Axial Ua (1) anomaly: A new mechanism to generate massless bosons

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. 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