The $\pi^0-\eta-\eta'$ mixing in a generalized multi-quark interaction scheme

0.26 MB·English

by  Alexander A. Osipov

The π0 η η′ mixing in a generalized multi-quark interaction scheme.
− −
A. A. Osipov , B. Hiller and A. H. Blin
∗ † ‡
CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal
We investigate the isospin symmetry breaking effects within a recently derived Nambu-Jona-
Lasiniorelatedmodelbyfittingthemeasuredpseudoscalarmesonmassesandweakdecaycouplings
fπ, fK. Our model contains the next to leading order terms in the 1/Nc expansion of the effective
multi-quarkLagrangian, includingtheonesthatbreakthechiralsymmetryexplicitly. Weshowthe
important phenomenological role of these interactions: (1) they lead to an accurate fit of the low-
lyingpseudoscalar nonet characteristics; (2) they account for a verygood agreement of thecurrent
quark masses with the present PDG values; (3) they reduce by 40% the ratio ǫ/ǫ′ of the π0 −η
andπ0−η′ mixingangles, ascompared tothecase thatcontemplates explicit breakingonly in the
6 leading order, bringing it in consonance with the quoted values in the literature. The conventional
1 NJL-typemodels fail in the joint description of these parameters.
0
2 PACSnumbers: 11.30.Rd,11.30.Qc,12.39.Fe,12.40.Yx,14.40.Aq,14.65.Bt

n
Ju I. Introduction ModelpredictionoftheCPviolationrelatedratio(ǫǫ′)CP
[14–17]representinga substantialcorrectiontothe QCD
3
1 The strong isospin symmetry is considered to be a penguin contributions [16]. It affects the value of the
very good approximation in the empirical description of K0 → π0π0 transition through the dominant QCD Q6
] penguin operator, which is one of the sources of uncer-
ph alatleadrgetobuthlkeohfiesrtarrocnhgyininterwahcticiohnbprreoackeinssgeso.fTthheiscihsirrae-l tainties in the determination of (ǫǫ′)CP [18], for a recent
- symmetry SU(3) SU(3) by different current quark review see [19].
p masses occurs, doLw×n to SUR(2) U(1) flavor symme- In chiral perturbation theory (ChPT)[20],[21],[2] the
[he tmryesoifnsmiut,misdac≪curmatse. aItntthheeoIcra×dseeroofftYthheeprsaetuiodoosfcatlhaer fiπr0s−t eηvamluixaitnegdatnogloerdoecrcupr4s ainlretahdeycaotntoerxdteropf2Kaln3dfowrams
v3 l[i1g]h,[t2]aannddsetrxapnlagienscuprarertnltyqtuhaerksmmalalssmeses(omnum−asms dd)i/ffmers- facItnortshiena[2n1a]l.ysis of η − η′ mixing of [22] the U(1)A
enceswithinchargedisospinmultiplets. Afurthersource anomaly is described by the gluon transition matrix el-
6
4 ofisospinbreakingis due to the electromagneticinterac- ement < 0|4απsGG˜|ηi > and the quark flavor basis has
1 tions, which are expected to be suppressed at the scale been used. As shown in [23] this basis is favored, as one
6 of strong interactions. ofthe twomixing anglesisindeedsmall. The decaycon-
0 stants follow the pattern of particle state mixing in that
A detailed quantitative analysis however requires
1. isospinbreakingcorrectionstobetakenintoaccountina basis. It has been shown that this approach leads to re-
0 series oflow energyphenomena, suchas: the description sults consistent with many observables related to η η′
−
5 mixing [22]. In [24],[25] it has been extended to include
of mass splittings of mesons, and Dashen’s theorem [3];
1 the mixing to the neutral pion.
sum rules for quark condensates [2], [4]; kaon decays [5];
v: π π [6],[7] and π K scattering [8],[9] in relation to Inthe presentworkweaddressthe π0 η η′ mixings
i me−sonic atoms, [10],−[11], ρ and τ decays involving η(η ) resulting from a recently proposed Lagr−ang−ian [26],[27],
X ′
mesons [12], a f mixing [13] in the scalar meson sec- which is reviewed below. In this effective Lagrangian
ar tor. 0− 0 approach built from all spin 0 and non-derivative multi-
quark interactions relevant at the scale of spontaneous
Strongisospinbreakingeffectsbecomeparticularlyrel-
chiral symmetry breaking, the complete set of interac-
evantif a certainprocessdepends crucially onthe differ-
tions which break explicitely the chiral symmetry was
ences of the light quark masses. If in addition the elec-
included for the first time. This Lagrangian repre-
tromagnetic interactions are a subleading effect, these
sentsageneralizationoftheoriginalNambu-Jona-Lasinio
processes provide for ideal tools in a quantitative analy-
[28, 29] model extended to the realistic three flavor and
sis of quark mass ratios. In the latter category are the
η,η 3π decays,theπ0 η andπ0 η mixings,aswell color case with U(1)A breaking six-quark ’t Hooft in-
′ ′
→ − − teractions [30–44] and an appropriate set of eight-quark
as the ρ ω mixing in the vector channels.
− interactions [45]. The last ones complete the number
Isospin breaking associated with the π0 η η sys-
− − ′ of vertices which are important in four dimensions for
tem has long been known to play a role in the Standard
dynamical SU(3) SU(3) chiral symmetry breaking
L R
×
[46, 47]. The Lagrangian considers all interactions rele-
vantatthesameorderinlargeN countingastheU(1)
c A
anomaly term.
∗Emailaddress: [email protected]
†Emailaddress: brigitte@teor.fis.uc.pt The role of the new interactions contained in the ex-
‡Emailaddress: alex@teor.fis.uc.pt plicit symmetry breaking (ESB) vertices has been an-
2
alyzed at meson tree level approximation and in the a detailed discussion of the large N counting scheme
c
isospinlimitinconnectionwiththelowlyingcharacteris- which complies with the counting rules based on powers
ticsofthepseudoscalarandscalarmesonnonets[26],[27] of the scale Λ of spontaneous breaking of chiral symme-
and in the T µ phase diagram associated with chiral try. Rephrasing it, means that the terms which vanish
−
transitions [48]. An unprecedented accuracy for the de- as Λ are the same that also vanish on grounds of
→ ∞
scription of the spectra has been achieved. One should N . Applying these rules, summarized after eq. 4,
c
→∞
stress that the present Lagrangian is able to account the followingset ofterms involvinginteractions with the
properly for the SU(3) breaking effects in the descrip- sourcesχ emerge,whichactalsoupto the sameorderin
tion of the weak decay constants f and f , in addition N counting as the ’t Hooft term
π K c
to yieldthe correctempirical η, η and K mesonmasses,
′
as well as the anomalous two photon decays of π,η,η , 10
′
L = L , (3)
inanunified description,whichwasanopenproblemfor χ i
model versions without the ESB terms. Xi=0
The paper is organized as follows. In the next section
where
is presented the effective multiquark Lagrangian and its
bosonized form, the associated Nc counting is reviewed, L0 = tr Σ†χ+χ†Σ
in section III we address the mixing in the π η η −
system, the choicesof representationof states,t−he de−cay′ L = κ¯1(cid:0)e e Σ (cid:1)χ χ +h.c.
1 ijk mnl im jn kl
−Λ
parametersinthe flavorbasis,andthe complianceofthe
κ¯
model in the approximation considered with the decay L = 2e e χ Σ Σ +h.c.
2 Λ3 ijk mnl im jn kl
parameters transforming as the states. In section IV we
g¯
presentanddiscussthenumericalfitsofthemassspectra L = 3tr Σ ΣΣ χ +h.c.
3 Λ6 † †
and decay parameters. We end with a summary of the
main results. L4 = Λg¯46tr(cid:0)Σ†Σ tr (cid:1)Σ†χ +h.c.
g¯5 (cid:0) (cid:1) (cid:0) (cid:1)
II. Survey of the model Lagrangian L5 = Λ4tr Σ†χΣ†χ +h.c.
II a. Multiquark picture
g¯6 (cid:0) (cid:1)
L6 = Λ4tr ΣΣ†χχ†+Σ†Σχ†χ
The Langrangian considered is built from all spin 0 g¯7 (cid:0) 2 (cid:1)
L = trΣ χ+h.c.
chiral SU(3)L SU(3)R symmetric and parity conserv- 7 Λ4 †
×
ing combinations relevant at the scale Λ of spontaneous g¯8 (cid:0) (cid:1)2
breakingofchiralsymmetry. Thismeansthatinthecor- L8 = Λ4 trΣ†χ−h.c.
respondingeffectivepotentialarekeptalltheinteractions g¯9(cid:0) (cid:1)
L = tr Σ χχ χ +h.c.
which do not vanish in the Λ limit. These con- 9 −Λ2 † †
→ ∞
sist of vertices involving non-derivative quark-antiquark g¯10 (cid:0) (cid:1)
fields, denoted by Σ, with Σ = 21(sa − ipa)λa and L10 = −Λ2tr χ†χ tr χ†Σ +h.c. (4)
s = q¯λ q, p = q¯iγ λ q, λ being the standard Gell-
a a a 5 a a (cid:0) (cid:1) (cid:0) (cid:1)
Mann matrices for a=1...8 and λ0 = 2/3 1, as well We recall that the Nc book keeping is as follows:
×
asoftheirinteractionswiththe externalsourcesχwhich
p
transform as χ=(3,3 ) under SU(3) SU(3) ,
L R
∗ × Σ N , Λ N0 =1, G 1/N , κ 1/N3,
L=q¯iγµ∂µq+Lint+Lχ. (1) κ1∼,g9,gc10 1∼/Ncc, ∼ c ∼ c
∼
κ ,g ,g ,g ,g 1/N2, g ,g 1/N3. (5)
The term 2 5 6 7 8 ∼ c 3 4 ∼ c
G¯ κ¯ The scale Λ which gives the right dimensionality to
L = tr Σ Σ + detΣ+detΣ
int Λ2 † Λ5 † the multiquark vertices is related with the cutoff on the
g¯1 (cid:0) (cid:1)2 g¯2(cid:0) (cid:1) divergentquarkone-loopintegrals,itsNc countingisdic-
+ trΣ†Σ + tr Σ†ΣΣ†Σ . (2) tatedbythegapequationsinthephaseofspontaneously
Λ8 Λ8
broken chiral symmetry, eqs. (21),(22),(23) below.
(cid:0) (cid:1) (cid:0) (cid:1)
is well known. Here and elsewhere we use barred quan- The terms L ...L of Lagrangian(4) have been intro-
1 10
tities for any dimensionless coupling, these are related duced recently in the model [26],[27] and generalize to
to the dimensionful ones through powers of the scale Λ, NLO in N the explicit symmetry breaking (ESB) stan-
c
gi = g¯i/Λγ. Lint contains the leading order (LO) in Nc dard LO mass term L0.
four quark (q) NJL interactions with coupling G¯, gener- Since all the blocks L ...L conform with the sym-
0 10
alized to the 3 flavor case, the NLO 6q ’t Hooft U(1)A metry pattern of the model one is free to choose for the
breakingflavordeterminantwithcouplingκ¯,andthetwo source the constant valued matrix χ= /2,
M
possible8qinteractions g¯ ,g¯ ,whichhavethesameN
1 2 c
∼
counting as the ’t Hooft term. We refer to [26],[27] for =µ λ =diag(µ ,µ ,µ ). (6)
a a u d s
M
3
TABLE I: The pseudoscalar masses and weak decay constans (all in MeV) in the isospin limit used as input (marked with *)
fordifferentsetsofthemodel. Parametersets(a),(b)containexplicitsymmetrybreakinginteractions(seeTableIII)andallow
for a fit of the scalar masses and strong decays as well, mσ =550 MeV, mκ =850 MeV, ma0 =mf0 =980 MeV [27]; set (c)
does not. Set (a) corresponds to an octet-singlet mixing angle in the scalar sector of θS =27.5◦, set (b) to θS =25◦.
Sets mπ mK mη mη′ fπ fK
a 138* 494* 547* 958* 92* 113*
b 138* 494* 547* 958* 92* 113*
c 138* 494* 475* 958* 92* 115.7
TABLE II:Parameter sets of themodel: mu =md =mˆ,ms, and Λare given in MeV. The couplings havethe following units:
[G]= GeV−2, [κ]= GeV−5, [g1]= [g2]= GeV−8. We also show here the values of constituent quark masses Mu =Md = Mˆ
and Ms in MeV. See also caption of Table I.
Sets mˆ ms Mˆ Ms Λ G −κ g1 g2
a 4.0* 100* 373 544 828 10.48 122. 3284 173
b 4.0* 100* 372 542 829 9.83 118.5 3305 -158
c 6.1 190 375 569 836 9.79 138.2 2500* 100*
TABLE III: Explicit symmetry breaking interaction couplings. The couplings have the following units: [κ2]= GeV−3, [g3]=
[g4]= GeV−6, [g5]=[g6]=[g7]=[g8]= GeV−4. See also caption of Table I.
Sets κ2 −g3 g4 g5 −g6 −g7 g8
a 6.17 6497 1235 213 1642 13.3 -64
b 5.61 6472 702 210 1668 100 -38
c 0 0 0 0 0 0 0
TABLE IV: The mixing angles in the η−η′ system in isospin limit, and related weak decay constants for the sets discussed
and in comparison with different approaches, see also main text (the systematic error estimates given in [81],[83] have been
omitted here).
Sets θ◦ θ◦ θ◦ f0 f8
P 0 8 fπ fπ
a -12* -1.42 -21.37 1.172 1.318
b -15* -4.42 -24.37 1.172 1.322
c -14.5 -2.82 -24.78 1.197 1.365
[22] phen. -13.3 -6.8 -19.4 1.10 1.19
[22] phen. -15.4 -9.2 -21.2 1.17 1.26
[79] CHPT -10.5 -1.5 -20.0 1.24 1.31
[54] CHPT - -4. -20.5 1.10 1.28
[80] sum rules - -15.6 -10.8 1.39 1.39
[81] Pad´e approximants (η) -16.4 -11.3 -21.3 1.15 1.22
[81] Pad´e approximants (η′) -13.3 -1.5 -24.2 1.28 1.46
[83] BABAR (from [81] (η)) -21.7 -26.7 -16.5 1.04 0.98
[83] BABAR (from [81] (η′)) -17.7 -15.6 -19.9 1.14 1.11
TABLE V:Empirical input used in thefits with isospin breaking, sets A and B with ESB interactions, set C without. Primes
indicate which masses of the pion and kaon multiplets have been used for the fit, the other being output. Masses in units of
MeV, angle ψ in degrees.
Sets m0π m±π mη m′η m0K m±K fπ fK ψ
A,B 136’ 136.6 547 958 500 494’ 92 113 39.7
C 136’ 137.0 477 958 501 497’ 92 116 39.7
4
TABLEVI:Parametersetswithisospinbreaking,setAwithESBinteractions,setBwithout. Thecouplingshavethefollowing
units: [G]=GeV−2,[κ]=GeV−5,[g1]=[g2]=GeV−8,[κ2]=GeV−3,[g3]=[g4]=GeV−6,[g5]=[g6]=[g7]=[g8]=GeV−4.
Λ is given in MeV. See also caption of Table I.
Sets G −κ g1 g2 Λ κ2 g3 g4 g5 g6 g7 g8
A 10.48 116.8 3284 1237 828.5 6.24 2365 1182 160 712 580 44
B 10.48 116.8 3284 1252 828.5 6.26 2481 1182 151 745 591 49
C 9.79 137.4 2500 117 835.7 0 0 0 0 0 0 0
TABLE VII: Isospin breaking parameter r= mmdd+−mmuu, current and constituent quark masses mu, md,ms Mu, Md, Ms in MeV
and π0−η, π0−η′ mixing angles ǫ and ǫ′.
Sets r mu md ms Mu Md Ms ǫ ǫ′ ǫǫ′
A 0.372* 2.179 4.760 95* 372 375 544 0.014* 0.0037* 3.78
B 0.372* 2.166 4.733 95* 372 375 544 0.017* 0.0045* 3.95
C 0.372* 3.774 8.246 194 373 380 573 0.022 0.0025 8.78
Whenever convenient we use in the following the equiv- and powers of 1/N , as well as the counting associated
c
alent redefinition of flavor indices from a = 0,3,8 to withtheθ anglerelatedwiththeU (1)sector,λisfound
A
i=u,d,s for any observable A [49] to be suppressed to all orders in the large N limit [54].
c
Regarding our model Lagrangian, the new couplings,
√2 √2 √2 the primed ones in (9), must not dominate over the N
1 c
Aa =eaiAi, eai = 2√3 √3 −√3 0 . (7) dtheepiernldeaendcinegocfotnhteribuuntpiorinmsetdheoynsecsa.leTahtismmosetanass tthheatunin-
1 1 2
 −  primed couplings, c 1/N , see (5) . Attributing this
  i ∼ c
The terms L1,L9,L10 are related to the Kaplan- counting to the ci suffices to warrant that one can use
Manoharambiguity[50–53]inthedefinitionofthequark the reparametrization freedom (9), in particular to ob-
mass. In the present case this corresponds to the follow- tain κ¯′1 = g¯9′ = g¯1′0 = 0. Indeed, remembering that our
ing freedom in transforming the external source [26] model Lagrangian has been constructed to incorporate
theclassesofmultiquarkinteractionswhichdonotvanish
χ(ci) = χ+ cΛ1 detχ† χ χ†χ −1+ Λc22χχ†χ intheeffectivepotentialasNc →∞inthephaseofspon-
taneously broken chiral symmetry, one gets for this case
c
+ 3tr χ(cid:0)χ χ (cid:1) (cid:0) (cid:1) (8) that κ¯ = c /2, and that up to subleading corrections
Λ2 † 1 − 1
(not considered at this order of the effective potential),
(cid:0) (cid:1)
withthree independent constantsc ,whichhas the same g¯ = c ,g¯ = c , g¯ =g¯ ,g¯ =g¯ ,g¯ =g¯ .
i 9 − 2 10 − 3 5′ 5 7′ 7 8′ 8
symmetrytransformationpropertyasχ. Thedescription
in terms of χ(ci) in the place of χ is equivalent, leading II b. Bosonized version
to the same Lagrangian, with some of the couplings re-
defined as
The low energy meson characteristicaare obtained af-
c
1 ter path integral bosonization of the quark Lagrangian
κ¯1 →κ¯′1 =κ¯1+ 2 , g¯5 →g¯5′ =g¯5−κ¯2c1, (1). Following [34] one may equivalently use the intro-
κ¯ κ¯
2 2 duceds ,p asauxiliaryfields,andafurthersetofphys-
g¯7 →g¯7′ =g¯7+ 2 c1, g¯8 →g¯8′ =g¯8+ 2 c1, icalscalaaraandpseudoscalarfieldsσ =σ λ , φ=φ λ to
a a a a
g¯9 →g¯9′ =g¯9+c2−2κ¯1c1, obtain the vacuum persistence amplitude of the theory
g¯ g¯ =g¯ +c +2κ¯ c . (9) as
10 → 1′0 10 3 1 1
Notethattheredefinitionofthe χfields(8)inthemodel Z= q q¯ σ φ exp i d4xL (q¯,q,σ,φ)
a a q
contains more terms than the one usually considered, in D D D D
Z a a (cid:18)Z (cid:19)
Y Y
the context of second order ChPT, which leads to the
+
∞
following redefinition of the current quark mass [50]
s p exp i d4xL (σ,φ;s,p) .
a a aux
× D D
(λ)= +λ ( † )−1det †. (10) Z Ya Ya (cid:18)Z (cid:19)
M→M M M M M M −∞
(11)
It has been reported that the Kaplan-Manohar ambi-
guity maybe inconflict withthe largeNc counting rules In these variables the Lagrangianreads
of ChPT [54]. Following from a threefold chiral expan-
sioninthenumberofderivatives,powersofquarkmasses L=L +L
q aux
5
TABLE VIII:ǫ and ǫ′ values in the literature.
ǫ ǫ′ ǫ
ǫ′
[63] phen. 0.014 0.0037 3.78
[25] phen. 0.017±0.002 0.004±0.001 4.25±1.17
[79] ChPt NLO 0.014÷0.016 - -
[84] phen. 0.021 - -
[85] Exp. 0.030±0.002 - -
[86] Exp. 0.026±0.007 - -
L =q¯(iγµ∂ (σ+M) iγ φ)q that measure the strength of the meson-meson interac-
q µ 5
− −
L =s (σ +M m )+p φ +L (s,p) tions. From eq. (13) and using (7) one obtains the fol-
aux a a a a a a int
− lowing system of cubic equations for the one index coef-
8
ficients h , (i= u,d,s )
+ L′i(s,p,m). (12) i { }
i=2
X κ h g
M m + t h h + i 2G+g h2+g mh + 2h3
HereL containstheESBtermsandL appearsass m i− i 4 ijk j k 2 1 4 2 i
′i 0 a a
m
inLaux. Theexternalscalarfieldsσ havebeenshiftedto + i 3g h2+g h2+2(g +(cid:0) g )m h +4g mh(cid:1)
σ σ+M, so that the expectation value of the shifted 4 3 i 4 5 6 i i 7
fie→lds in the vacuum corresponding to dynamically bro- +κ2ti(cid:2)jkmjhk =0. (cid:3) (16)
kenchiralsymmetryvanish. Theexpectationvalueofthe
unshiftedscalarfield<σ >=Maλa =diag(Mu,Md,Ms) Here tijk is a totally symmetric quantity, whose nonzero
corresponds to the point where the effective potential of components are tuds = 1; there is no summation over
the theory achievesits minimum, with M being the con- the open index i but we sum over the dummy indices,
stituent quark masses. In (12) the bilinear form in the e.g. h2 = h2u +h2d +h2s,mh = muhu +mdhd +mshs.
quarkfields L canbe integratedoutfromthe pathinte- Regarding the h coefficients with more than one index,
q
gral, and results in the fermion determinant (see (18) we indicate explicitly only the expression needed for the
below), which generates the kinetic terms of the σ,φ present study of the pseudoscalar masses (the complete
fields. The remaining integrations are over a static non- expressions up to 3 indices can be found in [26],[27])
Gaussian system of s,p fields, and are done in the sta-
tionaryphaseapproximation(SPA).Firstwepresentthe 2 h(2) −1 = 2G+g h2+g mh δ
results for the SPA integration, obtained from the ex- − ab 1 4 ab
tremum conditions 3A(cid:16)abc((cid:17)κhc+2(cid:0)κ2mc)+g2hrhc(da(cid:1)bedcre+2farefbce)
−
+g m h (d d +f f +f f )
∂L ∂L 3 r c abe cre are bce ace bre
=0, =0, (13)
g m m (d d f f )
∂s ∂p 5 r c are bce are bce
a a − −
+g m m d d 4g m m . (17)
6 r c abe cre 8 a b
which must be fulfilled in the neighbourhood of the uni- −
form vacuum state of the theory. The solutions of eq. which can be readily inverted. These coefficients are to-
(13) are seeked in the form: tally defined in terms of h and the parameters of the
a
model.
ssat = ha+h(a1b)σb+h(a1b)cσbσc+h(a2b)cφbφc+... Now to the fermion determinant related to the inte-
pst = h(2)φ +h(3)φ σ +... (14) gration over the fermion fields: we expand it using a
a ab b abc b c heat-kernel technique that takes appropriately into ac-
Eqs. (13)determineallcoefficientsofthisexpansiongiv- count the quark mass differences, being chiral covariant
ing rise to a system of cubic equations to obtain h , eq. at each order of the expansion [55–57],
a
(16), and a full set of recurrence relations to find higher
order coefficients in (14). The result is cast in the form 1 ∞ dt
W[Y] = ln|detD|=−2 t ρ(t)exp −tDE†DE ,
1 1 Z0 (cid:16) (cid:17)
Laux=haσa+ 2h(a1b)σaσb+ 2h(a2b)φaφb (15) DE†DE = M2−∂2+Y, Y =iγµ(∂µ+iγ5∂µφ)
+1σ h(1)σ σ + h(2) +h(3) φ φ +... + σ2+{M,σ}+φ2+iγ5[σ+M,φ], (18)
3 a abc b c abc bca b c
h (cid:16) (cid:17) i or
Here h are related to the quark condensates, h(1), h(2)
contribautetothemassesofscalarandpseudoscalaarbstataebs W[Y]= d4xE ∞ I tr[b ] (19)
respectively, and higher indexed h’s are the couplings − 32π2 i−1 i
Z i=0
X
6
swphaecree. DWEe csotnasniddserfothrethexepDanirsaiocnouppertaototrheinthEirudclmidoedain- + 2(Mu−Ms)2+MuMs−Md2 φ24+φ25
fied Seeley-DeWitt coefficient bi + h2(Md−Ms)2+MdMs−Mu2i(cid:0)φ26+φ27(cid:1)
b0 = 1Y,2 b1λ=−Y, λ + 1h2h(a1b)σaσb+ 12h(a2b)φaφb. i(cid:0) (cid:1)o (24)
3 8
b = + ∆ Y + (∆ +∆ )Y, (20)
2 ud us ds
2 2 2√3 The kinetic term requires a redefinition of the meson
fields,
with ∆ =M2 M2. This order of the expansiontakes
ij i − j
into account the dominant contributions of the quark 4π2
one-loop integrals Ii (i = 0,1,...); these are the arith- σa =gσaR, φa =gφRa, g2 = N I , (25)
metic average values I = 1[J (M2)+J (M2)+J (M2)] c 1
i 3 i u i d i s
where toobtainthestandardfactor1/4. Theflavorandcharged
pseudoscalar fields are related through
Ji(m2)= ∞t2dtiρ(tΛ2)e−tm2, (21) φu π+ K+
Z0 − √λa2φa = π√−2 √φd2 K0  (26)
with the Pauli-Villars regularizationkernel [58, 59] K K¯0 φs
 − √2 
ρ(tΛ2)=1 (1+tΛ2)exp( tΛ2). (22)  
− − and similarly for the scalar fields. In the following we
which is equivalent to the sharp4D cutoff regularization concentrate on the diagonal components of (26), which
for the scalar integrals considered. Both terms propor- according to eq. (7) induce mixing between the 0,3,8
tional to b and b have contributions to the gap equa- field components of (24), in general. Indicating also the
1 2
tions and meson masses, but only b contributes to the resultofthetransformationsdiscussedbelowin(29),(30),
2
kinetic andmesoninteractionterms. By excluding the σ we arrive at the following useful relations among fields
tadpole from the total Lagrangian, one obtains the gap
√2φ +φ
equations φ = φ + 0 8 =φ +η
u 3 3 ns
√3
N
h + c M 3I 3M2 M2 I =0. (23) √2φ +φ
i 6π2 i 0− i − 1 φ = φ + 0 8 = φ +η
d 3 3 ns
− √3 −
(cid:2) (cid:0) (cid:1) (cid:3)
whereM2 =M2+M2+M2. Wenowseethat(23)must
u d s 2 2φ
be solvedself-consistently with the SPA equations (16). φ = φ 8 =√2η . (27)
s 0 s
3 − √3
r
III. Mixing in the π0,η,η′ system Theneutralphysicalstatesπ0,η,η arerelatedtothe3 3
′
III a. Choices of representations ×
symmetric pseudoscalar meson mass matrix of elements
B emerginginthei,j = 0,3,8 channelsofL bya
ij mass
{ }
Finally one is ready to combine the terms of the total sequenceoftwotransformations = thatdiagonalize
S UV
Lagrangian L that contribute to the kinetic terms L it
kin
and meson masses L
mass
B B B φ
33 03 38 3
Lkin+Lmass (φ3,φ0,φ8)S−1S B03 B00 B08 S−1S φ0 ,
N I N I
= 16cπ21 tr (∂µσ)2+(∂µφ)2 + 4πc20(σa2+φ2a)  B38 B08 B88   φ8 
   (28)
NcI1 (cid:2)2(M +M )2 (cid:3)M M M2 (σ2+σ2)
− 12π2 u d − u d− s 1 2
first a rotation to the strange-nonstrange basis through
nh i
+ 2(M +M )2 M M M2 σ2+σ2 the orthogonalinvolutory matrix
u s − u s− d 4 5 V
+ h2(M +M )2 M M M2 i(cid:0)σ2+σ2 (cid:1)
d s − d s− u 6 7 φ3 φ3
+ (cid:2)21 σu2 8Mu2−Md2−Ms2 +σd2(cid:3)(cid:0)8Md2−M(cid:1) u2−Ms2  ηns =V φ0  (29)
η φ
s 8
+ σ2(cid:2) 8M(cid:0) 2 M2 M2 (cid:1) (cid:0) (cid:1)    
s s − u− d    
1 with
+ φ(cid:0)2 2M2 M2 M(cid:1)(cid:3)2 +φ2 2M2 M2 M2
2 u u− d − s d d − u − s
√3 0 0
+ φ2(cid:2) 2M(cid:0) 2 M2 M2 (cid:1) (cid:0) (cid:1) 1
s s − u− d = 0 √2 1 , (30)
+ 2((cid:0)Mu−Md)2+MuM(cid:1)(cid:3)d−Ms2 φ21+φ22 V √3 0 1 −√2 
h i(cid:0) (cid:1)  
7
and then through the unitary transformation to the having a single mixing angle involved in the determina-
U
physical states [25] tion of the decay constants associated with the η and η
′
mesons. We show in the next subsection that our model
π0 φ fulfills this condition at the approximationconsidered.
3
 ηη =U(ǫ1,ǫ2,ψ)ηηns , (31) moAdtelt’shims ηp′ocinotntoaninessahoteurlmd rreemlaitneddwthitehrtehaedetrophoolwogitchael
 ′   s  vacuum susceptibility. The generalized NJL Lagrangian
   
whichcombinestheU (1)breakingbythe’tHooft(2N )
where A f
determinantal Lagrangian with the 4q and 8q interac-
tionshasbeenshownin[62]tobeincorrespondencewith
1 ǫ +ǫ cosψ ǫ sinψ
1 2 2
= ǫ ǫ cosψ cosψ − sinψ (32) the Witten-Veneziano formula [64] which relates mη′, in
U − 2− 1 −  the large Nc limit of QCD with massive quarks, to the
ǫ sinψ sinψ cosψ
 − 1  topological susceptibility χYM of pure Yang-Mills,
 
Theconventionaldefinitionsǫ=ǫ +ǫ cosψ,ǫ =ǫ sinψ 6
forthe mixinganglesareusedint2he t1ables. T′he u1nitary m2η′ +m2η−2m2K =−f2χYM. (36)
π
matrix hasbeenlinearizedintheπ0 ηandπ0 η mix-
′
inganglUesǫ ,ǫ (δ),δ 1. Thisc−anbedone−because Inour modelthe topologicalsusceptibility is obtainedin
1 2
φ3 couples weak∼lyOto the≪ηns and ηs states, decoupling the large Nc limit as the following combination of model
in the isospin limit, while the mixing for the η η sys- parameters
′
−
tem is strong. Nevertheless we have tested numerically
κ M
thelinearizationbyobtainingalsotheexactEulerangles χ = ( )3. (37)
YM
4 2G
associated with the transformation (31), [60], the differ-
ences lying within the one to two percent level for the The cutoff Λ, which is the approximate scale at which
cases studied. dynamical chiral symmetry breaking sets in, does not
In the isospin limit in (32) leads to the 2 2 or- appear explicitely in the relation for χ , only hidden
U × YM
thogonal tranformation Rψ in the constituent quark mass M. This expression shows
a judicious interplay of the subleading in N counting
c
η =R ηns , (33) UA(1) breaking parameter κ and the LO in Nc 4q cou-
ψ pling G,which combinein a relationthat survivesin the
η′ ! ηs !
large N limit. This shows that the roles of chiral sym-
c
metry breaking and the breaking throughthe Adler Bell
with
Jackiwanomalyareintertwinedtoequipη withitslarge
′
cosψ sinψ mass, which does not vanish in the chiral limit.
Rψ = − , (34)
sinψ cosψ !
III b. Decay parameters
Weremindthattheangleψisrelatedtothemixingangle
θp At the order of the heat kernel considered, we obtain
the model’s axial-vector current as [62] [88]
φ0R = cosθp −sinθp η′ . (35) a =
(cid:18)φ8R(cid:19) sinθp cosθp !(cid:18)η(cid:19) A1µ
tr[( σR+Mg−1,∂µφR ∂µσR,φR )λa]+ (b3)
of the physical states η,η in the singlet-octet renormal- 4 { }−{ } O
′
ized basis states φ0,φ8 as ψ = θ +arctan√2, with the (38)
R R p
principal value of the angle θ comprised in the interval
p
intermsoftheanti-commutatorsinvolvingthebosonized
(π/4) θ (π/4), for details please see Appendix B
− ≤ p ≤ and renormalized fields σR,φR (25) and the constituent
of [61] and [62].
quark mass matrix M =M λ . From here it is straight-
We emphasize that one can freely choose among the a a
forward to calculate the matrix elements in the singlet-
different orthogonal bases to address the mixing of
octet basis
states. The reasonit is convenient to adopt the strange-
nonstrange basis is that it allows to infer whether the <0 a(0)φb >=ifabp . (39)
mixing-parameters are determined in a process indepen- |Aµ | R µ
dent way; in the context of ChPT it has been shown to One has
be so if certain OZI violating processes are suppressed,
M +M +M M +M
[63],[25][54], and with the exception of those originating f00 = u d s, f11 =f22 =f33 = u d
fromtopolgicaleffectsduetotheU(1) anomaly. Atthis 3g 2g
A
levelofaccuracythedecayconstantsfollowthepatternof f88 = Mu+Md+4Ms, f08 = Mu+Md−2Ms
particlestatemixinginthisbasiswhichistantamountto 6g 3√2g
8
M M
f03 =√2f38 = u− d (40) with
√6g
N I 2 N I 2
M +M M +M = c 0g2 ξ , = c 0g2 ζ ,
f44 =f55 = u s, f66 =f77 = d s. (41) Ci 2π2 − 3 i Bi 2π2 − 3 i
2g 2g
g2 g2
ξ = h(1)e e , ζ = h(2)e e , (48)
Inparticularoneobtains<0 1+i2(0)π(p)>=i√2f p ij 2 ab ai bj ij 2 ab ai bj
|Aµ | π µ
and < 0|A4µ+i5(0)|K(p) >=i√2fKpµ with fπ = Mu2+gMd whereξij,ζij aresymmetricquantities. Thequantitiesζi
andf = Mu+Ms,atthe orderofthe heatkernelexpan- are
K 2g
sion considered.
ζ =2M2 M2 M2, i=j =k. (49)
The neutral axial vector currents can alternatively be i i − j − k 6 6
taken in the strange non-strange basis (27)
Inserting them in and using the gap equations (23),
i
B
one obtains
2 1
ns = 0 + 8,
Aµ r3Aµ r3Aµ =g2 hi . (50)
i
B M
1 2 i
s = 0 8, (42)
Aµ r3Aµ−r3Aµ Similar expressions can be obtained for the ξi, i, but
C
they are not needed in the following. The divergence of
for which one obtains the decay constants
the axial current (38) reads in this basis
<0 σ(0)φτ >=ifστp , σ,τ =3,ns,s (43)
|Aµ | R µ { } ∂ i = (σ + Mi)∂2φ (∂2σ )φ . (51)
µAµ iR g iR− iR iR
M +M M
fns =f3 = u d, fs = s, Usingnowthe equationsofmotionforthe σ ,φ fields
iR iR
2g g
f3,ns = Mu−Md, f3,s =fns,s =0, (44) 0 = ∂2σiR−CiσiR−(ξijσjR+ξijσiR)
2g 0 = ∂2φ φ (ζ φ +ζ φ ) (52)
iR i iR ij jR ij iR
−B −
where f3,fns,fs are short-hand notations for
one obtains for the matrix elements
f3,3,fns,ns,fs,s. The elements of (43) are collected
in the following matrix <0∂ i φ >= Mi( δ +2ζ ), (53)
| µAµ| jR g Bi ij ij
f zf 0
π π
whichencodeallthechiralandU (1)symmetrybreaking
= zf f 0 (45) A
F  π π  terms. The off-diagonal elements of the inverse matrix
0 0 f
 s  pertinent to ζij are calculated to be
 
wisoitshpizns=ymMMmuue−+tMMrydd. I=tisf3fo,πnfsthmeaorrkdienrgofththeedreaptaiortiunrveolfvroinmg ζi−j1 = 14(hkκ+2mkκ2+2g8mimj), i6=j 6=k. (54)
ffuu+−ffdd ∼ O(δ), with ff = Mf/g the decay constants These contributions violate the OZI rule, the ones pro-
=diag f ,f ,f .
Ff { u d s} portionaltoκ,κ2 havetheiroriginintheUA(1)anomaly.
According to the idea behind the strange-nonstrange
Theterm g isacontributionoftheorderofthesquare
8
basis, the transformation to obtain the physical decay ∼
of the current quark masses.
constants
For comparison the transition elements of the diver-
genceoftheaxialcurrentinthestrangenon-strangebasis
= , P = π0,η,η , (46)
FP UF { ′} are
is the same that transforms the states, eq.(31). In the
Qi =<0∂ i φ >, i,j = 3,ns,s (55)
following we discuss how the obeservables obtained from j | µAµ| jR { }
ourmodelLagrangianfulfillthiscondition. Inordertodo
so,itisconvenienttoexpressthemesonmassLagrangian
f
Ln of the neutral states in the flavor basis Q33 = 2π(b++z(b−−2ζud))
f
Ln = 21 (∂µσiR)2+(∂µφiR)2+Ciσi2R+Biφ2iR Q3ns = 2π(b−+z(b++2ζud))
i=Xu,d,s(cid:26) (cid:2) (cid:3) Q3 = fπ (ζ ζ +z(ζ +ζ ))
s √2 us− ds us ds
+ (ξijσiRσjR+ζijφiRφjR) (47) Qns = fπ(b +z(b+ 2ζ ))
j=Xu,d,s  3 2 − − ud

9
Qnnss = f2π(b++z(b−+2ζud)) cnaoswe.oInnlythmeixistousrpeisnolfimthitetnhoenpsthryasnicgaelasntadtsetsrPan=geηc,oηm′paore-
Qns = fπ (ζ +ζ +z(ζ ζ )) nents,asfollowsfrom(33). Thiscasehasbeenconsidered
s √2 us ds us− ds in[62](where adetaileddiscussionofthe mixing scheme
f f in connection with the singlet octet basis in the isospin
Qs3 = √s2(ζus−ζds), Qsns = √s2(ζus+ζds) limit is also presented),
Qss =fs(B2s +ζss) (56) h0|Aiµ(0)|P(p)i=ifPipµ, (i=ns,s). (61)
The couplings can be represented in a way which is sim-
with
ilarto the one of Leutwyler– Kaiser[63],[54], who intro-
duce two mixing angles ϑ ,ϑ
u d ns s
b± =(B +ζuu) (B +ζdd)
2 ± 2
(57) fns fs f cosϑ f sinϑ
fi = η η = ns ns − s s
Note that the elements Qi, i,j = 3,ns,s are not sym- { P} fηn′s fηs′ ! fnssinϑns fscosϑs !
j { } (62)
metricunderexchangeof{i,j}differingbyterms∼zand Our calculations show that within our model, with M¯ =
y = fs. Obviously in the isospin limit the elements
∼ fπ Mu =Md,
Q3 ,Qns,Q3,Qs vanish. In this limit the Qs = yQns,
ns 3 s 3 ns s
y being a measure for flavor breaking in the light vs. M¯ M
fns = cosψ, fs = s sinψ,
strange sector. η g η − g
Finally using the relations (31),(32) and (56) we cal- M¯ M
cduivlaertegetnhceevoafcutuhme atxoiaplhycusircraelnpt,ar<tic0le∂tranbsiPtio>ns, ofbth=e fηn′s = g sinψ, fηs′ = gs cosψ. (63)
| µAµ|
3,ns,s , P = π0,η,η , discarding terms δ2, and
′ One thus obtains the relation for the mixing angles
{ } { } ∼
are able to show that it fulfills the following relation[25]
fnsfs =fnsfssin(ϑ ϑ )=0. (64)
<0|∂µAbµ|Pa >=M2aa′Ua′b′Fb′b. (58) P=η,η′ P P ns− s
X
wmiatthriFx,gwivheinchbyon(4t5h)earnedquMireama′etnhteopfhbyesiincgaldmiaegsoonnaml dases- Thisfollowsasthebasicparametersfns,fs,ϑns,ϑsofthe
matrix fi ,expressedintermsofmodelparameters(in
termines the mixing angles ψ,ǫ,ǫ′. Thus the decay con- the app{roPxi}mationconsidered)
stants transform as the states (31) within our model
calculations, at the order of the heat kernel considered. M¯ M
Higherordertermsinvolvederivativeinteractions,which f = =f , f = s , ψ =ϑ =ϑ . (65)
ns π s ns s
g g
are likely to change this behavior.
We obtain the follwing relations at (δ)
O Thereisadirectrelationbetweenthecommonmixingan-
gle ϑ =ϑ and the OZI-rule which has been discussed
f3f3 =f2, fnsfns =f2 ns s
P P π P P π in [63],[54]. The model predictions agree well with the
P=Xπ0,η,η′ P=Xπ0,η,η′ general requirements of chiral symmetry following from
fsfs =f2, chiral perturbation theory (ChPT), although the results
P P s
differ already at lowest order. For instance, we have
P=Xπ0,η,η′
(59) (M M¯)2
f2 =2f¯2 f¯2+ s− (66)
s K − π 2g¯2
for the diagonal elements, and
where barred quantities are a reminder that they are
fnsfs = f3fs =0,
P P P P taken in the isospin limit. The first two terms of this
P=Xπ0,η,η′ P=Xπ0,η,η′ formula are a well known low-energy relation which is
fP3fPns =zfπ2 (60) valid in standard ChPT. In the η′-extended version of
P=Xπ0,η,η′ CLahgPrTantghiaenre,wishaichOZcoI-nrturliebuvtieoslaatsinfg¯2tΛermtoinr.ht.hse. eofffe(c6t6i)ve.
π 1
for the crossed terms. The last relation is a consequence We have instead the term (M M¯)2/(2g¯2). Of course,
s
−
of (45). The vanishing of the other crossed terms indi- in our case the origin of this contribution is related with
cates that just one mixing angle is present for the η η theSU(3)flavoursymmetrybreakingeffectanddoesnot
′
−
mixing in the strange nonstrange basis at this order of have impact on the deviation from a single mixing angle
approximation. This can be seen in a simple way in the in the strange-nonstrange basis reported as consequence
isospinlimit,andtheargumentisthesameinthegeneral of the OZI-rule violating parameter Λ .
1
10
Away from the isospin limit one obtains
1
f2 =2f2 f2(1+2z)+ (Ms−Mu)2, (67) (ζns,s)(−1) = 2√2(hΣκ+2(κ2+g8ms)mΣ). (70)
s K − π 2g2
The first two elements vanish in the isospin symmetric
with a correctionof order δ in the f2 term, as compared
π case. The third element relates to the mixing in η η
′
to (66). The numerical values that we obtain are z −
4 10 3, whichamountsto a smallcorrectionof 0.8%∼ and is non-zero in this limit. In this basis one sees that
× − ∼ the mixing in the 3,ns sectorinvolvesother couplings as
in the f2 term.
π in the strangeness related sectors, in contrast to the off-
diagonal matrix elements in the u,d,s basis, (54). This
For completeness we indicate the decay constants and sets new constraints on these couplings, as compared to
mixing angles in the singlet-octet basis in the isospin the isospin symmetric case. It follows that the interplay
limit, used in Table (IV), for details see please [62], oftheseparameters(whichisconditionedbythefits),not
the actual magnitude of each of the terms, is relevant to
M¯
θ = ψ arctan(√2 ), obtain the size of isospincorrections. We remark that in
0
− Ms (ζns,s)(−1) isospinbreakingeffectsareabsent,andthatin
θ = ψ arctan(√2Ms) thediagonalelements(notshown)theyareovershadowed
8 − M¯ by the presence of ms,mΣ,hs,hΣ.
ψ = θ +arctan√2 (68) The current quark mass dependence in these expres-
P
sions enters together with the ESB couplings. In their
absence the effects of ESB come only through the differ-
f2 = 2f¯K2 +f¯π2 + f¯π2(Ms 1)2 ence in the light condensates h∆ which do not vanish if
0 3 6 M¯ − the conventional QCD mass term has m = m values,
u d
f2 = 4f¯K2 −f¯π2 + (Ms−M¯)2 (69) and if the couplings κ and g2 are not zero6 (if they also
8 3 3g¯2 vanish, only the heat kernel contribution to the meson
mass matrix carries the effects of isospin breaking). The
IV. Results and further discussion coupling κ is strongly correlated with the η η mass
′
splitting and it enters in the corresponding −(ζ )( 1)
ns,s −
The model has 15 parameters, 4 couplings G,κ,g1,g2 matrix element as factor of hΣ which remains approxi-
associated with L in (2), 7 non-vanishing couplings mately constant. Thus this parameteris notexpectedto
int
κ ,g ....g in in the ESB sector (4), the cutoff Λ, and varymuchinthefitofisospinbreakingeffects,whichhas
2 3 8
the 3 current quark masses. Before running a fit it is been also verified numerically.
convenient to understand to which parameters the dif-
ference of the light quark masses (mu md) is most Beforeshowingtheresultsforisospinbreaking,wedis-
−
sensitive. For that it is instructive to look at the ma- play relevant model observables in the isospin limit, in
trix components (17) in the strange-nonstrange basis, comparison to other approaches. We consider the cases
which vanish in the isospin limit. As mentioned before in which the ESB terms are present in the interaction
these elements belong to the SPA contribution to the Lagrangian, sets (a,b) in the Tables I-IV, and compare
meson mass Lagrangian, see last line in (24), which is with the parameter set (c) in which explicit symmetry
the part that carries the full information on the model breakingoccursonlythroughtheLOcurrentquarkmass
couplings. Note that the heat kernel contribution to the term. Table I indicates the mass spectra of the low ly-
meson masses, represented except for the kinetic terms ing pseudoscalar (and scalar meson nonets, see caption,
in the remaining of expression(24), only depends on the for sets (a,b)) used in the fit of parameters. Sets (a,b)
cutoff Λ of the Ii quark integrals. The dependence on havedifferentmixinganglesforthepseudoscalarsaswell
the model couplings of the heat kernel contribution only as for the scalars, leaving however the mass spectra and
enters implicitly through the constituent quark masses, weak decay constants f ,f inchanged. This is possible
π K
via the gap equations (23) which are solved selfconsis- because the fit is done simultaneously in both sectors,
tentlywiththelowestorderSPAequations(16). Defining leading to a readjustment of parameters. A comment
m∆ = 12(md−mu), mΣ = 21(md+mu), h∆ = 21(hd−hu) on the scalar mass spectrum: we were able to obtain a
and hΣ = 12(hd + hu), one has for the inverse matrix reasonable simultaneous fit for the σ mass and its large
elements of the ζij matrix, i,j = 3,ns,s decaywidth,attreelevelofthemesonicLagrangian. Itis
{ }
our understanding that this results partly from the fact
1
(ζ )( 1) = [h (2g h +g m ) that already at mesonic tree level there are signatures
3,ns − ∆ 2 Σ 3 Σ
4 of quark-antiquark as well as of admixtures of 2 quark-
+m∆(g3hΣ 2(g5 g6+2g8)mΣ)] 2 antiquark states present, from the underlying multi-
− −
quark Lagrangian,which contribute in the long distance
1 asymptotics to the internal structure of the mesons as
(ζ )( 1) = (h κ+2m (κ g m ))
3,s − 2√2 ∆ ∆ 2− 8 s well as to the mesonic interaction terms. In many other