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Light scalar susceptibilities and the π0−η mixing Ricardo Torres Andrés and Ángel Gómez Nicola DepartamentodeFísicaTeóricaII.Univ.Complutense.28040Madrid.Spain. Abstract. WehaveperformedathermalanalysisofthelightscalarsusceptibilitiesinthecontextofSU(3)-ChiralPerturbation Theory to one loop taking into account the QCD source of isospin breaking (IB), i.e corrections coming from mu (cid:54)=md. We find that the value of the connected scalar susceptibility in the infrared regime and below the critical temperature is 1 entirelydominatedbytheπ0−η mixing,whichleadstomodel-independentO(ε0)corrections,whereε∼md−mu,inthe 1 combinationχuu−χud offlavourbreakingsusceptibilities. 0 Keywords: ChiralPerturbationTheory,Scalarsusceptibilities,Isospinbreaking 2 PACS: 11.10.Wx,12.39.Fe,25.75.-q,21.65.Jk n a INTRODUCTION induces mass differences for the light mesons through J the presence of virtual photons. These corrections have 0 Thelow-energysectorofQCDhasbeensuccessfullyde- beenincludedintheChPTeffectivelagrangian[6,7]by 1 scribed within the chiral lagrangian framework. Chiral meansoftermslikeL ,L andsoon,withetheelec- e2 e2p2 PerturbationTheory(ChPT)isbasedonthespontaneous tric charge. These terms are easily incorporated in the ] h breaking of chiral symmetry and provides a consistent, ChPT power counting scheme by considering formally p systematic and model-independent scheme to calculate e2=O(p2/F2). - p low-energy observables [1, 2, 3]. This formalism has Theaimofthisworkistoexplorewithinthethermal e beenalsoextendedtoincludefinitetemperatureeffects, ChPTformalismtheIBcorrectionstothenext-to-leading h in order to describe meson gases and their evolution quark condensates and their corresponding light scalar [ towards chiral symmetry restoration [4, 5]. The effec- susceptibilities, both physical objects being directly re- 1 tive ChPT lagrangian is constructed as an expansion of latedtochiralsymmetryrestoration.Moredetailscanbe v the form L =L +L +... where p denotes a me- foundin[8]. p2 p4 2 son momentum or mass compared to the chiral scale 7 Λ ∼4πF(cid:39)1GeVwhereF isthepiondecayconstant 7 inχthechirallimit.Eachtermoftheexpansionisaccom- LIGHTQUARKCONDENSATESTO 1 paniedbyalowenergyconstant(LEC)whichhastobe ONELOOP . 1 determinedexperimentally. 0 ChPT can take into account both QCD (due to the We have calculated to one loop the light quark con- 1 1 light quark masses diference mu−md (cid:54)=0) and electro- densates (cid:104)u¯u(cid:105) and (cid:104)d¯d(cid:105) in SU(3)-ChPT taking into ac- : magneticIBbymeansofnewtermsthatimplementthe count both sources of IB. The main distinctive feature v chiralsymmetrybreakingpattern.Theformergenerates with respect to SU(2)-ChPT calculations is that, in this i X a π0−η mixing in the SU(3) lagrangian which intro- case, as commented above, a π0−η mixing term ap- r duces corrections of order (md−mu)/ms which will be pears through the tree-level mixing angle ε defined by a important when considering some combinations of the tan2ε=√3md−mu.Thesumanddifferenceofquarkcon- light scalar susceptibilities at finite temperature. On the 2 ms−mˆ densatesare otherhand,thepresenceofelectromagneticinteractions (cid:18) (cid:19) (cid:104)u¯u+d¯d(cid:105)(T3)=(cid:104)u¯u+d¯d(cid:105)(03)+2F2B0 13(cid:0)3−sin2ε(cid:1)gπ0(T)+2gπ±(T)+gK0(T)+gK±(T)+31(cid:0)1+sin2ε(cid:1)gη(T) (1) (cid:18) (cid:19) sin2ε (cid:104)u¯u−d¯d(cid:105)(3)=(cid:104)u¯u−d¯d(cid:105)(3)+2F2B √ [g (T)−g (T)]+g (T)−g (T) (2) T 0 0 3 π0 η K± K0 whereB = Mπ2 +O(ε),and differentiatingwithrespecttoε∼ md−mu,sothesuppres- 0 mu+md sionofthethermalfunctionsissmalmlesrinthecaseofthe 1 (cid:90) ∞ p2 1 susceptibilitiesthaninthequarkcondensate. g(T)= dp , i 4π2F2 0 EpeβEp−1 Because of the linearity in ε of (2) for a small mix- ingangle,thecombinations χ −χ and χ −χ re- uu ud dd du withE2=p2+M2andβ =T−1. ceiveanO(1)IBcorrectionduetoπ0−ηmixing,which p i Thesubscript0referstothezerotemperatureresults, wouldnotbefoundifmu=md istakenfromthebegin- which can be found in [9]. As a nontrivial check of our ning.Theanalysisoftheε-dependenceof(2)showsthat, calculation,onecanseethatthecondensates(1)-(2)are up to O(ε), χuu (cid:39)χdd, so combinations like χuu−χdd, finite and µ-scale independent with the renormalization whichalsovanishwithmu=md,arelesssensitivetoIB. oftheLEC,includingtheEMones,givenin[3,7]. One can also relate these flavour breaking suscepti- bilities with the connected and disconnected ones [10], often used in lattice analysis [11, 12]: χ = χ , and dis ud LIGHTSCALARSUSCEPTIBILITIES χ = 1(χ +χ −2χ ).Fromthepreviousanalysis, con 2 uu dd ud ANDTHEROLEOFTHEπ0−η MIXING wegetχcon(cid:39)χuu−χud. Therefore, our model-independent analysis including IBeffectsprovidestheleadingnonzerocontributionfor Different light quark masses allow to consider three in- the connected susceptibility which arises partially from dependentlightscalarsusceptibilitiesdefinedas π0−η mixing. This is particularly interesting for the ∂ ∂2 lattice, where artifacts such as taste-breaking mask the χ =− (cid:104)q¯ q (cid:105)= logZ(m (cid:54)=m ). (3) ij ∂m j j ∂m∂m u d behaviour of χcon with the quark mass and T when ap- i i j proaching the continuum limit [12]. In fact, our ChPT For the sake of simplicity we are setting e=0 from approachisusefultoexplorethechirallimit(m →0) u,d nowon,sinceelectromagneticcorrectionsaresmalland orinfrared(IR)regime,whichgivesaqualitativepicture theyarenotrelevantforourpresentdiscussion.Then,to ofthebehaviournearchiralsymmetryrestoration.Inthis leadingorderinthemixingangle,thecontributionofthe regime Mπ (cid:28) T (cid:28) MK, and therefore we can neglect π0−ηmixinginthequarkcondensatesum(1)isoforder thermal heavy particles, which are exponentially supp- ε2 whereas for (2) it goes like ε. The thermal functions resed. g(T,M),i=π0,η aresuppressedbythosecoefficients Theleadingorderresultsfortheconnectedanddiscon- i i andthequarkcondensatesdonotreceiveimportantcor- nectedsusceptibilitiesatzerotemperaturearethefollow- rections.Theimportantpointisthatdifferentiatingwith ing respect to a light quark mass is essentially the same as B2 (cid:18) M2(cid:19) B2 M2 χIR(T =0)=8B2[2Lr(µ)+Hr(µ)]− 0 1+log K − 0 log η +O(ε), (4) con 0 8 2 16π2 µ2 24π2 µ2 3B2 (cid:18) M2(cid:19) B2 (cid:32) M2 (cid:33) χIR(T =0)=32B2Lr(µ)− 0 1+log π + 0 5log η −1 +O(ε). (5) dis 0 6 32π2 µ2 288π2 µ2 Thelogtermofequation(5)isthedominantatT =0 butionsoforderO(1)inthemixingangle. andcanbefoundin[10],buttheconnectedIRsuscepti- If we consider the pion gas in a thermal bath, then bility(4)isnotzeroatT =0,becauseitreceivescontri- expressions(4)-(5)aremodifiedaccordingto B2 T2 (cid:32) T2 (cid:33) (cid:18) (cid:20) M (cid:21)(cid:19) [χ (T)−χ (0)]IR= 0 +O εB2 +O exp − η,K , (6) con con 18M2 0M2 T η η 3B2 T (cid:32) T2 (cid:33) (cid:18) (cid:20) M (cid:21)(cid:19) [χ (T)−χ (0)]IR= 0 +O εB2 +O exp − η,K . (7) dis dis 16π M 0M2 T π η Figure (1) and (2) show, respectively, the connected 0.012 susceptib√ility (6) for fixed tree level eta mass (propor- tionalto B m intheIRregime),andthedisconnected 2B0 0.010 one (7) for s0evesral values of the light quark mass ratio 0 0.008(cid:76)(cid:68)(cid:144) m/m ,andalsowithfixedtreeleveletamass.Theleading Χcon 0.006(cid:72) scalinsgwithT andthelightquarkmassinthisregimefor (cid:45) T (cid:76) thedisconnectedpiecegoeslike √T ,i.ethesamescaling Χcon 0.004(cid:72) calculatedin[10,11];whereasthemconnectedsusceptibil- 0.002(cid:64) itygrowsquadraticinT overamassscalemuchgreater 0.000 thantheSU(2)Goldstoneboson’sone.Therefore,inthe 0 50 100 150 200 continuum limit, we only expect χ to peak near the dis TMeV transition. FIGURE 1. Connected IR susceptibility normalized to B2, 0 ACKNOWLEDGMENTS forfixedtreeleveletamass. (cid:72) (cid:76) R.T.A would like to thank Cándida García Jiménez and Buenaventura Andrés López for invaluable advice. Work partially supported by the Spanish research con- tracts FPA2008-00592, FIS2008-01323, UCM-BSCH 0.20 2B0 mmmmss(cid:61)(cid:61)00..0015 G01R35687/20)8. 910309 and the FPI programme (BES-2009- 0Χdis 0.15(cid:72)(cid:76)(cid:68)(cid:144)(cid:42)(cid:42)(cid:42) mm(cid:144)(cid:144)mmss(cid:61)(cid:61)00..12 (cid:45) 0.10 T (cid:76) (cid:144) REFERENCES dis (cid:72) (cid:144) Χ 00..0005(cid:64)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) 12.. SJ..GWaesisnebreargn,dPHh.ysLiecuatAw9y6le,r3,2A7n(n1a9l7s9P)h.ys.158,(1984) 0 50 100 150 200 142. 3. J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 465 TMeV (1985). FIGURE 2. Disconnected IR susceptibility normalized to 4. J.GasserandH.Leutwyler,Phys.Lett.B184,83(1987). B2,forseverallightquarkmass(cid:72) rati(cid:76)osandfixedtreeleveleta 5. P.GerberandH.Leutwyler,Nucl.Phys.B321,387 0 mass. (1989). 6. G.Ecker,J.Gasser,A.PichandE.deRafael,Nucl.Phys. B321,311(1989). 7. R.Urech,Nucl.Phys.B433,234(1995). 8. A.G.NicolaandR.T.Andres,inpreparation. Notethat,aswehavealreadymentioned,theetamass 9. A.G.NicolaandR.T.Andres,arXiv:1009.2170[hep-ph]. terminequation(6)andinthesubleadingcorrectionsin 10. A.V.SmilgaandJ.J.M.Verbaarschot,Phys.Rev.D54, the mixing angle comes from the ε-analysis and the IR 1087(1996). expansionoftheg1(Mπ),anddoesnothaveanythingto 11. S.Ejirietal.,Phys.Rev.D80(2009)094505. dowiththermaletas. 12. C.E.DeTar,PoSLATTICE2008(2008)001.

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Light scalar susceptibilities and the $π^0-η$ mixing PDF

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Light scalar susceptibilities and the π0−η mixing Ricardo Torres Andrés and Ángel Gómez Nicola DepartamentodeFísicaTeóricaII.Univ.Complutense.28040Madrid.Spain. Abstract. WehaveperformedathermalanalysisofthelightscalarsusceptibilitiesinthecontextofSU(3)-ChiralPerturbation Theory to one loop taking into account the QCD source of isospin breaking (IB), i.e corrections coming from mu (cid:54)=md. We find that the value of the connected scalar susceptibility in the infrared regime and below the critical temperature is 1 entirelydominatedbytheπ0−η mixing,whichleadstomodel-independentO(ε0)corrections,whereε∼md−mu,inthe 1 combinationχuu−χud offlavourbreakingsusceptibilities. 0 Keywords: ChiralPerturbationTheory,Scalarsusceptibilities,Isospinbreaking 2 PACS: 11.10.Wx,12.39.Fe,25.75.-q,21.65.Jk n a INTRODUCTION induces mass differences for the light mesons through J the presence of virtual photons. These corrections have 0 Thelow-energysectorofQCDhasbeensuccessfullyde- beenincludedintheChPTeffectivelagrangian[6,7]by 1 scribed within the chiral lagrangian framework. Chiral meansoftermslikeL ,L andsoon,withetheelec- e2 e2p2 PerturbationTheory(ChPT)isbasedonthespontaneous tric charge. These terms are easily incorporated in the ] h breaking of chiral symmetry and provides a consistent, ChPT power counting scheme by considering formally p systematic and model-independent scheme to calculate e2=O(p2/F2). - p low-energy observables [1, 2, 3]. This formalism has Theaimofthisworkistoexplorewithinthethermal e beenalsoextendedtoincludefinitetemperatureeffects, ChPTformalismtheIBcorrectionstothenext-to-leading h in order to describe meson gases and their evolution quark condensates and their corresponding light scalar [ towards chiral symmetry restoration [4, 5]. The effec- susceptibilities, both physical objects being directly re- 1 tive ChPT lagrangian is constructed as an expansion of latedtochiralsymmetryrestoration.Moredetailscanbe v the form L =L +L +... where p denotes a me- foundin[8]. p2 p4 2 son momentum or mass compared to the chiral scale 7 Λ ∼4πF(cid:39)1GeVwhereF isthepiondecayconstant 7 inχthechirallimit.Eachtermoftheexpansionisaccom- LIGHTQUARKCONDENSATESTO 1 paniedbyalowenergyconstant(LEC)whichhastobe ONELOOP . 1 determinedexperimentally. 0 ChPT can take into account both QCD (due to the We have calculated to one loop the light quark con- 1 1 light quark masses diference mu−md (cid:54)=0) and electro- densates (cid:104)u¯u(cid:105) and (cid:104)d¯d(cid:105) in SU(3)-ChPT taking into ac- : magneticIBbymeansofnewtermsthatimplementthe count both sources of IB. The main distinctive feature v chiralsymmetrybreakingpattern.Theformergenerates with respect to SU(2)-ChPT calculations is that, in this i X a π0−η mixing in the SU(3) lagrangian which intro- case, as commented above, a π0−η mixing term ap- r duces corrections of order (md−mu)/ms which will be pears through the tree-level mixing angle ε defined by a important when considering some combinations of the tan2ε=√3md−mu.Thesumanddifferenceofquarkcon- light scalar susceptibilities at finite temperature. On the 2 ms−mˆ densatesare otherhand,thepresenceofelectromagneticinteractions (cid:18) (cid:19) (cid:104)u¯u+d¯d(cid:105)(T3)=(cid:104)u¯u+d¯d(cid:105)(03)+2F2B0 13(cid:0)3−sin2ε(cid:1)gπ0(T)+2gπ±(T)+gK0(T)+gK±(T)+31(cid:0)1+sin2ε(cid:1)gη(T) (1) (cid:18) (cid:19) sin2ε (cid:104)u¯u−d¯d(cid:105)(3)=(cid:104)u¯u−d¯d(cid:105)(3)+2F2B √ [g (T)−g (T)]+g (T)−g (T) (2) T 0 0 3 π0 η K± K0 whereB = Mπ2 +O(ε),and differentiatingwithrespecttoε∼ md−mu,sothesuppres- 0 mu+md sionofthethermalfunctionsissmalmlesrinthecaseofthe 1 (cid:90) ∞ p2 1 susceptibilitiesthaninthequarkcondensate. g(T)= dp , i 4π2F2 0 EpeβEp−1 Because of the linearity in ε of (2) for a small mix- ingangle,thecombinations χ −χ and χ −χ re- uu ud dd du withE2=p2+M2andβ =T−1. ceiveanO(1)IBcorrectionduetoπ0−ηmixing,which p i Thesubscript0referstothezerotemperatureresults, wouldnotbefoundifmu=md istakenfromthebegin- which can be found in [9]. As a nontrivial check of our ning.Theanalysisoftheε-dependenceof(2)showsthat, calculation,onecanseethatthecondensates(1)-(2)are up to O(ε), χuu (cid:39)χdd, so combinations like χuu−χdd, finite and µ-scale independent with the renormalization whichalsovanishwithmu=md,arelesssensitivetoIB. oftheLEC,includingtheEMones,givenin[3,7]. One can also relate these flavour breaking suscepti- bilities with the connected and disconnected ones [10], often used in lattice analysis [11, 12]: χ = χ , and dis ud LIGHTSCALARSUSCEPTIBILITIES χ = 1(χ +χ −2χ ).Fromthepreviousanalysis, con 2 uu dd ud ANDTHEROLEOFTHEπ0−η MIXING wegetχcon(cid:39)χuu−χud. Therefore, our model-independent analysis including IBeffectsprovidestheleadingnonzerocontributionfor Different light quark masses allow to consider three in- the connected susceptibility which arises partially from dependentlightscalarsusceptibilitiesdefinedas π0−η mixing. This is particularly interesting for the ∂ ∂2 lattice, where artifacts such as taste-breaking mask the χ =− (cid:104)q¯ q (cid:105)= logZ(m (cid:54)=m ). (3) ij ∂m j j ∂m∂m u d behaviour of χcon with the quark mass and T when ap- i i j proaching the continuum limit [12]. In fact, our ChPT For the sake of simplicity we are setting e=0 from approachisusefultoexplorethechirallimit(m →0) u,d nowon,sinceelectromagneticcorrectionsaresmalland orinfrared(IR)regime,whichgivesaqualitativepicture theyarenotrelevantforourpresentdiscussion.Then,to ofthebehaviournearchiralsymmetryrestoration.Inthis leadingorderinthemixingangle,thecontributionofthe regime Mπ (cid:28) T (cid:28) MK, and therefore we can neglect π0−ηmixinginthequarkcondensatesum(1)isoforder thermal heavy particles, which are exponentially supp- ε2 whereas for (2) it goes like ε. The thermal functions resed. g(T,M),i=π0,η aresuppressedbythosecoefficients Theleadingorderresultsfortheconnectedanddiscon- i i andthequarkcondensatesdonotreceiveimportantcor- nectedsusceptibilitiesatzerotemperaturearethefollow- rections.Theimportantpointisthatdifferentiatingwith ing respect to a light quark mass is essentially the same as B2 (cid:18) M2(cid:19) B2 M2 χIR(T =0)=8B2[2Lr(µ)+Hr(µ)]− 0 1+log K − 0 log η +O(ε), (4) con 0 8 2 16π2 µ2 24π2 µ2 3B2 (cid:18) M2(cid:19) B2 (cid:32) M2 (cid:33) χIR(T =0)=32B2Lr(µ)− 0 1+log π + 0 5log η −1 +O(ε). (5) dis 0 6 32π2 µ2 288π2 µ2 Thelogtermofequation(5)isthedominantatT =0 butionsoforderO(1)inthemixingangle. andcanbefoundin[10],buttheconnectedIRsuscepti- If we consider the pion gas in a thermal bath, then bility(4)isnotzeroatT =0,becauseitreceivescontri- expressions(4)-(5)aremodifiedaccordingto B2 T2 (cid:32) T2 (cid:33) (cid:18) (cid:20) M (cid:21)(cid:19) [χ (T)−χ (0)]IR= 0 +O εB2 +O exp − η,K , (6) con con 18M2 0M2 T η η 3B2 T (cid:32) T2 (cid:33) (cid:18) (cid:20) M (cid:21)(cid:19) [χ (T)−χ (0)]IR= 0 +O εB2 +O exp − η,K . (7) dis dis 16π M 0M2 T π η Figure (1) and (2) show, respectively, the connected 0.012 susceptib√ility (6) for fixed tree level eta mass (propor- tionalto B m intheIRregime),andthedisconnected 2B0 0.010 one (7) for s0evesral values of the light quark mass ratio 0 0.008(cid:76)(cid:68)(cid:144) m/m ,andalsowithfixedtreeleveletamass.Theleading Χcon 0.006(cid:72) scalinsgwithT andthelightquarkmassinthisregimefor (cid:45) T (cid:76) thedisconnectedpiecegoeslike √T ,i.ethesamescaling Χcon 0.004(cid:72) calculatedin[10,11];whereasthemconnectedsusceptibil- 0.002(cid:64) itygrowsquadraticinT overamassscalemuchgreater 0.000 thantheSU(2)Goldstoneboson’sone.Therefore,inthe 0 50 100 150 200 continuum limit, we only expect χ to peak near the dis TMeV transition. FIGURE 1. Connected IR susceptibility normalized to B2, 0 ACKNOWLEDGMENTS forfixedtreeleveletamass. (cid:72) (cid:76) R.T.A would like to thank Cándida García Jiménez and Buenaventura Andrés López for invaluable advice. Work partially supported by the Spanish research con- tracts FPA2008-00592, FIS2008-01323, UCM-BSCH 0.20 2B0 mmmmss(cid:61)(cid:61)00..0015 G01R35687/20)8. 910309 and the FPI programme (BES-2009- 0Χdis 0.15(cid:72)(cid:76)(cid:68)(cid:144)(cid:42)(cid:42)(cid:42) mm(cid:144)(cid:144)mmss(cid:61)(cid:61)00..12 (cid:45) 0.10 T (cid:76) (cid:144) REFERENCES dis (cid:72) (cid:144) Χ 00..0005(cid:64)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42)(cid:42) 12.. SJ..GWaesisnebreargn,dPHh.ysLiecuatAw9y6le,r3,2A7n(n1a9l7s9P)h.ys.158,(1984) 0 50 100 150 200 142. 3. J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 465 TMeV (1985). FIGURE 2. Disconnected IR susceptibility normalized to 4. J.GasserandH.Leutwyler,Phys.Lett.B184,83(1987). B2,forseverallightquarkmass(cid:72) rati(cid:76)osandfixedtreeleveleta 5. P.GerberandH.Leutwyler,Nucl.Phys.B321,387 0 mass. (1989). 6. G.Ecker,J.Gasser,A.PichandE.deRafael,Nucl.Phys. B321,311(1989). 7. R.Urech,Nucl.Phys.B433,234(1995). 8. A.G.NicolaandR.T.Andres,inpreparation. Notethat,aswehavealreadymentioned,theetamass 9. A.G.NicolaandR.T.Andres,arXiv:1009.2170[hep-ph]. terminequation(6)andinthesubleadingcorrectionsin 10. A.V.SmilgaandJ.J.M.Verbaarschot,Phys.Rev.D54, the mixing angle comes from the ε-analysis and the IR 1087(1996). expansionoftheg1(Mπ),anddoesnothaveanythingto 11. S.Ejirietal.,Phys.Rev.D80(2009)094505. dowiththermaletas. 12. C.E.DeTar,PoSLATTICE2008(2008)001.

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