Table Of ContentLight 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
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