symme y
S
S
Re iew
Axial UA(1)Anomaly: A New Mechanism o Gene a e
Massless Bosons
Vicen e Azcoi i


Ci a ion: Azcoi i, V. Axial UA(1)
Anomaly: A New Mechanism o
Gene a e Massless Bosons. Symme y
2021,13, 209. h ps://doi.o g/
10.3390/sym13020209
Academic Edi o : Angel Gómez
Nicola
Recei ed: 30 Decembe 2020
Accep ed: 25 Janua y 2021
Published: 28 Janua y 2021
Publishe ’s No e: MDPI s ays neu-
al wi h ega d o ju isdic ional clai-
ms in published maps and ins i u io-
nal a ilia ions.
Copy igh : © 2021 by he au ho . Li-
censee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and con-
di ions o he C ea i e Commons A -
ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
Depa amen o de Física Teó ica, Facul ad de Ciencias, and Cen o de As opa ículas y Física de Al as Ene gías
(CAPA), Uni e sidad de Za agoza, Ped o Ce buna 9, 50009 Za agoza, Spain; [email p o ec ed].es
Abs ac :
P io o he es ablishmen o
QCD
as he co ec heo y desc ibing had onic physics, i was
ealized ha he essen ial ing edien s o he had onic wo ld a low ene gies a e chi al symme y and
i s spon aneous b eaking. Spon aneous symme y b eaking is a non-pe u ba i e phenomenon, and,
hanks o massi e
QCD
simula ions on he la ice, we ha e a p esen a good unde s anding o he
acuum ealiza ion o he non-abelian chi al symme y as a unc ion o he physical empe a u e.
As a as he
UA(
1
)
anomaly is conce ned, and especially in he high empe a u e phase, he cu en
si ua ion is howe e a om sa is ac o y. The i s pa o his a icle is de o ed o e iewing he
p esen s a us o la ice calcula ions, in he high empe a u e phase o
QCD
, o quan i ies di ec ly
ela ed o he
UA(
1
)
axial anomaly. In he second pa , some ecen ly sugges ed in e es ing physical
implica ions o he
UA(
1
)
anomaly in sys ems whe e he non-abelian axial symme y is ul illed
in he acuum a e analyzed. Mo e p ecisely i is a gued ha , i he
UA(
1
)
symme y emains
e ec i ely b oken, he opological p ope ies o he heo y can be he basis o a mechanism, o he
han Golds one’s heo em, o gene a e a ich spec um o massless bosons a he chi al limi .
Keywo ds: chi al ansi ion; la ice QCD;U(1)anomaly; opology; massless bosons
1. In oduc ion
Nowadays, we know ha symme ies play an impo an ole in de e mining he
Lag angian o a quan um ield heo y. The e a e essen ially wo ypes o symme y, local
ones, o gauge symme ies, and global ones. The gauge symme ies a e cha ac e ized by
ans o ma ions which depend on he space- ime coo dina es, while, in global symme ies,
he ans o ma ions a e space- ime independen . In addi ion, gauge symme ies se e o
ix he couplings o he Lag angian and global symme ies allow us o o assign quan um
numbe s o he pa icles and o p edic he exis ence o massless bosons when a con inuous
global symme y is spon aneously b oken.
In wha conce ns
QCD
, he heo y o he s ong in e ac ion, and p io o he es ab-
lishmen o his heo y as he co ec heo y desc ibing had onic physics, i was ealized
ha he essen ial ing edien s o he had onic wo ld a low ene gies a e chi al symme y
and i s spon aneous b eaking. Indeed, hese wo p ope ies o he s ong in e ac ion ha e
impo an phenomenological implica ions and allow us o unde s and some puzzling
phenomena such as why pions ha e much smalle masses han he p o on mass and why
we do no see degene a e masses o chi al pa ne s in he boson sec o and pa i y pa ne s
in he ba yon sec o .
Chi al symme y b eaking by he acuum s a e o QCD is a non-pe u ba i e phe-
nomenon, which esul s om he in e ac ion o many mic oscopic deg ees o eedom and
can be in es iga ed mainly h ough la ice QCD simula ions. As a ma e o ac , la ice
QCD
is he mos powe ul echnique o in es iga ing non-pe u ba i e e ec s om i s
p inciples. Howe e , pu ing chi al symme y on o he la ice u ned ou o be a di icul
ask. The unde lying eason is ha a nai e la ice egula iza ion su e s om he doubling
p oblem. The addi ion o he Wilson e m o he nai e ac ion sol es he doubling p oblem
Symme y 2021,13, 209. h ps://doi.o g/10.3390/sym13020209 h ps://www.mdpi.com/jou nal/symme y
Symme y 2021,13, 209 2 o 27
bu b eaks chi al symme y explici ly, e en o massless qua ks. This is usually no con-
side ed o be a undamen al p oblem because we expec ha he symme y is es o ed in
he con inuum limi . Howe e , a ini e la ice spacing, chi al symme y may s ill be a he
s ongly iola ed by la ice e ec s.
On he o he hand, s agge ed e mions cope o he doubling p oblem educing he
numbe o species om six een o ou , and o educe he numbe o e mion species
om ou o one, a oo ing p ocedu e has been used. E en i con o e sial, he oo ing
p ocedu e has allowed ob aining e y accu a e esul s in la ice
QCD
simula ions wi h wo
and h ee la o s.
The doubling p oblem canno be simply o e come because he e is a undamen al
heo em by Nielsen and Ninomiya which s a es ha , on he la ice, one canno imple-
men chi al symme y as in he con inuum o mula ion, and a he same ime ha e a
heo y ee o double s. Howe e , despi e his di icul y, he p oblem o chi al symme y
on he la ice was sol ed a he end o he pas cen u y wi h a gene aliza ion o chi al
symme y, h ough he so-called Ginspa g–Wilson equa ion o he la ice Di ac ope a-
o , which eplaces he s anda d an icommu a ion ela ion o he con inuum o mula ion
Dγ5+γ5D=0 by Dγ5+γ5D=aDγ5D.
Wi h his new concep , a clean implemen a ion
o chi al symme y on he la ice has been achie ed. The axial ans o ma ions educe o
he con inuum ans o ma ions in he nai e con inuum limi , bu a ini e la ice spacing,
a
,
an axial ans o ma ion in ol es also he gauge ields, and his is how he Ginspa g–Wilson
o mula ion e ades he Nielsen–Ninomiya heo em.
All hese ea u es a e well es ablished in he la ice communi y, and he in e es ed
eade can ind in [1], o ins ance, a e y good guide.
Re u ning o he opic o
QCD
phenomenology, he e is also ano he puzzling phe-
nomenon which is known as he
U(
1
)
p oblem. The
QCD
Lag angian o massless qua ks is in-
a ian unde he chi al g oup
UV(N )×UA(N ) = SUV(N )×SUA(N )×UV(1)×UA(1),
wi h
V
and
A
deno ing ec o and axial ec o ans o ma ions espec i ely. Below 1
GeV
,
he la o index
uns om 1 o 3 (up, down, and s ange qua ks), and he chi al symme y
g oup is
UV(
3
)×UA(
3
)
. The ligh weigh pseudoscala s ound in Na u e sugges , as s a ed
abo e, ha he
UA(
3
)
axial symme y is spon aneously b oken in he chi al limi , bu in
such a case we would ha e nine Golds one bosons. The pions,
K
-meson, and
η
-meson a e
eigh o hem bu he candida e o he nin h Golds one boson, he
η0
-meson, has oo g ea
a mass o be a quasi-Golds one boson. This is he axial
U(
1
)
p oblem ha ’ Hoo sol ed
by ealizing ha he
UA(
1
)
axial symme y is anomalous a he quan um le el. ’ Hoo ’s
esolu ion o he U(1) p oblem sugges s in a na u al way he in oduc ion o a
CP
iola ing
e m in he QCD Lag angian, he θ- e m, hus gene a ing ano he long s anding p oblem,
he s ong CP p oblem.
Thanks o massi e QCD simula ions on he la ice, we ha e a p esen a good quali a-
i e and quan i a i e unde s anding on he acuum ealiza ion o he non-abelian
SUA(N )
chi al symme y, as a unc ion o he physical empe a u e, bu as a as
UA(
1
)
anomaly
and i s associa ed
θ
pa ame e a e conce ned, and especially in he high empe a u e phase,
he cu en si ua ion is a om sa is ac o y, and his makes unde s anding he ole o he
θ
pa ame e in QCD, as well as i s connec ion wi h he s ong CP p oblem, one o he bigges
challenges o high ene gy heo is s [2].
The aim o elucida e he exis ence o new low-mass weakly in e ac ing pa icles om a
heo e ical, phenomenological, and expe imen al poin o iew is in ima ely ela ed o his
issue. The ligh pa icle ha has ga he ed he mos a en ion has been he axion, p edic ed
by Weinbe g [
3
] and Wilczek [
4
], in he Peccei and Quinn mechanism [
5
], o explain he
absence o pa i y and empo al in a iance iola ions induced by he QCD acuum. The
axion is one o he mo e in e es ing candida es o make he da k ma e o he uni e se, and
he axion po en ial, which de e mines he dynamics o he axion ield, plays a undamen al
ole in his con ex .
The calcula ion o he opological suscep ibili y in QCD is al eady a challenge, bu cal-
cula ing he comple e po en ial equi es a s a egy o deal wi h he so called sign p oblem,
Symme y 2021,13, 209 3 o 27
ha is, he p esence o a highly oscilla ing e m in he pa h in eg al. Indeed, Euclidean la -
ice gauge heo y has no been able o help us much because o he imagina y con ibu ion
o he ac ion, coming om he
θ
- e m, which p e en s he applicabili y o he impo ance
sampling me hod [6].
The
QCD
axion model ela es he opological suscep ibili y
χT
a
θ=
0 wi h he
axion mass
ma
and decay cons an
a
h ough he ela ion
χT=m2
a 2
a
. The axion mass
is, on he o he hand, an essen ial ing edien in he calcula ion o he axion abundance
in he Uni e se. The e o e, a p ecise compu a ion o he empe a u e dependence o he
opological suscep ibili y in QCD becomes o p imo dial in e es in his con ex .
This a icle ocuses on he cu en s a us o he la ice calcula ions, in he high empe -
a u e chi ally symme ic phase o
QCD
, o quan i ies di ec ly ela ed o he
UA(
1
)
axial
anomaly, as he opological and axial
UA(
1
)
suscep ibili ies, and sc eening masses, as
well as discusses on some in e es ing physical implica ions o he
UA(
1
)
axial anomaly
in sys ems whe e he non-abelian axial symme y is ul illed in he acuum. In Sec ion 2,
some heo e ical p ejudices abou he e ec s o he axial anomaly in he high empe a u e
phase o
QCD
a e b ie ly e iewed, and wha he esul s o he nume ical simula ions
on he la ice sugges on he e ec i eness o he axial anomaly in his phase is analyzed.
In Sec ion 3, i is a gued ha he opological p ope ies o a quan um ield heo y, wi h
UA(
1
)
anomaly and exac non-abelian axial symme y, as o ins ance
QCD
in he high
empe a u e phase, can be he basis o a mechanism, o he han Golds one’s heo em, o
gene a e a ich spec um o massless bosons a he chi al limi . The wo- la o Schwinge
model, which was analyzed by Coleman [
7
] many yea s ago, is an excellen es bed o
e i ying he p edic ions o Sec ion 3, and Sec ion 4con ains he esul s o his es . The las
sec ion con ains a discussion o he esul s epo ed in his a icle.
2. Theo e ical Biases Ve sus Nume ical Resul s
The la ge mass o he
η0
meson should come om he e ec s o he
UA(
1
)
axial
anomaly and i s ela ed gauge ield opology, bo h p esen in
QCD
. Despi e he di icul y
o compu ing he con ibu ion o disconnec ed diag ams o he
η0
co ela o in la ice
simula ions, hese obs acles ha e been o e come and la ice calcula ions [
8
–
10
] gi e a mass
o he
η0
meson compa ible wi h i s expe imen al alue, and his can be seen as an indi ec
con i ma ion ha he e ec s o he anomaly a e p esen in he low empe a u e phase
o QCD.
Con e sely, he cu en si ua ion ega ding he a e o he axial anomaly in he high
empe a u e phase o
QCD
, whe e he non-abelian axial symme y is no spon aneously
b oken, is unclea , and his is qui e unsa is ac o y. The na u e o he chi al phase ansi ion
in wo- la o
QCD
, o ins ance, is a ec ed by he way in which he e ec s o he
UA(
1
)
axial anomaly mani es hemsel es a ound he c i ical empe a u e [
11
]. Indeed, i he
UA(
1
)
axial symme y emains e ec i ely b oken, we expec a con inuous chi al ansi ion
belonging o he h ee-dimensional
O(
4
)
ec o uni e sali y class, which shows a c i ical
exponen
δ=
4.789
(
6
)
[
12
], while, i
UA(
1
)
is e ec i ely es o ed, he chi al ansi ion is i s
o de o second o de wi h c i ical exponen s belonging o he
UV(
2
)×UA(
2
)→UV(
2
)
uni e sali y class (δ=4.3(1)) [13].
The i s in es iga ions on he a e o he
UA(
1
)
axial anomaly in he chi al symme-
y es o ed phase o
QCD
s a ed a long ime ago. The idea ha he chi al symme y
es o ed phase o wo- la o QCD could be symme ic unde
UV(
2
)×UA(
2
)
a he han
SUV(
2
)×SUA(
2
)
was aised by Shu yak in 1994 [
14
], based on an ins an on liquid-model
s udy. In 1996, Cohen [
15
] showed, using he con inuum o mula ion o wo- la o QCD,
and assuming he absence o he ze o mode’s con ibu ion, ha all he disconnec ed con-
ibu ions o he wo-poin co ela ion unc ions in he
SUA(
2
)
symme ic phase a high
empe a u e anish in he chi al limi . The main conclusion o his wo k is ha he eigh
scala and pseudoscala mesons should ha e he same mass in he chi al limi , he ypical
e ec s o he
UA(
1
)
axial anomaly being absen in his phase. In addi ion, Cohen a gued
in [
16
] ha he analy ici y o he ee ene gy densi y in he qua k mass
m
, a ound
m=
0,
Symme y 2021,13, 209 4 o 27
in he high empe a u e phase, imposes cons ain s on he spec al densi y o he Di ac
ope a o a ound he o igin which a e enough o gua an ee he p e ious esul s.
La e on, Aoki e al. [
17
] ob ained cons ain s on he Di ac spec um o o e lap
e mions, s ong enough o all o he
U(
1
)A
b eaking e ec s among co ela ion unc ions
o scala and pseudoscala ope a o s o anish, and hey concluded ha he e is no em-
nan o he
U(
1
)A
anomaly abo e he c i ical empe a u e in wo- la o
QCD
, a leas
in hese co ela ion unc ions. Thei esul s we e ob ained unde he assump ions ha
m
-independen obse ables a e analy ic unc ions o he squa e qua k-mass
m2
, a
m=
0,
and ha he Di ac spec al densi y can be expanded in Taylo se ies nea he o igin, wi h a
non- anishing adius o con e gence.
The ange o applicabili y o he assump ions made by [
17
] is howe e unclea . As
s a ed by he au ho s, hei esul s ongly elies on hei assump ion ha he acuum
expec a ion alues o qua k-mass independen obse ables, as he opological suscep ibili y,
a e analy ic unc ions o he squa e qua k-mass,
m2
, i he non-abelian chi al symme y is
es o ed. The wo- la o Schwinge model has a non-spon aneously b oken
SUA(
2
)
chi al
symme y and
UA(
1
)
axial anomaly, and Coleman’s esul o he opological suscep ibili y
in his model [7]
χT∝m4
3e2
3
shows explici ly a non-analy ic qua k-mass dependence, and hus cas s doub on he
gene al alidi y o he assump ions made in [17].
In [
18
] a Ginspa g–Wilson e mion la ice egula iza ion is used, and i is a gued
ha , i he acuum ene gy densi y is an analy ical unc ion o he qua k mass in he high
empe a u e phase o wo- la o
QCD
, all e ec s o he axial anomaly should disappea .
The main conclusion o [
18
] was ha ei he he ypical e ec s o he axial
UA(
1
)
anomaly
disappea in he symme ic high empe a u e phase o he acuum ene gy densi y shows a
singula beha io in he qua k mass a he chi al limi .
On he o he hand, an analysis o chi al and
UA(
1
)
symme y es o a ion based on
Wa d iden i ies and
U(
3
)
chi al pe u ba ion heo y is ca ied ou in [
19
,
20
]. The au ho s
showed in hei wo k ha , in he limi o exac
O(
4
)
es o a ion, unde s ood in e ms o
δ−η
pa ne degene a ion, he Wa d iden i ies analyzed yield also
O(4)×UA(1)
es o a-
ion in e ms o
π−η
degene a ion, and he pseudo-c i ical empe a u es o es o a ion o
O(4)and O(4)×UA(1) end o coincide in he chi al limi .
The i s la ice simula ions o in es iga e he a e o he
UA(
1
)
axial anomaly [
21
,
22
]
also s a ed in he 1990s. In Re . [
21
] he au ho s epo esul s o a nume ical simula ion
o he wo- la o model wi h s agge ed qua ks. They compu ed wo o de pa ame e s,
χπ−χσ
o he
SUA(
2
)
chi al symme y and
χπ−χδ
o he
UA(
1
)
axial symme y, whe e
χπ
,
χσ
, and
χδ
a e he pion,
σ
, and
δ
-meson suscep ibili ies, espec i ely, and hey showed
e idence o a es o a ion o he
SUV(
2
)×SUA(
2
)
chi al symme y, jus abo e he c osso e ,
bu no o he axial
UA(
1
)
symme y. Re . [
22
] con ains he esul s o a simila calcula ion
in wo- la o
QCD
using also a s agge ed e mion la ice egula iza ion. As s a ed by he
au ho s, he ela i ely coa se la ice spacing in hei simula ions,
a∼1
3
Fe mi, does no
allow o conclusi e esul s on he e ec i eness o he U(1)Aanomaly.
A e hese pionee ing wo ks, his issue has been ex ensi ely in es iga ed using
nume ical simula ions on he la ice, and he wo ks in [
23
–
43
] a e ep esen a i e o ha .
We ocus below on he mos ecen ly ob ained esul s.
In [
29
],
(
2
+
1
)
- la o
QCD
is simula ed, using chi al domain wall e mions, o
empe a u es be ween 139 and 196 MeV. The ligh -qua k mass is chosen so ha he pion
mass is held ixed a a hea ie - han-physical 200 MeV alue, while he s ange qua k
mass is se o i s physical alue. The au ho s epo ed esul s o he chi al condensa es,
connec ed and disconnec ed suscep ibili ies, and he Di ac eigen alue spec um and ind a
pseudoc i ical empe a u e
Tc∼
165 MeV and clea e idence o
UA(
1
)
symme y b eaking
abo e Tc.
Re . [
31
] also p o ided a s udy o
QCD
wi h
(
2
+
1
)
- la o s o highly imp o ed s ag-
ge ed qua ks. The au ho s in es iga ed he empe a u e dependence o he anomalous
UA(
1
)
Symme y 2021,13, 209 5 o 27
symme y b eaking in he high empe a u e phase, and o his end hey employed he o e lap
Di ac ope a o , exploi ing i s p ope y o p ese ing he index heo em e en a non- anishing
la ice spacing. The pion mass is ixed o 160 MeV, and, by quan i ying he con ibu ion o he
nea -ze o eigenmodes o
χπ−χδ
, he au ho s concluded ha he anomalous b eaking o he
axial symme y in QCD is s ill isible in he ange Tc⩽T⩽1.5Tc.
The he mal ansi ion o
QCD
wi h wo degene a e ligh la o s is analyzed in [
34
]
by la ice simula ions, using
O(a)
-imp o ed Wilson qua ks and he unimp o ed Wilson
plaque e ac ion. In his wo k, he au ho s in es iga ed he s eng h o he anomalous
b eaking o he
UA(
1
)
symme y in he chi al limi by compu ing he symme y es o a ion
pa e n o sc eening masses in a ious iso ec o channels, and, o quan i y he s eng h o
he
UA(
1
)
-anomaly, hey used he di e ence be ween scala and pseudoscala sc eening
masses. They concluded ha hei esul s sugges ha he
UA(
1
)
-b eaking is s ongly
educed a he ansi ion empe a u e, and ha his dis a o s a chi al ansi ion in he O(4)
uni e sali y class.
Resul s o mesonic sc eening masses in he empe a u e ange
140 MeV ⩽T⩽2500 MeV
in
(
2
+
1
)
- la o
QCD
, using he highly imp o ed s agge ed qua k ac ion, a e also epo ed
by he
Ho QCD
Collabo a ion in [
41
], wi h a physical alue o he s ange qua k mass, and
wo alues o he ligh qua k mass co esponding o pion masses o 160 and 140 MeV. Com-
pa ing sc eening masses o chi al pa ne s, ela ed h ough he chi al
SUL(
2
)×SUR(
2
)
and he axial
UA(
1
)
ans o ma ions, espec i ely, he au ho s ound, in he case o ligh –
ligh mesons, e idence o he degene acy o sc eening masses ela ed h ough he chi al
SUL(
2
)×SUR(
2
)
a o e y close o he pseudoc i ical empe a u e,
Tpc
, while sc eening
masses ela ed h ough an axial
UA(
1
)
ans o ma ion s a becoming degene a e only a
abou 1.3Tpc.
A ecen calcula ion in
(
2
+
1
)
- la o
QCD
[
42
], using also he highly imp o ed
s agge ed qua k ac ion, shows, a e con inuum and chi al ex apola ions, ha he axial
anomaly emains mani es ed in wo-poin co ela ion unc ions o scala and pseudoscala
mesons in he chi al limi , a a empe a u e o abou 1.6 imes he chi al phase ansi ion
empe a u e. The analysis is based on no el ela ions be ween he n h-o de ligh qua k
mass de i a i es o he Di ac eigen alue spec um,
ρ(λ
,
ml)
, and he
(n+
1
)
-poin co e-
la ions among he eigen alues o he massless Di ac ope a o , and he calcula ions we e
ca ied ou a he physical alue o he s ange qua k mass, h ee la ice spacings, and ligh
qua k masses co esponding o pion masses in he ange 55–160 MeV.
Re . [
43
] p o ided he la es esul s o he JLQCD collabo a ion. In his wo k, he
au ho s in es iga ed he a e o he
UA(
1
)
axial anomaly in wo- la o
QCD
a empe a u es
190–330 MeV using domain wall e mions, eweigh ed o o e lap e mions, a a la ice
spacing o 0.07 m. They measu ed he axial
UA(
1
)
suscep ibili y,
χπ−χδ
, and examined
he degene acy o
UA(
1
)
pa ne s in meson and ba yon co ela o s. Thei conclusion is
ha all he da a abo e he c i ical empe a u e indica e ha he axial
UA(
1
)
iola ion is
consis en wi h ze o wi hin s a is ical e o s.
All he esul s discussed hus a mainly e e o he empe a u e dependence o
he axial suscep ibili y
UA(
1
)
, sc eening masses, and ela ed quan i ies. The opological
suscep ibili y,
χT
, is ano he obse able ha can be use ul in in es iga ing he a e o he
axial anomaly in he high- empe a u e phase o
QCD
, and i s dependence on empe a u e
has also been ex ensi ely in es iga ed [35–37,39,43].
The au ho s o Re . [
35
] explo ed
N =
2
+
1
QCD
in a ange o empe a u es, om
Tc
o a ound 4
Tc
, and hei esul s o he opological suscep ibili y di e s ongly, bo h
in he size and in he empe a u e dependence, om he dilu e ins an on gas p edic ion,
gi ing ise o a shi o he axion da k-ma e window o almos one o de o magni ude
wi h espec o he ins an on compu a ion.
The au ho s o Re . [
36
], howe e , obse ed in he same model e y dis inc empe a-
u e dependences o he opological suscep ibili y in he anges abo e and below 250 MeV;
while, o empe a u es abo e 250 MeV, he dependence is ound o be consis en wi h

Symme y 2021,13, 209 6 o 27
he dilu e ins an on gas app oxima ion, a lowe empe a u es, he allo o opological
suscep ibili y is milde .
On he o he hand, a no el app oach is p oposed in [
37
], i.e., he ixed
Q
in eg a ion,
based on he compu a ion o he mean alue o he gauge ac ion and chi al condensa e a
ixed opological cha ge
Q
; he au ho s ound a opological suscep ibili y many o de s o
magni ude smalle han ha o Re . [
35
] in he cosmologically ele an empe a u e egion.
A mo e ecen la ice calcula ion [
39
] o he opological p ope ies o
N =
2
+
1 QCD
wi h physical qua k masses and empe a u es a ound 500 MeV gi es as a esul a small bu
non- anishing opological suscep ibili y, al hough wi h la ge e o ba s in he con inuum
limi ex apola ions, poin ing ha he e ec s o he
UA(
1
)
axial anomaly s ill pe sis a
hese empe a u es.
The JLQCD collabo a ion [
43
] also epo ed esul s o he opological suscep ibili y
in wo- la o
QCD
, in he empe a u e ange 195–330 MeV, o se e al qua k masses, and
hei da a show a supp ession o
χT(m)
nea he chi al limi . The au ho s claimed ha
hei esul s a e no accu a e enough o de e mine whe he
χT(m)
anishes a a ini e
qua k mass.
In sho , we see how, despi e he g ea e o de o ed o in es iga ing he a e o he
axial anomaly in he chi ally symme ic phase o
QCD
, he cu en si ua ion on his issue
is a om sa is ac o y.
3. Physical E ec s o he UA(1)Anomaly in Models wi h Exac SUA(N )
Chi al Symme y
We de o e he es o his a icle mainly o analyze he physical e ec s o he
UA(
1
)
anomaly in a e mion-gauge heo y wi h wo o mo e la o s, which exhibi s an exac
SUA(N )
chi al symme y in he chi al limi . Howe e , we also gi e a quick look o he one-
la o model and o he mul i- la o model wi h spon aneous non-abelian chi al symme y
b eaking. Al hough many o he esul s p esen ed he e can be ound in [
18
,
44
,
45
], we make
he es o his a icle sel -con ained o ease o eading.
We show in his sec ion ha a gauge- e mion quan um ield heo y, wi h
UA(
1
)
axial
anomaly, and in which he scala condensa e anishes in he chi al limi because o an exac
non-abelian
SUA(
2
)
chi al symme y, should exhibi a singula qua k-mass dependence o
he acuum ene gy densi y and a di e gen co ela ion leng h in he co ela ion unc ion
o he scala condensa e, i he
UA(
1
)
symme y is e ec i ely b oken. On he con a y,
i we assume ha all co ela ion leng hs a e ini e, and hence he acuum ene gy densi y
is an analy ical unc ion o he qua k mass, we show ha he acuum ene gy densi y
becomes, a leas up o second o de in he qua k masses,
θ
-independen . In he o me
case, he non-anomalous Wa d–Takahashi (W-T) iden i ies ell us ha se e al pseudoscala
co ela ion unc ions, hose o he
SUA(
2
)
chi al pa ne s o he la o single scala meson,
should exhibi a di e gen co ela ion leng h oo. We also a gue ha his esul can be
gene alized o any numbe o la o s N >2.
3.1. Some Backg ound
To begin, le us w i e he con inuum Euclidean ac ion o a ec o -like gauge heo y
wi h global UA(1)anomaly in he p esence o a θ- acuum e m
S=Zddx


N
∑
¯
ψ (x)γµDµ(x)+m ψ (x)+1
4Fa
µν(x)Fa
µν(x)+iθQ(x)


(1)
whe e
d
is he space- ime dimensionali y,
Dµ(x)
is he co a ian de i a i e,
N
is he
numbe o la o s, and
Q(x)
is he densi y o opological cha ge o he gauge con igu a ion.
The opological cha ge
Q
is he in eg al o he densi y o opological cha ge
Q(x)
o e he
space- ime olume, and i is an in ege numbe which in he case o
QCD
eads as ollows
Symme y 2021,13, 209 7 o 27
Q=g2
64π2Zd4xeµνρσFa
µν(x)Fa
ρσ(x). (2)
To keep ma hema ical igo , we a oid ul a iole di e gences wi h he help o a la ice
egula iza ion and use Ginspa g-Wilson (G-W) e mions [
46
], he o e lap e mions [
47
,
48
]
being an explici ealiza ion o hem. The mo i a ion o use G-W e mions is ha hey sha e
wi h he con inuum o mula ion all essen ial ing edien s. Indeed, G-W e mions show an
explici
UA(
1
)
anomalous symme y [
49
], good chi al p ope ies, a quan ized opological
cha ge, and allow us o es ablish and exac index heo em on he la ice [50].
The la ice e mionic ac ion o a massless G-W e mion can be w i en in a compac
o m as
SF=ad¯
ψDψ=ad∑
,w
¯
ψ( )D( ,w)ψ(w)(3)
whe e
and
w
con ain si e, Di ac, and colo indices, and
D
, he Di ac–Ginspa g–Wilson
ope a o , obeys he essen ial an icommu a ion equa ion
Dγ5+γ5D=aDγ5D(4)
abeing he la ice spacing.
Ac ion (3) is in a ian unde he ollowing la ice UA(1)chi al o a ion
ψ→eiαγ5(I−1
2aD)ψ,¯
ψ→¯
ψeiα(I−1
2aD)γ5(5)
which o
a→
0 educes o he s anda d con inuum chi al ans o ma ion. Howe e , he
in eg a ion measu e o G assmann a iables is no in a ian , and he change o
a iables (5)
induces a Jacobian
e−i2αa
2 (γ5D)(6)
whe e a
2 (γ5D)=n−−n+=Q(7)
is an in ege numbe , he di e ence be ween le -handed and igh -handed ze o modes, which can
be iden i ied wi h he opological cha ge
Q
o he gauge con igu a ion.
Equa ions (6) and (7)
show
us how Ginspa g–Wilson e mions ep oduce he UA(1)axial anomaly.
We can also add a symme y b eaking mass e m,
m¯
ψ1−a
2Dψ
o ac ion (3), so G-W
e mions wi h mass a e desc ibed by he e mion ac ion
SF=ad¯
ψDψ+adm¯
ψ1−a
2Dψ(8)
and i can also be shown ha he scala and pseudoscala condensa es
S=¯
ψ1−a
2DψP=i¯
ψγ51−a
2Dψ(9)
ans o m, unde he chi al
UA(
1
)
o a ions (5), as a ec o , jus in he same way as
¯
ψψ
and
i¯
ψγ5ψdo in he con inuum o mula ion.
In wha ollows, we use dimensionless e mion ields and a dimensionless Di ac–
Ginspa g–Wilson ope a o . In such a case, he e mion ac ion o he N - la o model is
SF=
N
∑
¯
ψ Dψ +m ¯
ψ1−1
2Dψ (10)
whe e
m
is he mass o la o
in la ice uni s. The pa i ion unc ion o his model, in he
p esence o a
θ
- acuum e m, can be w i en as he sum o e all opological sec o s,
Q
, o
he pa i ion unc ion in each opological sec o imes a θ-phase ac o ,
Symme y 2021,13, 209 8 o 27
Z=∑
Q
ZQeiθQ(11)
whe e
Q
, which akes in ege alues, is bounded a ini e olume by he numbe o deg ees
o eedom. A la ge la ice olume, he pa i ion unc ion should beha e as
Zβ,m ,θ=e−VE(β,m ,θ)(12)
whe e
Eβ,m ,θ
is he acuum ene gy densi y,
β
is he in e se gauge coupling,
m
is he
- la o mass, and V=Vs×L is he la ice olume in uni s o he la ice spacing.
3.2. Q =0Topological Sec o . The One-Fla o Model and he Mul i-Fla o Model wi h
Spon aneous Chi al Symme y B eaking
In ou analysis o he physical phenomena induced by he opological p ope ies o
he heo y, he
Q=
0 opological sec o plays an essen ial ole, and because o ha we
de o e his subsec ion o e iew some esul s conce ning he ela ion be ween acuum
expec a ion alues o local and non-local ope a o s compu ed in he
Q=
0 sec o , wi h
hei co esponding alues in he ull heo y, which akes in o accoun he con ibu ion
o all opological sec o s. In pa icula , we show ha he acuum ene gy densi y, as well
as he acuum expec a ion alue o any ini e ope a o , as o ins ance local o in ensi e
ope a o s, compu ed in he
Q=
0 opological sec o , is equal, in he in ini e olume limi ,
o i s co esponding alue in he ull heo y. We also show ha his p ope y is in gene al
no ue o non-local ope a o s, he la o -single pseudoscala suscep ibili y being a
pa adigma ic example o his. Howe e , he e a e non-local ope a o s, o ins ance he
second-o de e mion-mass de i a i es o he acuum ene gy densi y, he alues o which
in he
Q=
0 sec o ma ch hei co esponding alues in he ull heo y, in he in ini e la ice
olume limi .
We also analyze in his subsec ion he one- la o case, as well as he mul i- la o case
wi h spon aneous chi al symme y b eaking, and show how, al hough he a o emen ioned
p ope ies imply ha he
UA(
1
)
symme y is spon aneously b oken in he
Q=
0 opological
sec o , he Golds one heo em is no ealized because he di e gence o he la o -single
pseudoscala suscep ibili y, in his sec o , does no o igina e om a di e gen co ela ion
leng h [18].
The pa i ion unc ion and he mean alue o any ope a o
O
, o ins ance he scala
and pseudoscala condensa es, o any co ela ion unc ion, in he
Q=
0 opological sec o ,
can be compu ed, espec i ely, as
ZQ=0=1
2πZdθZ(β,m ,θ)(13)
hOiQ=0=RdθhOiθZ(β,m ,θ)
RdθZ(β,m ,θ)(14)
whe e
hOiθ
, which is he mean alue o
O
compu ed wi h he la ice egula ized in eg a ion
measu e (1), is a unc ion o he in e se gauge coupling
β
, la o masses
m
, and
θ
, and we
es ic ou sel es o he case in which i akes a ini e alue in he in ini e la ice olume
limi . Since he acuum ene gy densi y (12), as a unc ion o
θ
, has i s absolu e minimum
a
θ=
0 o non- anishing e mion masses, he ollowing ela ions hold in he in ini e
olume limi
EQ=0β,m =Eβ,m ,θθ=0(15)
hOiQ=0=hOiθ=0(16)
Symme y 2021,13, 209 9 o 27
whe e EQ=0β,m is he acuum ene gy densi y o he Q=0 opological sec o .
Taking in mind hese esul s, le us s a wi h he analysis o he one- la o model a
ze o empe a u e. The esul s ha ollow apply, o ins ance, o one- la o
QCD
in ou
dimensions o o he one- la o Schwinge model.
In he one la o model, he only axial symme y is an anomalous
UA(
1
)
symme y.
The s anda d wisdom on he acuum s uc u e o his model in he chi al limi is ha i
is unique a each gi en alue o
θ
, he
θ
- acuum. Indeed, he only plausible eason o
ha e a degene a e acuum in he chi al limi would be he spon aneous b eakdown o
chi al symme y, bu , since i is anomalous, ac ually he e is no symme y. Fu he mo e,
due o he chi al anomaly, he model shows a mass gap in he chi al limi , and he e o e
all co ela ion leng hs a e ini e in physical uni s. Since he model is ee om in a ed
di e gences, he acuum ene gy densi y can be expanded in powe s o he e mion mass
mu
, ea ing he qua k mass e m as a pe u ba ion [
51
]. This expansion is hen an o dina y
Taylo se ies
E(β,mu,θ)=E0(β)−Σ(β)mucos θ+O(m2
u), (17)
gi ing ise o he ollowing expansions o he scala and pseudoscala condensa es
hSui=−Σ(β)cos θ+O(mu)(18)
hPui=−Σ(β)sin θ+O(mu)(19)
whe e
Su
and
Pu
a e he scala and pseudoscala condensa es (9) no malized by he
la ice olume
Su=1
V¯
ψ1−1
2DψPu=i
V¯
ψγ51−1
2Dψ(20)
The opological suscep ibili y
χT
is gi en, on he o he hand, by he ollowing expansion
χT=Σ(β)mucos θ+O(m2
u)(21)
The esolu ion o he
UA(
1
)
p oblem is ob ious i we se down he W-T iden i y which
ela es he pseudoscala suscep ibili y
χη=∑xhPu(x)Pu(0)i
, he scala condensa e
hSui
,
and he opological suscep ibili y χT
χη=−hSui
mu−χT
m2
u
. (22)
Indeed, he di e gence in he chi al limi o he i s e m in he igh -hand side o (22)
is canceled by he di e gence o he second e m in his equa ion, gi ing ise o a ini e
pseudoscala suscep ibili y, and a ini e non- anishing mass o he pseudoscala
η
boson.
In wha conce ns he
Q=
0 opological sec o , we wan o no ice wo ele an ea u es:
1. The global UA(1)axial symme y is no anomalous in he Q=0 opological sec o .
2.
I we apply Equa ion (16) o he compu a ion o he acuum expec a ion alue o
he scala condensa e, we ge ha he
UA(
1
)
symme y is spon aneously b oken in
he
Q=
0 sec o because he chi al limi o he in ini e olume limi o he scala
condensa e, he limi s aken in his o de , does no anish.
Equa ion (14) allows us o w i e o he in ini e olume limi o he wo-poin pseu-
doscala co ela ion unc ion, hPu(x)Pu(0)i, he ollowing ela ion
hPu(x)Pu(0)iQ=0=hPu(x)Pu(0)iθ=0. (23)
This equa ion implies ha he mass o he pseudoscala boson,
mη
, which can be
ex ac ed om he long dis ance beha io o he wo-poin co ela ion unc ion, compu ed
in he
Q=
0 sec o , is equal o he alue we should ge in he ull heo y, aking in o accoun
he con ibu ion o all opological sec o s. On he o he hand, he opological suscep ibili y,
Symme y 2021,13, 209 16 o 27
Landau’s heo y o phase ansi ions p edic s ha he end poin placed a he o igin
o coo dina es in he
(mu
,
md)
plane is a c i ical poin , he scala condensa e should show
a non-analy ic dependence on he e mion masses
mu
and
md
as we app oach he c i ical
poin , and hence he scala suscep ibili y should di e ge. Howe e , since he acuum
ene gy densi y in he
Q=
0 opological sec o , as well as i s e mion mass de i a i es,
ma ches he acuum ene gy densi y and e mion mass de i a i es in he ull heo y, and
he same is ue o he c i ical equa ion o s a e, Landau’s heo y o phase ansi ions
p edic s a non-analy ic dependence o he la o single scala condensa e on he e mion
mass, and a di e gen co ela ion leng h in he chi al limi o ou ull heo y, in which we
ake in o accoun he con ibu ion o all opological sec o s.
Mo e p ecisely, we can apply he Landau app oach o analyze he c i ical beha io
a ound he wo i s -o de ansi ion lines in Figu e 1nea he end poin , o c i ical poin .
In he analysis o he
md=
0 ansi ion line, we conside
md
as an ex e nal “magne ic ield”
and
mu
as he “ empe a u e”, and ice e sa o he analysis o he
mu=
0 line. Then,
he s anda d Landau app oach ells us ha he up and down condensa es e i y he wo
ollowing equa ions o s a e
−muhSui−3=−2C1mdhSui−2+4C2
−mdhSdi−3=−2C1muhSdi−2+4C2(45)
whe e
C1
and
C2
a e wo posi i e cons an s. I we ix he a io o he up and down masses
mu
md=λ
, he equa ions o s a e (45) allow us o w i e he ollowing expansions o de up
and down condensa es
hSui=−m
1
3
u
1
4C21
3+C1
32C2
21
3λ
m
1
3
u+. . .

hSdi=−m
1
3
d
1
4C21
3+C1λ
32C2
21
3
m
1
3
d+. . .
(46)
Equa ion (46) shows explici ly he non analy ical beha io o he up and down con-
densa es. In he degene a e la o case,
mu=md=m
, he scala condensa e and he
la o -single scala suscep ibili y nea he c i ical poin scale as
hSi=hSui+hSdi=−2
C21
3m1
3+. . .
χσ(m)=1
32
C21
3m−2
3+. . . (47)
showing up explici ly he di e gence o he la o single scala suscep ibili y in he chi al
limi .
We see ha he c i ical beha io o he chi al condensa e in he Landau
app oach (47)
is desc ibed by he mean ield c i ical exponen
δ=
3. Mean ield c i ical exponen s
a e expec ed o be co ec in high dimensions, while, in low dimensions, he e ec o
luc ua ions can change hei mean ield alues. This means ha , in he la e case, he
Landau app oach gi e us a good quali a i e desc ip ion o he phase diag am bu ails in
i s quan i a i e p edic ions o c i ical exponen s.
To inish he Landau app oach analysis, we wan o poin ou ha all hese esul s can
be gene alized in a s aigh o wa d way o a numbe o la o s N >2.

Symme y 2021,13, 209 17 o 27
3.5. C i ical Beha io o he Two-Fla o Model wi h an Isospin B eaking Te m
Beyond he Landau app oach, we can pa ame e ize he c i ical beha io o he la o
single scala condensa e and o he mass-dependen con ibu ion o he acuum ene gy
densi y, in he wo degene a e la o model, wi h a c i ical exponen δ>1
hSim→0≃ −Cm1
δ. (48)
E(β,m)−E(β, 0)≃ − Cδ
δ+1mδ+1
δ. (49)
whe e
C
is a dimensionless posi i e cons an ha depends on he in e se gauge coupling
β
. Equa ion (48) gi es us a di e gen scala suscep ibili y,
χσ(m)∼C
δm1−δ
δ
, and hence a
massless scala boson as m→0.
I , on he o he hand, we w i e he W-T iden i y o he iso iple o “pions” which
ollows om he SUA(2)non-anomalous chi al symme y
χ¯
π(m)=−hSi
m, (50)
we ge ha also
χ¯
π(m)
di e ges when
m→
0 as
Cm1−δ
δ
, and a ich spec um o massless
bosons
(σ
,
¯
π)
eme ges in he chi al limi . The suscep ibili y o he la o single pseudoscala
condensa e ul ills he anomalous W-T iden i y (30), and, because o he
UA(
1
)
axial anomaly,
he η-boson mass is expec ed o emain ini e (non- anishing) in he chi al limi .
The hype scaling hypo hesis, which a ises as a na u al consequence o he block-spin
eno maliza ion g oup app oach, says ha he only ele an leng h nea he c i ical poin
o a magne ic sys em, in wha conce ns he singula pa
Es(β,m)
o he ee o acuum
ene gy densi y, is he co ela ion leng h
ξ
. Since Equa ion (49) con ains only he singula
con ibu ion o he acuum ene gy densi y, we can w i e
Es(β,m)≃ − Cδ
δ+1mδ+1
δ∼ξ−d(51)
and he ollowing ela ionship be ween he co ela ion leng h and he e mion mass
ξ∼m−δ+1
dδ(52)
which implies ha he pion and sigma-meson masses scale wi h he e mion mass as ollows
m¯
π,mσ∼mδ+1
dδ(53)
In he p esence o an isospin b eaking e m, he e mion ac ion can be w i en in a
compac o m as
SF=mu+md
2¯
ψ1−1
2Dψ−md−mu
2¯
ψ1−1
2Dτ3ψ+¯
ψDψ(54)
whe e
ψ
is a G assmann ield ca ying si e, Di ac, colo , and la o indices and
τ3
is he
hi d Pauli ma ix ac ing in la o space.
I we also include a
θ
- acuum e m in he ac ion, his
θ
- e m can be emo ed h ough
a chi al
UA(
1
)
ans o ma ion, which lea es he
¯
ψDψ
in e ac ion e m in a ian . I nex we
also pe o m a sui able non-anomalous chi al ans o ma ion, we ge he e ec i e e mion
ac ion ha ollows
SF=M(mu,md,θ)¯
ψ1−1
2Dψ+A(mu,md,θ)i¯
ψγ51−1
2Dψ
Symme y 2021,13, 209 18 o 27
+B(mu,md,θ)¯
ψ1−1
2Dτ3ψ+¯
ψDψ(55)
whe e M(mu,md,θ),A(mu,md,θ)and B(mu,md,θ)a e gi en by
M(mu,md,θ)=1
2m2
u+m2
d+2mumdcos θ1
2(56)
A(mu,md,θ)=2mumdsin θ
2
(mu+md)1+m2
u+m2
d−2mumd
m2
u+m2
d+2mumd an2θ
21
2
(57)
B(mu,md,θ)=−md−mu
2 cos θ
21+m2
u+m2
d−2mumd
m2
u+m2
d+2mumd an2θ
21
2
(58)
Since we do no expec singula i ies a non- anishing e mion masses, he acuum
ene gy densi y
E(β,M,A,B)
can be expanded in powe s o
A
and
B
as an o dina y Taylo
se ies, and, aking in o accoun he symme ies o he e ec i e ac ion (55), we can w i e he
ollowing equa ion o his expansion up o second o de
E(β,mu,md,θ)≡E(β,M,A,B)=E(β,M, 0, 0)+1
2A2χη(β,M)+1
2B2χδ(β,M)+. . . (59)
whe e
χη(β,M)
and
χδ(β,M)
a e he la o single pseudoscala suscep ibili y and he
δ
-meson suscep ibili y in he heo y wi h wo degene a e la o s o mass
M(mu
,
md
,
θ)
,
espec i ely. No e ha his expansion should ha e a good con e gence i
θ
and
md−mu
a e small.
The acuum ene gy densi y, o he lowes o de o he expansion (59), is ha o he model
wi h wo degene a e la o s o mass
M(mu
,
md
,
θ)
, in he absence o a
θ
- acuum e m. We
show abo e ha his model should show a c i ical beha io
(48) and (49),
a ound he chi al
limi , and hence we ge , o he lowes o de o his expansion,
E(β,mu,md,θ)−E(β, 0, 0, 0)=−C
2δ+1
δ
δ
δ+1m2
u+m2
d+2mumdcos θδ+1
2δ+. . . (60)
The ee ene gy densi y depends on
mu
,
md
and
θ
h ough
m2
u+m2
d+2mumdcos θ1
2
,
and i s dominan con ibu ion in he chi al limi is gi en by he powe -law beha io o
Equa ion (60) (no e ha i we apply his expansion o he acuum ene gy densi y o wo-
la o
QCD
a
T=
0, whe e chi al symme y is spon aneously b oken, and hence
δ=∞
in (48) and (49), we ge he acuum ene gy densi y o he low ene gy chi al e ec i e
Lag angian model [51]).
The la o -single pseudoscala suscep ibili y,
χη(β,M)
, ul ills he anomalous W-T
iden i y (30), and hence i is expec ed o emain ini e in he chi al limi . Since he
SUA(
2
)
chi al symme y is exac in his limi , he same holds ue o
χδ(β,M)
. In such condi ions,
he ele ance o he second-o de co ec ion o he ze o-o de con ibu ion o he acuum
ene gy densi y (59), o wo degene a e la o s, u ns ou o be
A2(m,θ)
E(β,M, 0)−E(β, 0, 0)∼m1−1
δsin2θ
2
cos θ
21+1
δ
(61)
while, in he isospin b eaking case, and o small θ alues, we ha e
A2(mu,md,θ)
E(β,M, 0, 0)−E(β, 0, 0, 0)∼m2
um2
dθ2
(mu+md)3+1
δ
Symme y 2021,13, 209 19 o 27
B2(mu,md,θ)
E(β,M, 0, 0)−E(β, 0, 0, 0)∼(md−mu)2
(mu+md)1+1
δ
(62)
Since
δ>
1 (
δ=
3 in he mean ield model), we see ha he c i ical beha io o he
model, which desc ibes he low ene gy heo y, is ully con olled in bo h cases by he ze o-
o de con ibu ion o he acuum ene gy densi y (63), and he second-o de con ibu ion
can be neglec ed in wha conce ns he chi al limi o he heo y.
Le us now look a some in e es ing physical consequences ha can be ob ained om
Equa ion (60). In he degene a e la o case,
mu=md=m
, Equa ions (52), (53), and
(60) become
E(β,m,θ)−E(β, 0, 0)=−Cδ
δ+1mcos θ
2δ+1
δ+. . . (63)
ξ∼mcos θ
2−δ+1
dδ(64)
m¯
π,mσ∼mcos θ
2δ+1
dδ(65)
Fo non-degene a e la o s, he acuum ene gy densi y (60) a
θ=
0 is a unc ion o
mu+md
; hence, he acuum expec a ion alues o he up and down condensa es a e equal,
and he same holds ue o hei suscep ibili ies:
hSui=hSui=−C
2δ+1
δ
(mu+md)1
δ
∑
x
(hSu(x)Su(0)i−hSu(x)ihSu(0)i)=∑
x
(hSd(x)Sd(0)i−hSd(x)ihSd(0)i)=
∑
x
(hSu(x)Sd(0)i−hSu(x)ihSd(0)i)=C
2δ+1
δ
1
δ(mu+md)1−δ
δ. (66)
We do no see any dependency on
md−mu
, and isospin b eaking e ec s a e he e o e
absen in hese quan i ies, which on he o he hand show a singula beha io in he chi al
limi . The no malized la o single scala suscep ibili y, χσ,
χσ=C
21
δ
1
δ(mu+md)1−δ
δ. (67)
di e ges in he chi al limi , while he
δ
-meson suscep ibili y,
χδ
, anishes in he ze o-o de
app oxima ion o he acuum ene gy densi y, indica ing ha i is a good app oxima ion
when he a io o he σand δmeson masses is small, mσ
mδ1.
The opological suscep ibili y is gi en by
χT=C
2δ+1
δ
(mu+md)1−δ
δmumd(68)
showing ha his quan i y is sensi i e o he isospin b eakdown.
The W-T iden i y o he cha ged pions, π±, eads
χπ±=−hSui+hSdi
mu+md
(69)
and hence we ge
χπ±=C
21
δ
(mu+md)1−δ
δ. (70)
Simila o he
σ
-suscep ibili y, he cha ged pions suscep ibili y di e ges in he
chi al limi .
Symme y 2021,13, 209 20 o 27
To calcula e he suscep ibili y o he neu al pion, we use he ollowing
W−T
iden i ies
∑
xhPu(x)Pu(0)i=−hSui
mu−χT
m2
u
∑
xhPd(x)Pd(0)i=−hSdi
md−χT
m2
d
∑
xhPu(x)Pd(0)i=−χT
mumd
(71)
which gi e us
∑
xhPu(x)Pu(0)i=∑
xhPd(x)Pd(0)i=−∑
xhPu(x)Pd(0)i=C
2δ+1
δ
(mu+md)1−δ
δ(72)
and, o he no malized neu al pion suscep ibili y, we ge
χπ0=C
21
δ
(mu+md)1−δ
δ. (73)
Equa ions (70) and (73) show ha he
π±
and
π0
suscep ibili ies a e equal and inde-
penden o
md−mu
. Again, isospin b eaking e ec s a e absen in hese quan i ies, and,
e en hough
md−mu6=
0, he h ee pions ha e he same mass. In wha conce ns he
la o -single pseudoscala suscep ibili y, χη, Equa ion (72) shows ha i anishes.
Finally, i o simplici y we conside wo degene a e la o s, Equa ions (53) and (68) imply
ha he pion mass
m¯
π
(o he
σ
-meson mass) and he opological suscep ibili y
χT
e i y he
ollowing ela ion m¯
π
(χT)1
d
=k(β,L )(74)
whe e
k
is a dimensionless quan i y ha depends on he in e se gauge coupling
β
, and
e en ually, a ini e empe a u e
T
, on he la ice empo al ex en
L
, bu ha is independen
o he e mion mass m.
In summa y, i is shown ha , in he ze o-o de app oxima ion o he acuum ene gy
densi y, which accoun s o he chi al c i ical beha io o he heo y, isospin b eaking e ec s
only mani es in he opological suscep ibili y. The h ee pions ha e he same mass, he
a io o he pion (73) and
σ
-meson (67) suscep ibili ies is equal o he c i ical exponen
δ
,
and he pion (o
σ
-meson) mass is ela ed wi h he opological suscep ibili y, as shown in
Equa ion (74).
4. Two-Fla o Schwinge Model as a Tes Bed
Quan um Elec odynamics in
(
1
+
1
)
-dimensions is a good labo a o y o es he
esul s epo ed in he p e ious sec ion. The model is con ining [
53
], exac ly sol able
a ze o e mion mass, has non- i ial opology, and shows explici ly he
UA(
1
)
axial
anomaly [54]
. Besides ha , he Schwinge model does no equi e in ini e eno maliza ion,
and his means ha , i we use a la ice egula iza ion, he ba e pa ame e s emain ini e in
he con inuum limi .
On he o he hand, he
SUA(N )
non-anomalous axial symme y in he chi al limi
o he mul i- la o Schwinge model is ul illed in he acuum, and his p ope y makes
his model a pe ec candida e o check ou p edic ions on he exis ence o quasi-massless
scala and pseudoscala bosons in he spec um o he model, he mass o which anishes
in he chi al limi .
The Euclidean con inuum ac ion o he wo- la o heo y is
S=Zd2x{¯
ψu(x)γµ∂µ+iAµ(x)ψu(x)+¯
ψd(x)γµ∂µ+iAµ(x)ψd(x)}+
Symme y 2021,13, 209 21 o 27
Zd2x{mu¯
ψu(x)ψu(x)+md¯
ψd(x)ψd(x)+1
4e2F2
µν(x)+iθQ(x)}(75)
whe e
mu
and
md
a e he e mion masses and
e
is he elec ic cha ge o gauge coupling,
which has mass dimensions.
Fµν(x) = ∂µAν(x)−∂νAµ(x)
, and
γµ
a e 2
×
2 ma ices
sa is ying he algeb a
{γµ,γν}=2gµν (76)
A he classic le el his heo y has an in e nal
SUV(
2
)×SUA(
2
)×UV(
1
)×UA(
1
)
symme y in he chi al limi . Howe e , he
UA(
1
)
-axial symme y is b oken a he quan um
le el because o he axial anomaly. The di e gence o he axial cu en is
∂µJA
µ(x) = 1
2πeµνFµν(x), (77)
whe e
eµν
is he an isymme ic enso , and hence does no anish. The axial anomaly
induces he opological θ- e m iθQ=iθRd2xQ(x)in he ac ion, whe e
Q(x)=1
4πeµνFµν(x)(78)
is he densi y o opological cha ge, he opological cha ge Qbeing an in ege numbe .
The Schwinge model was analyzed yea s ago by Coleman [
7
], compu ing some
quan i a i e p ope ies o he heo y in he con inuum o bo h, weak coupling
e
m
1, and
s ong coupling o chi al limi e
m1.
Fo he one- la o case, Coleman compu ed he pa icle spec um o he model, which
shows a mass gap in he chi al limi , and conjec u ed he exis ence o a phase ansi ion a
θ=π
and some in e media e e mion mass
m
sepa a ing a weak coupling phase (
e
m
1),
whe e he
Z2
symme y o he model a
θ=π
is spon aneously b oken, om a s ong
coupling phase (
e
m
1), in which he
Z2
symme y is ul illed in he acuum. This
quali a i e esul has ecen ly been con i med by nume ical simula ions o he Euclidean-
la ice e sion o he model [55].
Wha is howe e mo e in e es ing o he con en o his a icle is he Coleman anal-
ysis o he wo- la o model. As s a ed abo e, he heo y (75) has an in e nal
SUV(
2
)×
SUA(
2
)×UV(
1
)×UA(
1
)
symme y in he chi al limi , and he
UA(
1
)
axial symme y is
anomalous. Since con inuous in e nal symme ies canno be spon aneously b oken in a
local ield heo y in wo dimensions [
56
], he
SUA(
2
)
symme y has o be ul illed in he
acuum, and he scala condensa e, which is an o de pa ame e o his symme y, an-
ishes in he chi al limi . Hence, he wo- la o Schwinge model e i ies all he condi ions
assumed in Sec ion 3.
We summa ize he e he main Coleman’s indings o he wo- la o model wi h
degene a e masses mu=md=m:
1.
Fo weak coupling,
e
m
1, he esul s on he pa icle spec um a e almos he same
as o he massi e Schwinge model.
2.
Fo s ong coupling,
e
m
1, he low-ene gy e ec i e heo y depends only on one
mass pa ame e , m2
3e1
3cos2
3θ
2; he acuum ene gy densi y is hen p opo ional o
E(m,e,θ)∝e2
3mcos θ
24
3; (79)
and he chi al condensa e, a θ=0, is he e o e
h¯
ψψi∝m1
3e2
3(80)
3.
The ligh es pa icle in he heo y is an iso iple , and he nex ligh es is an isosingle .
The isosingle /iso iple mass a io is
√3
. I he e a e o he s able pa icles in he

Symme y 2021,13, 209 22 o 27
model, hey mus be
Oe
m2
3
imes hea ie han hese. The ligh boson mass,
M
,
has a ac ional powe dependence on he e mion mass m:
M∝e1
3mcos θ
22
3(81)
Many o hese esul s ha e been co obo a ed by se e al au ho s bo h in he
con inuum [57–61]
and using he la ice app oach [
62
,
63
]. Coleman concluded his pape
[
7
] by asking some ques ions conce ning hings he did no unde s and, and we ci e he e wo
o hem:
1.
Why a e he ligh es pa icles in he heo y a degene a e iso iple , e en i one qua k
is 10 imes hea ie han he o he ?
2. Why does he nex -ligh es pa icle has IPG =0++, a he han 0−−?
The esul s o Sec ion 3allow us o quali a i ely unde s and he main Coleman’s
indings o he wo- la o model wi h degene a e masses in he s ong coupling limi , as
well as o gi e a eliable answe o he p e ious ques ions.
In Sec ion 3.5, we p edic , om he in e play be ween he
UA(
1
)
anomaly and he exac
SUA(
2
)
chi al symme y, a singula beha io o he acuum ene gy densi y
(49) and (63)
in
he chi al limi limi as
E∼Cmcos θ
2δ+1
δ
. In he Schwinge model, a simple dimensional
analysis ell us ha
C
mus be p opo ional o
eδ−1
δ
. The e o e, ou esul ma ches pe ec ly
Coleman’s esul (79) i we choose δ=3.
In wha conce ns he masses o he ligh bosons, ou p edic ion (65),
m¯
π
,
mσ∼
mcos θ
2δ+1
dδma ches, o δ=3, Colemans’s esul (81) oo.
In Sec ion 3.5, we also p edic ha he la o -single scala suscep ibili y (67) and
he “pion” suscep ibili y (70) and (73), should di e ge in he chi al limi as
K
δm1−δ
δeδ−1
δ
and
Km1−δ
δeδ−1
δ
, espec i ely ( he ac o
eδ−1
δ
comes again om dimensional analysis in he
Schwinge model), and o δ=3 we ha e
χσm→0=K
3m−2
3e2
3∼| h0|ˆ
Oσ|σi |2
mσ
χπ0m→0=Km−2
3e2
3∼| h0|ˆ
Oπ0|π0i |2
mπ0
(82)
whe e Kis a dimensionless cons an .
We also show ha he
σ
and
¯
π
meson masses, in he s ong-coupling limi , scale wi h
he qua k mass as
m¯
π,mσ∼m2
3e1
3. (83)
Taking in o accoun ha he
SUA(
2
)
symme y is exac in he chi al limi ,
Equa ions (82)
and (83) imply ha
lim
m→0| h0|ˆ
Oσ|σi |2=lim
m→0| h0|ˆ
Oπ0|π0i |2∼e(84)
and he e o e we ha e
lim
m→0
χπ0(m,e)
χσ(m,e)=lim
m→0
mσ(m,e)
mπ0(m,e)=3 (85)
These esul s show ha indeed he ligh es pa icle in he heo y is an iso iple , and
he nex ligh es is an isosingle
IPG =
0
++
. Howe e , ou esul o he a io
mσ
m¯
π=
3 [
45
]
is in disag eemen wi h Coleman’s esul mσ
m¯
π=√3 [7].
In wha conce ns he i s Coleman’s ques ion, we a gue in Sec ion 3.5 ha he s ong
coupling limi pe o med by Coleman co esponds o he ze o-o de con ibu ion o he
acuum ene gy densi y expansion (59). This ze o-o de con ibu ion depends on he
qua k masses only h ough he combina ion
mu+md
, and we show ha in such a case
Symme y 2021,13, 209 23 o 27
only he opological suscep ibili y is sensi i e o isospin b eaking e ec s. The h ee pion
suscep ibili ies (70) and (73) and masses a e equal, and o see isospin b eaking e ec s we
should go o he second-o de con ibu ion. The ele ance o he second-o de co ec ion
o he ze o-o de con ibu ion o he acuum ene gy densi y is also es ima ed (62), and
o
θ=
0 u ns ou o be o he o de o
(md−mu)2
(mu+md)4
3e2
3
, a esul ha jus i ies he alidi y
o he ze o-o de app oxima ion in he s ong-coupling (
e
mu,d
1) limi (Geo gi ecen ly
a gued [
64
] ha isospin b eaking e ec s a e exponen ially supp essed in he wo- la o
Schwinge model as a consequence o con o mal coalescence).
The analysis done in his sec ion s ongly sugges s ha he exis ence o quasi-massless
chi al bosons in he spec um o he wo- la o Schwinge model, nea he chi al limi ,
does no o igina es in some unin e es ing peculia i ies o wo-dimensional models, bu i
should be a consequence o he in e play be ween exac non-abelian chi al symme y, and
an e ec i ely b oken
UA(
1
)
anomalous symme y. Wha is a wo-dimensional peculia i y
is he ac ha , in he chi al limi , when all e mion masses anish, hese quasi-massless
bosons become uns able, and he low-ene gy spec um o he model educes o a massless
non-in e ac ing boson, in acco dance wi h Coleman’s heo em [
56
] which o bids he
exis ence o massless in e ac ing bosons in wo dimensions.
5. Conclusions and Discussion
Thanks o massi e
QCD
simula ions on he la ice, we ha e a p esen a good quali a-
i e and quan i a i e unde s anding o he acuum ealiza ion o he non-abelian
SUA(N )
chi al symme y, as a unc ion o he physical empe a u e. As a as he
UA(
1
)
anomaly
and i s associa ed
θ
pa ame e a e conce ned, and especially in he high empe a u e phase,
he cu en si ua ion is howe e a om sa is ac o y. Wi h he aim o cla i ying he cu en
s a us conce ning his issue, we de o e he i s pa o his a icle o analyzing he p esen
s a us o he in es iga ions on he e ec i eness o he
UA(
1
)
axial anomaly in
QCD
, a em-
pe a u es a ound and abo e he non-abelian chi al ansi ion c i ical empe a u e. We show
ha heo e ical p edic ions equi e assump ions whose alidi y is no always p o en, and
la ice simula ions using di e en disc e iza ion schemes lead o appa en ly con adic o y
conclusions in se e al cases. Hence, despi e he g ea e o de o ed o in es iga ing he
a e o he axial anomaly in he chi ally symme ic phase o
QCD
, we s ill do no ha e a
clea answe o his ques ion.
In he second pa o he a icle we analyze some ecen ly sugges ed [
45
] in e es ing
physical implica ions o he
UA(
1
)
anomaly, in sys ems whe e he non-abelian axial sym-
me y is ul illed in he acuum. The s anda d wisdom on he o igin o massless bosons in
he spec um o a Quan um Field Theo y, desc ibing he in e ac ion o gauge ields coupled
o ma e ields, is based on wo well known ea u es: gauge symme y and spon aneous
symme y b eaking o con inuous symme ies. We show ha he opological p ope ies o
he heo y can be he basis o an al e na i e mechanism, o he han Golds one’s heo em,
o gene a e massless bosons in he chi al limi , i he
UA(
1
)
symme y emains e ec i ely
b oken, and he non-abelian SUA(N )chi al symme y is ul illed in he acuum.
The wo- la o Schwinge model, o Quan um Elec odynamics in wo space- ime
dimensions, is a good es -bed o ou p edic ions. Indeed, he Schwinge model shows a
non- i ial opology, which induces he
UA(
1
)
axial anomaly. Mo eo e , in he wo- la o
case, he non-abelian
SUA(
2
)
chi al symme y is ul illed in he acuum, as equi ed by
Coleman’s heo em [
56
] on he impossibili y o b eak spon aneously con inuous symme-
ies in wo dimensions. This model was analyzed by Coleman long ago in [
7
], whe e
he compu ed some quan i a i e p ope ies o he heo y in he con inuum o bo h weak
coupling,
e
m
1, and s ong coupling
e
m
1. In wha conce ns he s ong-coupling
esul s, he main Coleman indings a e quali a i ely in ag eemen wi h ou p edic ions.
The acuum ene gy densi y, and he chi al condensa e shows a singula dependence on
he e mion mass,
m
, in he chi al limi , and he la o single scala suscep ibili y di e ges
Symme y 2021,13, 209 24 o 27
when
m→
0. In addi ion, ou esul s p o ide a eliable answe o some ques ions ha
Coleman asked himsel .
I is wo h wonde ing i he eason o he ich spec um o ligh chi al bosons nea he
chi al limi , ound in he Schwinge [
7
] and
U(N)
[
65
] models, lies in some unin e es ing
peculia i ies o wo-dimensional models, o i he e is a deepe and gene al explana ion o
his phenomenon. We wan o ema k, conce ning his, ha he analysis done in
Sec ion 4
s ongly sugges s ha he exis ence o quasi-massless chi al bosons in he spec um o he
wo- la o Schwinge model, nea he chi al limi , does no o igina es in some unin e es ing
peculia i ies o wo-dimensional models bu i should be a consequence o he in e play
be ween exac non-abelian chi al symme y, and an e ec i ely b oken
UA(
1
)
anomalous
symme y. Wha is a wo-dimensional peculia i y is he ac ha , in he chi al limi , when all
e mion masses anish, hese quasi-massless bosons become uns able, and he low-ene gy
spec um o he model educes o a massless non-in e ac ing boson [
66
,
67
], in acco dance
wi h Coleman’s heo em [
56
] which o bids he exis ence o massless in e ac ing bosons in
wo dimensions.
In wha conce ns
QCD
, he analysis o he e ec s o he
UA(
1
)
axial anomaly in
i s high empe a u e phase, in which he non-abelian chi al symme y is es o ed in he
g ound s a e, has a oused much in e es in ecen ime because o i s ele ance in axion
phenomenology. Mo eo e , he way in which he
UA(
1
)
anomaly mani es s i sel in he
chi al symme y es o ed phase o
QCD
a high empe a u e could be es ed when p obing
he QCD phase ansi ion in ela i is ic hea y ion collisions.
We a gue in Sec ion 3 ha a quan um ield heo y, wi h an exac non-abelian
SUA(
2
)
symme y, and in which he
UA(
1
)
axial symme y is e ec i ely b oken, should exhibi a
singula qua k-mass dependence in he acuum ene gy densi y and a di e gen co ela ion
leng h in he co ela ion unc ion o he scala condensa e, in he chi al limi . On he
con a y, i all co ela ion leng hs a e ini e, and hence he acuum ene gy densi y is an
analy ical unc ion o he qua k mass, we show ha he acuum ene gy densi y becomes, a
leas up o second o de in he qua k masses,
θ
-independen . The opological suscep ibili y
ei he anish o is a leas o ou h o de in he qua k masses and, in such a case, all ypical
e ec s o he
UA(
1
)
anomaly a e los .
QCD
in he chi ally symme ic phase,
T'Tc
, shows
an exac non-abelian axial symme y, and, hence, ei he he acuum ene gy densi y is an
analy ical unc ion o he qua k masses and
QCD
becomes
θ
-independen o he sc eening
mass spec um o he model shows se e al quasi-massless chi al bosons, whose masses
anish in he chi al limi . Which o he wo a o emen ioned possibili ies ac ually happens
in he high empe a u e phase o
QCD
is a di icul ques ion, as ollows om he cu en
s a us o la ice simula ions epo ed in his a icle.
A ecen la ice calcula ion [
39
] o he opological p ope ies o h ee- la o
QCD
wi h
physical qua k masses and empe a u es a ound 500 MeV gi es as a esul a small bu
non- anishing opological suscep ibili y, al hough wi h la ge e o ba s in he con inuum
limi ex apola ions, sugges ing ha he e ec s o he
UA(
1
)
axial anomaly s ill pe sis a
hese empe a u es. I we assume his o be ue, and hence ha he e is a empe a u e
in e al in he high empe a u e phase whe e he
UA(
1
)
anomalous symme y emains
e ec i ely b oken, we can apply o his empe a u e in e al he main conclusions o
Sec ion 3.
Taking in o accoun la ice de e mina ion o he ligh qua k masses [
68
] (
mu≃2 MeV
,
md≃
5 MeV,
ms≃
94 MeV), we can conside
QCD
wi h wo quasi-massless qua ks
as a good app oach. The esul s o Sec ion 3p edic hen a spec um o ligh
σ
and
¯
π
mesons a
T'Tc
. The p esence o hese ligh scala and pseudoscala mesons in he
chi ally symme ic high empe a u e phase o
QCD
could, on he o he hand, signi ican ly
in luence he dilep on and pho on p oduc ion obse ed in he pa icle spec um [
69
] a
hea y-ion collision expe imen s.
La ice calcula ions o mesonic sc eening masses in wo- [
34
] and h ee- la o [
41
]
QCD
, a ound and abo e he c i ical empe a u e, gi e esul s ha a e un o una ely no
enough o allow a good check o ou spec um p edic ion. Howe e , he esul s o [
41
]
Symme y 2021,13, 209 25 o 27
show a small change o he pion sc eening-mass when c ossing he c i ical empe a u e
and a dec easing sc eening mass, a T'Tc, when going om he ¯
us o he ¯
ud channel.
Funding:
This wo k was unded by FEDER/Minis e io de Ciencia e Inno ación unde G an No.
PGC2018-095328-B-I00 (MCI/AEI/FEDER, UE).
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Con lic s o In e es : The au ho decla es no con lic o in e es .
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