µ3Hµναβuν∂αSβ−µ3Pµ
5.(4.14)
In he case o he axial- ec o cu en s, we ha e
uµh∆Jµ
0Aico =Ncµ5
6π2Sµωµ,
Pµ
σh∆Jσ
0Aico =Ncµ5
12π2−2µSωµ+µναβuν∂αSβ,(4.15)
o he 0- la o componen s, whe eas in he case o he 3- la o he esul is
uµh∆Jµ
3Aico =Nc
24π2S6,
Pµ
σh∆Jσ
3Aico =−Nc
24π2µSPµ
2−TPµ
1,S.(4.16)
Le us s ess once mo e ha all he exp essions gi en he e only ep esen he linea o sional
con ibu ions o he longi udinal and ans e se componen s o he co a ian cu en s, ha
should be added o he espec i e non o sional e ms ound in [20]. As al eady poin ed
ou , o sion couples o Nambu-Golds one bosons only h ough he 3- la o cu en s. The
o sional con ibu ions o he 0- la o co a ian ec o and axial- ec o cu en s a e jus
gi en by he co esponding e ms o he BZ cu en s linea in he o sion. This implies
ha he e a e no linea o sional con ibu ions o he 0- la o componen o he consis en
cu en s and, as a consequence, no linea o sional e ms in he WZW ac ion depending
on V0io A0i.
– 17 –
JHEP05(2021)209
To ind he exp ession o he co a ian cu en s in e ms o he physical elec omagne ic
ields, we expand he KK-in a ian componen s o he ec o ield in e ms o he cha ge
ma ix Qde ined in (4.9) and he gene a o 3acco ding o (see [20])
V0µ 0+V3µ 3= 3V0µQ+V3µ−3V0µ 3.(4.17)
As usual, he unb oken U(1)V ac o is iden i ied as he one coupling o he Qma ix,
whe eas he ield coupling o 3is se o ze o, which implies Vµ≡3V0µ=V3µ. We w i e
now he o sional e ms in he co a ian elec omagne ic cu en
hJµ
emico =e
3hJµ
0Vico +ehJµ
3Vico ,(4.18)
in e ms o he pion ields as
h∆Ji
emico =−e2Nc
12π2 2
π
ijkSjEkπ+π−−µe2Nc
12π2 2
π
Bi
Sπ+π−
−ieNc
12π2 2
π
Tijk∂kµS
Tπ+∂jπ−−π−∂jπ+−e2Nc
6π2 2
π
TijkVj∂kµS
Tπ+π−
+µe2Nc
12π2 2
π
ijkSj∂kπ+π−+5e2Nc
18π2µSBi+ijkSjEk+O(π3),(4.19)
whe e he elec ic and magne ic ields a e de ined in eq. (2.36) and Bi
Sis gi en in (2.37), all
ields he e being KK-in a ian . The las , pion-independen e m is he BZ elec omagne ic
cu en o he unb oken heo y, and eplica es he s uc u e ound in eq. (2.41) o he
Abelian case. We see ha he e is no o sion-media ed elec omagne ic coupling o he
neu al pion. The e exi s none heless a o sional pion-dependen con ibu ion o he chi al
elec ic e ec gi en by he i s e m in eq. (4.19), his ime induced by he T-odd spa ial
sc ew- o sion ield.
As o he ans e se axial- ec o cu en s, we see ha he only coupling o o sion o
pions a ises om he 3- la o componen
h∆Ji
3Aico =−Nc
12π2 π
Tijk∂jπ0∂kµS
T+O(π3).(4.20)
The e is he e o e no o sional co ec ions o he pion-media ed chi al elec ic, magne ic,
and o ical sepa a ion e ec s ound in [20]. In e s ingly, he coupling showed in eq. (4.20)
is he only o sion-induced e m in ol ing he neu al pion in he cons i u i e ela ions a
his o de .
5 Closing ema ks
In his pape we ha e analyzed he linea e ec o backg ound o sion in he pa i ion unc-
ion o a cha ged luid minimally coupled o g a i y. The e ms s udied a e hose induced
by he ’ Hoo anomaly associa ed wi h he sc ew o sion, dual o he an isymme ic pa
o he o sion enso . In he Abelian case, ou esul s show he exis ence o magne ic and
o ical chi al o sional e ec s, whe eas he axial- ec o cu en does no ha e any linea
o sional co ec ions.
– 18 –
JHEP05(2021)209
In his same model, he co a ian spin ene gy po en ial and ene gy-momen um enso
ha e been compu ed in e ms o he axial- ec o co a ian cu en , wi h coe icien s ha
only depend on he me ic unc ions. Since he axial- ec o cu en has been shown no
o depend on he o sion a linea o de , we conclude ha he spin ene gy po en ial does
no exhibi linea o sional con ibu ions. The si ua ion is qui e di e en in he case o he
co a ian ene gy-momen um enso . Al hough i s componen s a e also w i en in e ms o
he co a ian axial- ec o cu en s, he coe icien s now do depend linea ly on he o sion
enso . Thus, we ind linea o sional co ec ions o he ene gy-momen um enso , which
ac ually include componen s o he o sion enso ha do no appea in he e ec i e ac ion.
This la e si ua ion is analogous o he one al eady ound in 2 + 1 dimension [45]. I is
impo an o s ess ha he o sional con ibu ions o he ene gy-momen um enso and
he spin ene gy po en ial a e associa ed wi h he implici dependence o he e ec i e axial-
ec o gauge ield on bo h he me ic and he o sion componen s. This is wha makes he
o sional heo y genuinely di e en om he heo y o a Di ac e mion axially coupled o
an ex e nal gauge ield, as i is e lec ed in he cons i u i e ela ions.
We ha e also s udied linea o sional chi al e ec s in a wo- la o had onic supe luids
s udied in e . [20]. We ound ha no new couplings o he Nambu-Golds one bosons o
o sion eme ge om he 0- la o componen s o he co a ian ec o and axial- ec o cu -
en s. The analysis o he elec omagne ic ans e se cu en , on he o he hand, shows he
exis ence o o sional chi al elec ic e ec media ed by he wo cha ged Nambu-Golds one
bosons π±, whe eas no o sional o ical e ec appea s. In e es ingly, he e a e no o sional
co ec ions o he pion-media ed elec ic, magne ic, and o ical chi al sepa a ion e ec s
ha we e ound in [20].
In his pape we ha e exploi ed he analogy be ween backg ound o sion and axial-
ec o couplings o compu e he linea e ec s o o sion, which come om iangle diag ams
wi h an axial- ec o cu en coupled o he backg ound sc ew o sion ield. A di e en
sec o is he one associa ed wi h he Nieh-Yan anomaly [34,35], which explici ly depends
on he UV cu o scale o he heo y. This Nieh-Yan e m has been shown o be ele an
in condensed ma e , whe e his cu o a ises na u ally. I would be in e es ing o u he
explo e he physical implica ions o his anomaly along he lines ollowed in he p esen
pape o he iangle con ibu ions. This issue will be add essed elsewhe e.
Acknowledgmen s
We hank Ka l Lands eine o discussions. This wo k has been suppo ed by Span-
ish Science Minis y g an s PGC2018-094626-B-C21 (MCIU/AEI/FEDER, EU) and
PGC2018-094626-B-C22 (MCIU/AEI/FEDER, EU), as well as by Basque Go e nmen
g an IT979-16.
A Basics o space ime o sion
In his appendix, we gi e a b ie o e iew o he main ma hema ical ea u es o space imes
wi h o sion. The ocus will lie on he basic di e en ial geome ic aspec s, wi h u he
– 19 –
JHEP05(2021)209
de ails being a ailable in a numbe o e iews (see, o example, [26,27,50,51,54,55]).
Le us conside a ou -dimensional cu ed mani old and an o hono mal e ad basis {eA=
eA
µdxµ}
ηABeA
µeB
ν=gµν,(A.1)
wi h ηAB he la Lo en z me ic and gµν he space ime me ic.8The spin connec ion ωA
B
de ines he no ion o pa allel anspo , allowing he cons uc ion o he co a ian de i a i e
ope a o , ha in he pa icula case o p- o m enso o he ype τA
B akes he o m
∇τA
B=dτA
B+ωA
CτC
B+ (−1)p+1τA
CωC
B.(A.2)
In wha ollows, we assume he connec ion ωA
BC o be me ic compa ible, ∇ηAB = 0.
To sion is de ined by he i s Ca an s uc u e equa ion
TA=deA+ωA
BeB,(A.3)
while he second one gi es he cu a u e
RA
B=dωA
B+ωA
CωC
B.(A.4)
Bo h o sion and cu a u e a e geome ical quan i ies ela ed o he beha io o ec o s
unde (in ini esimal) pa allel anspo .
The o sion wo- o m can be expanded in he e ad basis as
TA=1
2TA
BCeBeC,(A.5)
whe e he componen s on he igh -hand side a e an isymme ic in he lowe wo indices,
TA
(BC)= 0. To pa ame ize o sion, i is con enien o in oduce an auxilia y o sionless
connec ion ωA
Bassocia ed wi h he same e ad basis eAand sa is ying
deA+ωA
BeB= 0.(A.6)
This auxilia y connec ion is also me ic compa ible, ∇ηAB = 0, whe e he e and elsewhe e
in he pape we indica e all quan i ies associa ed wi h his Le i-Ci i a connec ion by an
o e line. The con o sion one- o m is de ined by
κA
B≡ωA
B−ωA
B,(A.7)
which, being he di e ence o wo connec ions, ans o ms as a enso unde local Lo en z
ans o ma ions. Combining eqs. (A.3) and (A.6), we w i e he o sion wo- o m TAin
e ms o he con o ion one- o m as
TA=κABeB=−κA
BCeBeC,(A.8)
8In his wo k, Lo en z indices a e deno ed by capi al La in le e s, while space ime indices a e indica ed
by G eek le e s. Lowe case La in indices a e ese ed o spa ial componen s. To make no a ion ligh e ,
we omi he wedge (∧) o deno e ex e io p oduc s.
– 20 –
JHEP05(2021)209
whe e in he second equali y we ha e expanded κA
B=κA
BCeC. This iden i y shows ha
he an isymme ic pa o he con o sion in he wo lowe indices is de e mined by he
componen s o he o sion enso
κA
[BC]=−1
2TABC .(A.9)
Using me ic compa ibili y, he symme ic piece can be compu ed in e ms o he o sion
componen s as
κA
(BC)=1
2TA
B C +1
2TA
C B.(A.10)
Simila exp essions a e ob ained using a coo dina e basis, wi h he o sion being iden i ied
wi h he an isymme ic pa o he connec ion acco ding o
Tµνσ =−2Γµ
[νσ]=−2κµ
[νσ].(A.11)
B A summa y o exp essions om e . [20]
Fo he sake o comple eness, we lis in his appendix he scala and enso s uc u es
in oduced in e . [20] o w i e he cons i u i e ela ions o he wo- la o chi al had onic
supe luid s udied in sec ion 4. In he case o he longi udinal componen s o he cu en s,
hese a e exp essed in e ms o he ollowing i e scala s uc u es
S1,a ≡µναβIµuν∂αVaβ =IµBµ
a(a= 0,3),
S2≡1
2µναβIµuν∂αuβ=Iµωµ,
S3≡µναβuµV3ν∂αIβ−i
3T LνLαLβ,(B.1)
S4,a ≡µναβTµuν∂αVaβ =TµBµ
a(a= 0,3),
S5≡1
2µναβTµuν∂αuβ=Tµωµ,
whe e he magne ic ield is de ined in eq. (4.13), and Iµand Tµa e gi en in eq. (4.8),
whose expansions in e ms o he pion ields a e gi en by
H=−2
2
π
π+π−+O(π4),
Iµ=−2
π
∂µπ0+O(π3),(B.2)
Tµ=2i
2
ππ+∂µπ−−π−∂µπ+−4
2
π
π+π−V3µ+O(π3).
Finally, he ans e se componen s o he co a ian cu en s ound in [20] a e exp essed in
e ms o he ou enso s uc u es
Pµ
1,a ≡µναβuνIα∂βµa
T(a= 0,3),
Pµ
2≡µναβuν∂αIβ,
Pµ
3,a ≡µναβuνTα∂βµa
T(a= 0,3),(B.3)
Pµ
4≡µναβuν∂αTβ.
– 21 –
JHEP05(2021)209
These exp essions ha e been w i en in e ms o he chemical po en ials
µa=e−σVa0.(B.4)
Open Access. This a icle is dis ibu ed unde he e ms o he C ea i e Commons
A ibu ion License (CC-BY 4.0), which pe mi s any use, dis ibu ion and ep oduc ion in
any medium, p o ided he o iginal au ho (s) and sou ce a e c edi ed.
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