Effective multi-quark interactions with explicit breaking of chiral symmetry
EEffective multi-quark interactions with explicit breaking of chiral symmetry
A. A. Osipov ∗ , B. Hiller † and A. H. Blin ‡ Centro de F´ısica Computacional, Departamento de F´ısica da Universidade de Coimbra, 3004-516 Coimbra, Portugal
In a long distance Lagrangian approach to the low lying meson phenomenology we present anddiscuss the most general spin zero multi-quark interaction vertices of non-derivative type whichinclude a set of effective interactions proportional to the current quark masses, breaking explicitelythe chiral SU (3) L × SU (3) R and U A (1) symmetries. These vertices are of the same order in N c counting as the ’t Hooft flavor determinant interaction and the eight quark interactions which extendthe original leading in N c four quark interaction Lagrangian of Nambu and Jona-Lasinio. The N c assignements match the counting rules based on arguments set by the scale of spontaneous chiralsymmetry breaking. With path integral bosonization techniques which take appropriately intoaccount the quark mass differences we derive the mesonic Lagrangian up to three-point mesonicvertices. We demonstrate that explicit symmetry breaking effects in interactions are essential toobtain the correct empirical ordering and magnitude of the splitting of certain states such as m K I. INTRODUCTION A long history of applying the Nambu – Jona-Lasinio(NJL) model in hadron physics shows the importanceof the concept of effective multi-quark interactions formodelling QCD at low energies. Originally formulatedin terms of the γ gauge invariant nonlinear four-fermioncoupling [1, 2], the model has been extended to the re-alistic three flavor and color case with U (1) A breakingsix-quark ’t Hooft interactions [3–17] and an appropri-ate set of eight-quark interactions [18]. The last onescomplete the number of vertices which are important infour dimensions for dynamical SU (3) L × SU (3) R chiralsymmetry breaking [19, 20].The explicit breaking of chiral symmetry in the NJLmodel is described by introducing the standard lightquark mass term of the QCD Lagrangian (light meansconsisting of u, d and s quarks), e.g. [21, 22]. The currentquark mass dependence is of importance for several rea-sons, in particular for the phenomenological descriptionof meson spectra and meson-meson interactions, and forthe critical point search in hot and dense hadronic mat-ter, where it has a strong impact on the phase diagram[23]. The values of the current quark masses are deter-mined in the Higgs sector of the Standard Model. In thisregard they are foreign to QCD and, at an effective de- ∗ Email address: [email protected] † Email address: brigitte@teor.fis.uc.pt ‡ Email address: alex@teor.fis.uc.pt scription, can be included through the external sources,interacting with the originally massless quark fields. Thisis why the explicit chiral symmetry breaking (ChSB) bythe standard mass term of the free Lagrangian is only apart of the more complicated picture arising in effectivemodels beyond leading order [24]. Chiral perturbationtheory [25–28] gives a well-known example of a self con-sistent accounting of the mass terms, order by order, inan expansion in the masses themselves. In fact, extendedNJL-type models should not be an exception from thisrule. If one considers multi-quark effective vertices, tothe extent that 1 /N c suppressed ’t Hooft and eight-quarkterms are included in the Lagrangian, certain mass de-pendent multi-quark interactions must be also taken intoaccount.The aim of the present work is precisely to analyzethese higher order terms in the quark mass expansion.Our consideration proceeds along the following steps.We start from the three-flavor NJL-type model with self-interacting massless quarks. The SU (3) L × SU (3) R chiralsymmetry of the Lagrangian is known to be dynamicallybroken to its SU (3) V subgroup at some scale Λ, withΛ being one of the model parameters. There is also ex-plicit symmetry breaking due to the bare quark masses χ , which are taken to transform as χ = (3 , ∗ ) under SU (3) L × SU (3) R . Since the Lagrangian contains, in gen-eral, an unlimited number of non-renormalizable multi-quark and χ -quark interactions (scaled by some powersof Λ), we formulate the power counting rules to classifythese vertices in accordance with their importance for dy-namical symmetry breaking. Then we bosonize the the-ory by using the path-integral method. The functional a r X i v : . [ h e p - ph ] S e p integrals are calculated in the stationary phase approx-imation and by using the heat kernel technique. As aresult one obtains the low-energy meson Lagrangian. Atlast we fix the parameters of the model by confronting itto the experimental data. In particular, we show the abil-ity of the model to describe the spectrum of the pseudoGoldstone bosons, including the fine tuning of the η − η (cid:48) splitting, and the spectrum of the light scalar mesons: σ or f (500), κ (800), f (980), and a (980).The coupling constants of multi-quark vertices, fixedfrom mass-spectra, enter the expressions for meson de-cay amplitudes and lead to a bulk of model predictions.It is interesting to note that certain multi-quark verticesof the model encode implicitly in the couplings of thetree level bosonized Lagrangian the signature of q ¯ q andmore complex quark structures which are elsewhere ob-tained by considering explicitly meson loop corrections,tetraquark configurations and so on [29–41]. It seemsappropriate, therefore, to examine the possible physicsopportunities connected with the discovery and study ofsuch multi-quark structures in hadrons. For instance, bycalculating the mass spectra and the strong decays of thescalars, one can realize which multi-quark interactionsare most relevant at the scale of spontaneous ChSB. Onthe other hand, by analyzing the two photon radiativedecays, where a different scale, associated with the elec-tromagnetic interaction, comes into play, one can studythe possible recombinations of quarks inside the hadron.We will show, for example, that the a (980) meson cou-ples with a large strength of the multi-quark componentsto the two kaon channel in its strong decay to two pions,but evidences a dominant q ¯ q component in its radiativedecay. As opposed to this, the σ and f (980) mesons donot display an enhanced q ¯ q component neither in theirtwo photon decays nor in the strong decays.There are several direct motivations for this work. Inthe first place, the quark masses are the only parametersof the QCD Lagrangian which are responsible for the ex-plicit ChSB, and it is important for the effective theoryto trace this dependence in full detail. In this paper itwill be argued that it is from the point of view of the1 /N c expansion that the new quark mass dependent in-teractions must be included in the NJL-type Lagrangianalready when the U (1) A breaking ’t Hooft determinantalinteraction is considered. This important point is some-how completely ignored in the current literature.A second reason is that nowadays it is getting clearthat the eight-quark interactions, which are almostinessential for the mesonic spectra in the vacuum, canbe important for the quark matter in a strong magneticbackground [42–46]. The simplest next possibility is toadd to that picture a set of new effective quark-mass-dependent interactions, discussed in this work. Such fea-ture of the quark matter has not been studied yet, butprobably contains interesting physics.Further motivation comes from the hadronic matterstudies in a hot and dense environment. It is known thatlattice QCD at finite density suffers from the numerical sign problem. This is why the phase diagram is noto-riously difficult to compute “ab initio”, except for theextremely high density regime where perturbative QCDmethods are applicable. In such circumstances effectivemodels designed to shed light on the phase structure ofQCD are valuable, especially if such models are known tobe successful in the description of the hadronic matter atzero temperature and density. Reasonable modificationsof the NJL model are of special interest in this contextand our work aims also at future applications in thatarea.The paper is organized as follows. In section II theeffective Lagrangian in terms of quark degrees of free-dom and bosonic sources with specific quantum numbersis derived using a classification scheme which selects allpossible non-derivative vertices according to the symme-tries of the strong interaction and which are relevant atthe scale Λ of spontaneous chiral symmetry breaking. Itis then shown that this scheme can be equally organizedin terms of the large N c counting rules, which in turnallow to attribute to the couplings of the interactionsencoded signatures of q ¯ q and more complex structuresinvolving four fermions. We obtain in this section alsothat a set of interactions lead to the Lagrangian specificKaplan-Manohar ambiguity associated with the currentquark masses.In section III we proceed to bosonize the multi-quarkLagrangian in two steps. First, we introduce in sectionIII-A a set of auxiliary scalar fields. By these new vari-ables the multi-quark interactions can be brought to theYukawa form that is quadratic in Fermi fields. Conse-quently one obtains a Gauss-type integral over quarks,and a set of integrals over auxiliary fields. The latterare evaluated by the stationary phase method. We ob-tain here the vertices up to the cubic power in the mesonfields, needed for the study of the meson spectra andof the two-body decays. Then, in section III-B, we inte-grate over quark fields. The arising quark determinant ofthe Dirac operator is a complicated non-local functionalof the collective meson fields. We calculate it in the low-energy regime by using the Schwinger-DeWitt technique,based on the heat kernel expansion. In this approxima-tion one can adequately incorporate the effect of differentquark masses contained in the modulus of the one-loopquark determinant. We derive the kinetic terms of thecollective meson fields, as well as the heat kernel part ofcontributions to meson masses and interactions. In theend of this section we present the complete bosonizedLagrangian, give the mixing angle conventions used, andthe expressions for the strong decay widths. In sectionIII-C we obtain the expressions for the radiative widthsof the pseudoscalars and scalars.In section IV we present the numerical results and dis-cussion, in IV-A for the meson mass spectra and weakdecay constants, in IV-B for the strong decays and inIV-C for the radiative decays.We conclude in section V with a summary of the mainresults. II. EFFECTIVE MULTI-QUARKINTERACTIONS The chiral quark Lagrangian has predictive power forthe energy range which is of order Λ (cid:39) πf π ∼ q , the scale Λ, and the external sources χ , whichgenerate explicit symmetry breaking effects – resulting inmass terms and mass-dependent interactions.The color quark fields possess definite transformationproperties with respect to the chiral flavor U (3) L × U (3) R global symmetry of the QCD Lagrangian with threemassless quarks (in the large N c limit). It is con-venient to introduce the U (3) Lie-algebra valued fieldΣ = ( s a − ip a ) λ a , where s a = ¯ qλ a q , p a = ¯ qλ a iγ q ,and a = 0 , , . . . , λ = (cid:112) / × λ a being the stan-dard SU (3) Gell-Mann matrices for 1 ≤ a ≤ 8. Un-der chiral transformations: q (cid:48) = V R q R + V L q L , where q R = P R q, q L = P L q , and P R,L = (1 ± γ ). Hence,Σ (cid:48) = V R Σ V † L , and Σ † (cid:48) = V L Σ † V † R . The transformationproperty of the source is supposed to be χ (cid:48) = V R χV † L .Any term of the effective multi-quark Lagrangian with-out derivatives can be written as a certain combinationof fields which is invariant under chiral SU (3) R × SU (3) L transformations and conserves C, P and T discrete sym-metries. These terms have the general form L i ∼ ¯ g i Λ γ χ α Σ β , (1)where ¯ g i are dimensionless coupling constants (startingfrom eq. (21) the dimensional couplings g i = ¯ g i / Λ γ willbe also considered). Using dimensional arguments wefind (in four dimensions) α + 3 β − γ = 4, with integervalues for α, β and γ .We obtain a second restriction by considering only thevertices which make essential contributions to the gapequations in the regime of dynamical chiral symmetrybreaking, i.e. we collect only the terms whose contribu-tions to the effective potential survive at Λ → ∞ . We getthis information by contracting quark lines in L i , findingthat this term contributes to the power counting of Λ inthe effective potential as ∼ Λ β − γ , i.e. we obtain that2 β − γ ≥ ).Combining both restrictions we come to the conclusionthat only vertices with α + β ≤ α = 0; those are four, six and eight-quark in-teractions, corresponding to β = 2 , β = 1 class is forbidden by chiral symmetry require-ments; (ii) there are only six classes of vertices depend-ing on external sources χ , they are: α = 1 , β = 1 , , α = 2 , β = 1 , 2; and α = 3 , β = 1.Let us consider now the structure of multi-quark ver-tices in detail [48]. The Lagrangian corresponding to thecase (i) is well known L int = ¯ G Λ tr (cid:0) Σ † Σ (cid:1) + ¯ κ Λ (cid:0) det Σ + det Σ † (cid:1) + ¯ g Λ (cid:0) tr Σ † Σ (cid:1) + ¯ g Λ tr (cid:0) Σ † ΣΣ † Σ (cid:1) . (3)It contains four dimensionful couplings G, κ, g , g .The second group (ii) contains eleven terms L χ = (cid:88) i =0 L i , (4)where L = − tr (cid:0) Σ † χ + χ † Σ (cid:1) L = − ¯ κ Λ e ijk e mnl Σ im χ jn χ kl + h.c.L = ¯ κ Λ e ijk e mnl χ im Σ jn Σ kl + h.c.L = ¯ g Λ tr (cid:0) Σ † ΣΣ † χ (cid:1) + h.c.L = ¯ g Λ tr (cid:0) Σ † Σ (cid:1) tr (cid:0) Σ † χ (cid:1) + h.c.L = ¯ g Λ tr (cid:0) Σ † χ Σ † χ (cid:1) + h.c.L = ¯ g Λ tr (cid:0) ΣΣ † χχ † + Σ † Σ χ † χ (cid:1) L = ¯ g Λ (cid:0) trΣ † χ + h.c. (cid:1) L = ¯ g Λ (cid:0) trΣ † χ − h.c. (cid:1) L = − ¯ g Λ tr (cid:0) Σ † χχ † χ (cid:1) + h.c.L = − ¯ g Λ tr (cid:0) χ † χ (cid:1) tr (cid:0) χ † Σ (cid:1) + h.c. (5)Each term in the Lagrangian L is hermitian by itself, butbecause of the parity symmetry of strong interactions,which transforms one of them into the other, they havea common coupling ¯ g .Some useful insight into the Lagrangian above can beobtained by considering it from the point of view of the1 /N c expansion. Indeed, the number of color componentsof the quark field q i is N c , hence summing over colorindices in Σ gives a factor of N c , i.e. one counts Σ ∼ N c .The cut-off Λ that gives the right dimensionality tothe multi-quark vertices scales as Λ ∼ N c = 1, as adirect consequence of the gap equations (see eq. (37)below), which imply 1 ∼ N c G Λ ; on the other hand, sincethe leading quark contribution to the vacuum energy isknown to be of order N c , the first term in (3) is estimatedas N c , and we conclude that G ∼ /N c .Furthermore, the U (1) A anomaly contribution (thesecond term in (3)) is suppressed by one power of 1 /N c ,it yields κ ∼ /N c .The last two terms in (3) have the same N c count-ing as the ’t Hooft term. They are of order 1. Indeed,Zweig’s rule violating effects are always of order 1 /N c with respect to the leading order contribution ∼ N c .This reasoning helps us to find g ∼ /N c . The termwith g ∼ /N c is also 1 /N c suppressed. It representsthe next to the leading order contribution with one in-ternal quark loop in N c counting. Such vertex containsthe admixture of the four-quark component ¯ qq ¯ qq to theleading quark-antiquark structure at N c → ∞ .Next, all terms in eq. (5), except L , are of order 1.The argument is just the same as before: this part ofthe Lagrangian is obtained by succesive insertions of the χ -field ( χ counts as χ ∼ 1) in place of Σ fields in thealready known 1 /N c suppressed vertices. It means that κ , g , g ∼ /N c , κ , g , g , g , g ∼ /N c , and g , g ∼ /N c .There are two important conclusions here. The first isthat at leading order in 1 /N c only two terms contribute:the first term of eq. (3), and the first term of eq. (5).This corresponds exactly to the standard NJL model pic-ture, where mesons are pure ¯ qq states with constituentswhich have a non-zero bare mass. At the next to leadingorder we have thirteen terms additionally. They trace theZweig’s rule violating effects ( κ, κ , κ , g , g , g , g , g ),and an admixture of the four-quark component to the ¯ qq one ( g , g , g , g , g ). Only the phenomenology of thelast three terms from eq. (3) has been studied until now.We must still understand the role of the other ten termsto be consistent with the generic 1 /N c expansion of QCD.The second conclusion is that the N c counting justifiesthe classification of the vertices made above on the basisof the inequality (2). This is seen as follows: the equiva-lent inequality (cid:100) ( α + β ) / (cid:101) ≤ N c → ∞ (it follows from (1) that β − (cid:100) γ/ (cid:101) ≥ g i ∼ /N (cid:100) γ/ (cid:101) c , where (cid:100) γ/ (cid:101) is the nearestinteger greater than or equal to γ/ L = ¯ qiγ µ ∂ µ q + L int + L χ . (6)In this SU (3) L × SU (3) R symmetric chiral Lagrangianwe neglect terms with derivatives in the multi-quark in-teractions, as usually assumed in the NJL model. Wefollow this approximation, because the specific questionsfor which these terms might be important, e.g. the radialmeson excitations, or the existence of some inhomoge-neous phases, characterized by a spatially varying orderparameter, are not the goal of this work.Finally, having all the building blocks conform withthe symmetry pattern of the model, one is now free to choose the external source χ . Putting χ = M / 2, where M = diag( µ u , µ d , µ s ) , we obtain a consistent set of explicitly breaking chiralsymmetry terms. This leads to the following mass de-pendent part of the NJL Lagrangian L χ → L µ = − ¯ qmq + (cid:88) i =2 L (cid:48) i (7)where the current quark mass matrix m is equal to m = M + ¯ κ Λ (det M ) M − + ¯ g M + ¯ g (cid:0) tr M (cid:1) M , (8)and L (cid:48) = ¯ κ e ijk e mnl M im Σ jn Σ kl + h.c.L (cid:48) = ¯ g tr (cid:0) Σ † ΣΣ † M (cid:1) + h.c.L (cid:48) = ¯ g tr (cid:0) Σ † Σ (cid:1) tr (cid:0) Σ † M (cid:1) + h.c.L (cid:48) = ¯ g tr (cid:0) Σ † M Σ † M (cid:1) + h.c.L (cid:48) = ¯ g tr (cid:2) M (cid:0) ΣΣ † + Σ † Σ (cid:1)(cid:3) L (cid:48) = ¯ g (cid:0) trΣ † M + h.c. (cid:1) L (cid:48) = ¯ g (cid:0) trΣ † M − h.c. (cid:1) (9)Let us note that there is a definite freedom in the def-inition of the external source χ . In fact, the sources χ ( c i ) = χ + c Λ (cid:0) det χ † (cid:1) χ (cid:0) χ † χ (cid:1) − + c Λ χχ † χ + c Λ tr (cid:0) χ † χ (cid:1) χ (10)with three independent constants c i have the same sym-metry transformation property as χ . Therefore, we couldhave used χ ( c i ) everywhere that we used χ . As a result,we would come to the same Lagrangian with the follow-ing redefinitions of couplings¯ κ → ¯ κ (cid:48) = ¯ κ + c , ¯ g → ¯ g (cid:48) = ¯ g − ¯ κ c , ¯ g → ¯ g (cid:48) = ¯ g + ¯ κ c , ¯ g → ¯ g (cid:48) = ¯ g + ¯ κ c , ¯ g → ¯ g (cid:48) = ¯ g + c − κ c , ¯ g → ¯ g (cid:48) = ¯ g + c + 2¯ κ c . (11)Since c i are arbitrary parameters, this corresponds to acontinuous family of equivalent Lagrangians. This familyreflects the known Kaplan – Manohar ambiguity [49–52]in the definition of the quark mass, and means that sev-eral different parameter sets (11) may be used to repre-sent the data. In particular, without loss of generalitywe can use the reparametrization freedom to obtain theset with ¯ κ (cid:48) = ¯ g (cid:48) = ¯ g (cid:48) = 0.The effective multi-quark Lagrangian can be writtennow as L = ¯ q ( iγ µ ∂ µ − m ) q + L int + (cid:88) i =2 L (cid:48) i . (12)It contains eighteen parameters: the scale Λ, three pa-rameters which are responsible for explicit chiral sym-metry breaking µ u , µ d , µ s , and fourteen interaction cou-plings ¯ G, ¯ κ, ¯ κ , ¯ κ , ¯ g , . . . , ¯ g . Three of them, ¯ κ , ¯ g , ¯ g ,contribute to the current quark masses m . Seven moredescribe the strength of multi-quark interactions with ex-plicit symmetry breaking effects. These vertices containnew details of the quark dynamics which have not beenstudied yet in any NJL-type models. We shall now seehow important they are. III. BOSONIZATION: MESON MASSES ANDDECAYSA. Stationary phase contribution The model can be solved by path integral bosoniza-tion of the quark Lagrangian (12). Indeed, following[7] we may equivalently introduce auxiliary fields s a =¯ qλ a q, p a = ¯ qiγ λ a q , and physical scalar and pseudoscalarfields σ = σ a λ a , φ = φ a λ a . In these variables the La-grangian is a bilinear form in quark fields (once the re-placement has been done the quarks can be integratedout giving us the kinetic terms for the physical fields φ and σ ) L = ¯ q ( iγ µ ∂ µ − σ − iγ φ ) q + L aux ,L aux = s a σ a + p a φ a − s a m a + L int ( s, p )+ (cid:88) i =2 L (cid:48) i ( s, p, µ ) . (13)It is clear, that after the elimination of the fields σ, φ by means of their classical equations of motion, one canrewrite this Lagrangian in its original form (12). Theterm bilinear in the quark fields in (13) will be integratedout using the heat kernel technique in the next subsec-tion. The remaining higher order quark interactions col-lected in L aux will be integrated in the stationary phaseapproximation (SPA). In terms of auxiliary bosonic vari-ables one has L int ( s, p ) = L q + L q + L (1)8 q + L (2)8 q ,L q ( s, p ) = ¯ G (cid:0) s a + p a (cid:1) ,L q ( s, p ) = ¯ κ A abc s a ( s b s c − p b p c ) , (14) L (1)8 q ( s, p ) = ¯ g (cid:0) s a + p a (cid:1) , L (2)8 q ( s, p ) = ¯ g [ d abe d cde ( s a s b + p a p b ) ( s c s d + p c p d )+ 4 f abe f cde s a s c p b p d ] , and the quark mass dependent part is as follows L (cid:48) = 3¯ κ A abc µ a ( s b s c − p b p c ) ,L (cid:48) = ¯ g µ a [ d abe d cde s b ( s c s d + p c p d ) − f abe f cde p b p c s d ] ,L (cid:48) = ¯ g µ b s b (cid:0) s a + p a (cid:1) ,L (cid:48) = ¯ g µ b µ d ( d abe d cde − f abe f cde ) ( s a s c − p a p c ) ,L (cid:48) = ¯ g µ a µ b d abe d cde ( s c s d + p c p d ) ,L (cid:48) = ¯ g Λ ( µ a s a ) ,L (cid:48) = − ¯ g Λ ( µ a p a ) , (15)where A abc = 13! e ijk e mnl ( λ a ) im ( λ b ) jn ( λ c ) kl , (16)and the U (3) antisymmetric f abc and symmetric d abc con-stants are standard.Our final goal is to clarify the phenomenological role ofthe mass-dependent terms described by the Lagrangiandensites of eq. (15). We can gain some understandingof this by considering the low-energy meson dynamicswhich follows from our Lagrangian. For that we mustexclude quark degrees of freedom in (13), e.g., by in-tegrating them out from the corresponding generatingfunctional. The standard Gaussian path integral leads usto the fermion determinant, which we expand by using aheat-kernel technique [53–56]. The remaining part of theLagrangian, L aux , depends on auxiliary fields which donot have kinetic terms. The equations of motion of sucha static system are the extremum conditions ∂L∂s a = 0 , ∂L∂p a = 0 , (17)which must be fulfilled in the neighbourhood of the uni-form vacuum state of the theory. To take this into ac-count one should shift the scalar field σ → σ + M .The new σ -field has a vanishing vacuum expectationvalue (cid:104) σ (cid:105) = 0, describing small amplitude fluctuationsabout the vacuum, with M being the mass of constituentquarks. We seek solutions of eq. (17) in the form: s sta = h a + h (1) ab σ b + h (1) abc σ b σ c + h (2) abc φ b φ c + . . .p sta = h (2) ab φ b + h (3) abc φ b σ c + . . . (18)Eqs. (17) determine all coefficients of this expansion giv-ing rise to a system of cubic equations to obtain h a , andthe full set of recurrence relations to find higher ordercoefficients in (18). We can gain some insight into thephysical meaning of these parameters if we calculate theLagrangian density L aux on the stationary trajectory. Infact, using the recurrence relations, we are led to the re-sult L aux = h a σ a + 12 h (1) ab σ a σ b + 12 h (2) ab φ a φ b (19)+ 13 σ a (cid:104) h (1) abc σ b σ c + (cid:16) h (2) abc + h (3) bca (cid:17) φ b φ c (cid:105) + . . . Indicated are all the terms which are necessary to analyzethe mass spectra and two particle decays. Here h a definethe quark condensates, h (1) ab , h (2) ab contribute to the massesof scalar and pseudoscalar states, and higher order h ’s arethe couplings that measure the strength of the meson-meson interactions. The transition from the Lagrangian L aux ( s, p ) in (13) to its form L aux ( σ, φ ) in (19) can beviewed as a Legendre transformation.We proceed now to explain the details of determining h . We address first the coefficients h a , h (1) ab , and h (2) ab .In particular, eq. (17) states that h a = 0, if a (cid:54) = 0 , , h α ( α = 0 , , i = u, d, sh α = e αi h i , e αi = 12 √ √ √ √ √ −√ − , (20)satisfy the following system of cubic equations∆ i + κ t ijk h j h k + h i (cid:0) G + g h + g µh (cid:1) + g h i + µ i (cid:2) g h i + g h + 2( g + g ) µ i h i + 4 g µh (cid:3) + κ t ijk µ j h k = 0 . (21)Here ∆ i = M i − m i ; t ijk is a totally symmetric quantity,whose nonzero components are t uds = 1; there is no sum-mation over the open index i but we sum over the dummyindices, e.g. h = h u + h d + h s , µh = µ u h u + µ d h d + µ s h s .In particular, eq. (8) reads in this basis m i = µ i (cid:16) g µ i + g µ (cid:17) + κ t ijk µ j µ k . (22)For the set g = g = κ = 0 the current quark mass m i coincides precisely with the explicit symmetry breakingparameter µ i .Note that the factor multiplying h i in the third termof eq. (21) is the same for each flavor. This quantity alsoappears in all meson mass expressions, and there is nofurther dependence on the couplings G, g , g involvedfor meson states with a = 1 , , . . . , 7. Thus there is afreedom of choice which allows to vary these couplings,condensates and quark masses µ i , without altering thispart of the meson mass spectrum.To obtain the coefficients h ( i ) ab , ( i = 1 , 2) in the La-grangian L aux (19), it is sufficient to collect in the sta-tionary phase equations (17) only the terms linear in thefields, as can be seen from the structure of the solutions (18). Moreover, for any coefficient multiplying a certainnumber n of fields in L aux it is required to consider termsonly up to order n − h (1) ab and h (2) ab are − (cid:16) h (1) ab (cid:17) − = (cid:0) G + g h + g µh (cid:1) δ ab + 4 g h a h b +3 A abc ( κh c + 2 κ µ c ) + g h r h c ( d abe d cre + 2 d ace d bre )+ g µ r h c ( d abe d cre + d ace d bre + d are d bce )+2 g ( µ a h b + µ b h a ) + g µ r µ c ( d are d bce − f are f bce )+ g µ r µ c d abe d cre + 4 g µ a µ b . (23) − (cid:16) h (2) ab (cid:17) − = (cid:0) G + g h + g µh (cid:1) δ ab − A abc ( κh c + 2 κ µ c ) + g h r h c ( d abe d cre + 2 f are f bce )+ g µ r h c ( d abe d cre + f are f bce + f ace f bre ) − g µ r µ c ( d are d bce − f are f bce )+ g µ r µ c d abe d cre − g µ a µ b . (24)These coefficients are totally defined in terms of h a andthe parameters of the model. Eqs. (23)-(24) can be easilyconverted into explicit formulae for h ( i ) ab , ( i = 1 , h ( i ) abc , ( i = 1 , , L aux , one equates the fac-tors of σ a σ b , φ a φ b , φ a σ b in (17) independently to zero.After some algebra, this results into the following expres-sions h (1) abc = (cid:20) κ A ¯ a ¯ b ¯ c + g ( h ¯ a δ ¯ b ¯ c + 2 h ¯ c δ ¯ a ¯ b )+ g h ¯ r ( d ¯ a ¯ b ¯ ρ d ¯ r ¯ c ¯ ρ + 12 d ¯ a ¯ r ¯ ρ d ¯ b ¯ c ¯ ρ )+ g m ¯ r (2 d ¯ a ¯ c ¯ ρ d ¯ b ¯ r ¯ ρ + d ¯ b ¯ c ¯ ρ d ¯ a ¯ r ¯ ρ − f ¯ b ¯ c ¯ ρ f ¯ a ¯ r ¯ ρ )+ g m ¯ a δ ¯ b ¯ c + 2 m ¯ c δ ¯ a ¯ b ) (cid:105) h (1) a ¯ a h (1) b ¯ b h (1) c ¯ c (25) h (2) abc = (cid:20) − κ A ¯ a ¯ b ¯ c + g h ¯ a δ ¯ b ¯ c + g h ¯ r ( f ¯ a ¯ b ¯ ρ f ¯ c ¯ r ¯ ρ + 12 d ¯ a ¯ r ¯ ρ d ¯ b ¯ c ¯ ρ ) − g m ¯ r (2 f ¯ a ¯ c ¯ ρ f ¯ b ¯ r ¯ ρ + f ¯ b ¯ c ¯ ρ f ¯ a ¯ r ¯ ρ − d ¯ b ¯ c ¯ ρ d ¯ a ¯ r ¯ ρ )+ g m ¯ a δ ¯ b ¯ c (cid:105) h (1) a ¯ a h (2) b ¯ b h (2) c ¯ c (26) h (3) abc = (cid:20) − κ A ¯ a ¯ b ¯ c + 2 g h ¯ c δ ¯ b ¯ a + g h ¯ r ( d ¯ a ¯ b ¯ ρ d ¯ c ¯ r ¯ ρ + f ¯ r ¯ a ¯ ρ f ¯ c ¯ b ¯ ρ + f ¯ r ¯ b ¯ ρ f ¯ c ¯ a ¯ ρ )+ g m ¯ r ( d ¯ a ¯ b ¯ ρ d ¯ c ¯ r ¯ ρ + f ¯ b ¯ c ¯ ρ f ¯ a ¯ r ¯ ρ + f ¯ a ¯ c ¯ ρ f ¯ b ¯ r ¯ ρ )+ g m ¯ c δ ¯ b ¯ a ] h (2) a ¯ a h (2) b ¯ b h (1) c ¯ c . (27)Contracting with φ b φ c in eq. (19), one sees that the termgoing with h (2) abc is simply half the one going with h (3) bca ,and L aux simplifies to L aux = h a σ a + 12 h (1) ab σ a σ b + 12 h (2) ab φ a φ b + σ a (cid:18) h (1) abc σ b σ c + h (2) abc φ b φ c (cid:19) + . . . (28)Although there are five parameters κ, g , g , g , g whichappear explicitly in h ( i ) abc , they do not represent new free-dom to fit the meson interaction dynamics, since they oc-cur also in the h ( i ) ab ; through the latter the h ( i ) abc depend im-plicitly also on further six parameters G, κ , g , g , g , g .All will be fixed by fitting the mass spectra and weakdecay constants, see (38) and section IV below. B. The heat kernel contribution We now turn our attention to the total Lagrangian ofthe bosonized theory. To write down this Lagrangianwe should add the terms coming from integrating outthe quark degrees of freedom in (13) to our result (28).Fortunately, the technicalities are known. We use themodified heat kernel technique [54–56] developed for thecase of explicit chiral symmetry breaking. In the isospinlimit one can find all necessary details of such calculationsfor instance in [53]. For future reference we apply it hereto obtain the result for the more general case in whichthe strong isospin symmetry is broken.From the vacuum to vacuum persistence amplitude inthe spontaneous broken phase Z [ σ, φ ] = (cid:90) D q D ¯ q exp (cid:18) i (cid:90) d x L q ( σ, φ ) (cid:19) , L q ( σ, φ ) = ¯ q ( iγ µ ∂ µ − M − σ − iγ φ ) q (29)the heat kernel result for the integration over the quarkdegrees of freedom is W [ Y ] = ln | det D | = − (cid:90) ∞ dtt ρ ( t ) exp (cid:16) − tD † E D E (cid:17) ,D † E D E = M − ∂ + Y, Y = iγ µ ( ∂ µ + iγ ∂ µ φ )+ σ + { M, σ } + φ + iγ [ σ + M, φ ] , (30)or W [ Y ] = − (cid:90) d x E π ∞ (cid:88) i =0 I i − tr[ b i ] (31)where D E stands for the Dirac operator in Euclideanspace. We consider the expansion up to the third Seeley-DeWitt coefficient b i b = 1 , b = − Y,b = Y λ ud Y + λ √ us + ∆ ds ) Y, (32)with ∆ ij = M i − M j . This order of the expansion takesinto account the dominant contributions of the quark one-loop integrals I i ( i = 0 , , . . . ); these are the arith-metic average values I i = [ J i ( M u ) + J i ( M d ) + J i ( M s )]where J i ( m ) = ∞ (cid:90) d tt − i ρ ( t Λ ) e − tm , (33)with the Pauli-Villars regularization kernel [57, 58] ρ ( t Λ ) = 1 − (1 + t Λ ) exp( − t Λ ) . (34)In the following we need therefore only to know two ofthem (the lowest order ∼ b contributes to the effectivepotential and is not needed in the present study) J ( m ) = Λ − m ln (cid:18) m (cid:19) , (35)and J ( m ) = ln (cid:18) m (cid:19) − Λ Λ + m . (36)While both terms proportional to b and b have con-tributions to the gap equations and meson masses, only b contributes to the kinetic and interaction terms. The σ tadpole term must be excluded from the total La-grangian. This gives us a system of gap equations h i + N c π M i (cid:2) I − (cid:0) M i − M (cid:1) I (cid:3) = 0 . (37)Here N c = 3 is the number of colors, and M = M u + M d + M s . Combining all terms of the total Lagrangian L = L kin + L mass + L int that contribute to the kineticterms L kin and meson masses L mass one gets L kin + L mass = N c I π tr (cid:2) ( ∂ µ σ ) + ( ∂ µ φ ) (cid:3) + N c I π ( σ a + φ a ) − N c I π (cid:110)(cid:104) M u + M d ) − M u M d − M s (cid:105) ( σ + σ )+ (cid:104) M u + M s ) − M u M s − M d (cid:105) (cid:0) σ + σ (cid:1) + (cid:2) M d + M s ) − M d M s − M u (cid:3) (cid:0) σ + σ (cid:1) + 12 (cid:2) σ u (cid:0) M u − M d − M s (cid:1) + σ d (cid:0) M d − M u − M s (cid:1) + σ s (cid:0) M s − M u − M d (cid:1)(cid:3) + 12 (cid:2) φ u (cid:0) M u − M d − M s (cid:1) + φ d (cid:0) M d − M u − M s (cid:1) + φ s (cid:0) M s − M u − M d (cid:1)(cid:3) + (cid:104) M u − M d ) + M u M d − M s (cid:105) (cid:0) φ + φ (cid:1) + (cid:104) M u − M s ) + M u M s − M d (cid:105) (cid:0) φ + φ (cid:1) + (cid:104) M d − M s ) + M d M s − M u (cid:105) (cid:0) φ + φ (cid:1)(cid:111) + 12 h (1) ab σ a σ b + 12 h (2) ab φ a φ b . (38)The kinetic term requires a redefinition of meson fields, σ a = gσ Ra , φ a = gφ Ra , g = 4 π N c I , (39)to obtain the standard factor 1 / 4. The flavor and chargedfields are related through λ a √ φ a = φ u √ π + K + π − φ d √ K K − ¯ K φ s √ λ a √ σ a = σ u √ a +0 κ + a − σ d √ κ κ − ¯ κ σ s √ (40)and in particular for the diagonal components φ u = φ + √ φ + φ √ φ + η ns φ d = − φ + √ φ + φ √ − φ + η ns φ s = (cid:114) φ − φ √ √ η s (41)and similar for the scalar fields. Here we also introducethe η ns and η s which stand for the flavor components ofthe physical η, η (cid:48) states in the nonstrange and strangebasis. In addition to the flavor mixing in the η, η (cid:48) chan-nels the isospin breaking induces a coupling between the π and these states π = φ + (cid:15)η + (cid:15) (cid:48) η (cid:48) . (42)To get the physical π , η and η (cid:48) mesons and correspond-ingly the scalar a (980), σ and f (980) mesons one mayproceed as in [59]. Since φ couples weakly to the η ns and η s states (decoupling in the isospin limit) while the η − η (cid:48) mixing is strong, it is appropriate to use isoscalar η ns , η s and isovector φ combinations as a starting point foran unitary transformation to the physical meson states π , η, η (cid:48) . In this case the corresponding unitary matrix U can be linearized in the π − η and π − η (cid:48) mixing angles (cid:15) , (cid:15) ∼ O ( (cid:15) ) , (cid:15) (cid:28) 1. Precisely [59] π ηη (cid:48) = U ( (cid:15) , (cid:15) , ψ ) φ η ns η s , (43)where U = (cid:15) + (cid:15) cos ψ − (cid:15) sin ψ − (cid:15) − (cid:15) cos ψ cos ψ − sin ψ − (cid:15) sin ψ sin ψ cos ψ (44)In particular, in eq.(42) (cid:15) = (cid:15) + (cid:15) cos ψ, (cid:15) (cid:48) = (cid:15) sin ψ .In the isospin limit we use the mixing angle conven-tions summarized in the Appendix B of [58]. We have the following different possibilities of relating the physicalstates ( ¯ X, X ) with the states of the strange-nonstrangebasis (cid:18) ¯ XX (cid:19) = R ψ (cid:18) X ns X s (cid:19) = R ¯ ψ (cid:18) − X s X ns (cid:19) , (45)where the orthogonal 2 × R ψ is R ψ = (cid:18) cos ψ − sin ψ sin ψ cos ψ (cid:19) , (46)or of the singlet-octet basis (cid:18) ¯ XX (cid:19) = R θ (cid:18) X X (cid:19) . (47)Here θ , being a solution of the equation tan 2 θ = x , isthe principal value of arctan x , i.e. belongs to the interval − ( π/ ≤ θ ≤ ( π/ ψ is related with θ by theequation ψ = θ + ¯ θ id , where ¯ θ id ( θ id + ¯ θ id = π/ 2) is deter-mined by the equations sin ¯ θ id = (cid:112) / 3, cos ¯ θ id = 1 / √ ψ = θ + arctan √ θ + 54 . ◦ . It meansthat ψ is restricted to the range 9 . ◦ ≤ ψ ≤ . ◦ . Ifthe value of ψ leaves the range, we must resort to theangle ¯ ψ = ψ − ( π/ 2) = θ − θ id , taking values in theinterval − . ◦ ≤ ¯ ψ ≤ . ◦ . These two angles corre-spond to two alternative phase conventions for a strange¯ ss -component. As a result of the following numericalcalculations, in the case of the pseudoscalars the identi-fication of the physical states is ¯ X = η, X = η (cid:48) and forthe scalars ¯ X = f (980) , X = σ .We turn to the interaction terms of the heat kernelaction in (30). The only contribution comes from Y / b and reads L ( hk ) int = − N c π I M a [ d abρ d ceρ σ b ( σ c σ e + φ c φ e )+ 2 f acρ f beρ σ b φ c φ e ] , (48)which must be added to the interaction piece stemmingfrom (28), yielding the total interaction Lagrangian L int = L ( hk ) int + σ a (cid:18) h (1) abc σ b σ c + h (2) abc φ b φ c (cid:19) . (49)Note that all dependence on the parameters of the ex-plicit symmetry breaking quark interactions is explicitlyabsorbed in the bosonized Lagrangian through the ma-trices h (1 , ab for the meson mass spectra (38) and throughthe h (1 , , abc for the meson interaction Lagrangian (49).In other words, the formal structure of the Lagrangian(28) in comparison to the case without these interactionsremains unchanged. This differs from the heat kernel La-grangian where the information about the difference inconstituent quark masses leads to a resummation of theheat kernel series for the modified Seeley-DeWitt coeffi-cients b i [54–56]. The parameters of these two seeminglyseparated sectors of the Lagrangian, i.e. the constituentquark masses and scale parameter Λ for the heat kernelLagrangian on one hand, and the multiquark interactioncouplings for the SPA piece on the other hand, are con-nected through the gap equations (37) which must besolved self-consistently with the SPA equations (21).In the remaining of this subsection we discuss thescheme in which the strong decay widths of the scalarmesons are calculated. Given the complexity of the La-grangian, we will restrict our study of the decays to thetree level bosonic couplings (48), (49). To deal in an ap-proximate way with the proximity of particle thresholdsto the resonance mass we shall resort to the widely ac-cepted Flatt´e type distribution [60]. Other closed bosonicchannel contributions will not be taken into considerationfor simplicity, since the ratios of couplings in the concur-ring closed channels to the nominal one turn out to benumerically less relevant in our fits.The strong decay width of the scalar meson S in twopseudoscalars P , P are thus obtained asΓ β = | (cid:126)p β | πm S | g β | ≡ ¯ g β | (cid:126)p β | (50)with | (cid:126)p β | = (cid:115) [ m S − ( m + m ) ] [ m S − ( m − m ) ]4 m S where index β specifies all necessary kinematic char-acteristics of the channel S → P P , and the masses m S , m , m of the states. We introduce also a shorthandnotation for the dimensionless quantity ¯ g β in eq.(50). Inthis definition we include all flavor and symmetry factorsassociated with the final state.The so obtained widths are valid in the Breit-Wignerresonance scheme, which is known to be an incompletedescription for decays with the resonance mass close tothe threshold of particle emission. We use Flatt´e distri-butions in the cases of the a (980) and f (980) decays toaccomodate the threshold effects associated with the twokaon production, on grounds of analyticity and unitarityat the threshold. Close to this threshold the elastic scat-tering cross section for πη in the case of a or ππ for f is parametrized by a two-channel resonance σ el = 4 π | f el | ,f βel = 1 | (cid:126)p β | m R Γ β m R − s − im R (Γ β + Γ SK ¯ K ) (51)with the index β designating here either the a πη or the f ππ channels andΓ SK ¯ K = ¯ g SK (cid:113) s − m K above threshold i ¯ g SK (cid:113) m K − s below threshold . (52)where ¯ g SK stands for the coupling of S to the two kaons,in this case S = a or f . Here m R is the nominal res-onance mass and s = ( p + p ) , where p , p are the 4-momenta of P and P . Near the K ¯ K threshold onlythe width Γ SK ¯ K is expected to vary strongly; the widthsΓ β are approximated by a constant value in this region,taken to be (50) evaluated at s = m R , since the πη and ππ thresholds lie further away from the resonance. Thenumerical results are presented and discussed in the sec-tion IV. C. A note on radiative decays Additional information on the structure of the mesonsis obtained through the study of their radiative decays.We consider in this work the two photon decays atthe quark one-loop order of the scalar and pseudoscalarmesons. The corresponding integrals are finite. A di-rect extension of the heat kernel Lagrangian to incor-porate the coupling to the electromagnetic interactionshows that there is no contribution up to the order b of the Seeley-DeWitt coefficients for the scalar decays.The anomalous pseudoscalar - two photon decays belongto the imaginary part of the action and are not contem-plated by the heat kernel techniques considered, whichapply only to the real part. By the Adler-Bardeen theo-rem [61–63] they are fully determined by the three-pointfunction Feynman amplitudes involving one quark loop;higher orders only redefine the couplings. There is how-ever a source of uncertainty which resides in the modeldependent determination of the coupling of the η and η (cid:48) mesons to the quarks. In our approach they are calcu-lated within the heat kernel technique outlined in sec-tion III.B. Regarding the scalar meson two photon de-cays, they are also most simply evaluated through thethree-point Feynman amplitudes, keeping only the con-tribution corresponding to the first non-vanishing orderin the heat kernel action, that is the term involving theSeeley-DeWitt coefficient b . From now on we will con-sider the case with exact SU (2) isospin symmetry, i.e. µ u = µ d = ˆ µ (cid:54) = µ s , and M u = M d = ˆ M (cid:54) = M s . Withthe standard electromagnetic coupling to quarks L γ = − e ¯ qγ µ QqA µ , Q = ( λ + √ λ ) and using the Pauli-Villars regularization, the scalar meson photon photonamplitude A : S ( s ) → γ ( p , (cid:15) ∗ µ ) + γ ( p , (cid:15) ∗ ν ) is obtained interms of the gauge invariant tensor L µν = ( p µ p ν − sg µν ),with s = ( p + p ) A µνSγγ = L µν A Sγγ ; S = σ, f (980) , a (980) A σγγ = 59 T u cos ¯ ψ − √ T s sin ¯ ψA f γγ = − T u sin ¯ ψ − √ T s cos ¯ ψA a γγ = 13 T u (53)where T i = 32 παgM i Q ( s, M i ) , i=(u,s)0 Q ( s, M i ) = iN c π (cid:90) dx (cid:90) − x dy (1 − xy ) × (cid:90) ∞ dtρ ( t Λ ) e − t ( M i − xys ) (54) α = e π is the fine structure constant and g the field nor-malization defined in (39). The factors of T i result fromthe flavor traces and projection to the physical stateswith the angle ¯ ψ defined in (45). The result for the in-tegral Q ( s, M i ) with the Pauli-Villars kernel ρ ( t Λ ), eq.(34), has been evaluated in [64]. To obtain the dominantcontribution, i.e. the first non-vanishing order in the heatkernel series, one needs to express the integrals Q ( s, M i )as the following averaged sum evaluated at s = 0 [55, 56] Q (0 , M i ) → Q (0 , M u , M s )= 13 (2 Q (0 , M u ) + Q (0 , M s ))+ O ( b ) (55)where the term O ( b ) is discarded as it belongs to thenext order in the heat kernel series (30), and Q (0 , M i ) = − N c π M i (cid:18) Λ Λ + M i (cid:19) , (56)or, in the notation of (33), we have that Q (0 , M i ) = − N c π J ( M i ) . (57)Finally the decay widths for the scalar mesons in thenarrow width approximation are given as (see also 64)Γ Sγγ = m S π | A Sγγ | (58)The anomalous decay of the pseudoscalars P =( π , η, η (cid:48) ) in two photons P ( p ) → γ ( p , (cid:15) ∗ µ ) + γ ( p , (cid:15) ∗ ν )has the same Lorentz structure in all channels and reads A µνP γγ = (cid:15) µναβ p α p β A P γγ A ηγγ = − T Pu sin ¯ ψ P − √ T Ps cos ¯ ψ P A η (cid:48) γγ = 59 T Pu cos ¯ ψ P − √ T Ps sin ¯ ψ P A π γγ = 13 T Pu (59)where ¯ ψ P stands for the mixing angle in the pseudoscalarchannels, eq. (45) and T Pi ( s, M i ) = 32 παgM i I P ( s, M i ) I P ( s, M ) = − N c π (cid:90) dx (cid:90) − x dy (cid:90) ∞ dte − t ( M − xys ) (60)and the contribution to the imaginary part of the heatkernel action is I P (0 , M ) = − N c π M . (61) At this stage one sees that the only parameter depen-dence in the radiative decays of the scalars and pseu-doscalars enters through the wave function normaliza-tion g , common to all decays considered, and throughthe constituent quark masses; there is also an explicit de-pendence on the scale Λ in the case of the scalar decaysthrough the factor ( Λ Λ + M ) in (56). The PCAC hypoth-esis establishes a relation between g , the weak pion andkaon decay couplings and the constituent quark masses(see also (66) below) f π = ˆ Mg ; f K = ˆ M + M s g . (62)These identities allow to eliminate all dependence on theconstituent quark masses from the pseudoscalar radiativedecays, leading to T P (0 , ˆ M ) = N c απf π , T P (0 , M s ) = N c απ (2 f K − f π ) . (63)One obtains then the celebrated relation A πγγ = απf π forthe π decay amplitude [61]. The Adler-Bardeen theoremallows to infer that the study and measurement of theanomalous decays are a reliable means of determinationof the mixing angle of the η and η (cid:48) mesons, which mustcomply with the mixing angle determination extractedfrom the mass spectrum. One should also stress thatwith the present model Lagrangian one is able to accountproperly for the SU (3) breaking effects in the descriptionof the weak decay constants f π and f K , in addition tohaving the correct empirical η and η (cid:48) meson masses (seesection IV), which has been an open problem until now.This is important for the numerical consistency in theamplitudes (63).The respective widths are calculated asΓ P γγ = | (cid:126)p | π | A P γγ | (64)with | (cid:126)p | = (cid:112) m P / m P the pseudoscalar mass. Thenumerical results are presented in section IV. IV. FIXING PARAMETERS, NUMERICALRESULTS AND DISCUSSIONA. Meson Spectra and weak decays In the chiral limit, m u = m d = m s = 0, the Lagrangian(38) leads to the conserved vector, V aµ , and axial-vector, A aµ , currents. The matrix elements of axial-vector cur-rents (cid:104) |A aµ (0) | φ bR ( p ) (cid:105) = ip µ f ab (65)define the weak and electromagnetic decay constants ofphysical pseudoscalar states (see details in [53]). Nowlet us fix the values of the various quantities intro-duced. After choosing the set κ = g = g = 0 we1still have to fix fourteen parameters: Λ , ˆ m, m s , G, κ, κ and g , . . . , g . There are two intrinsic restrictions ofthe model, namely, the stationary phase (21) and thegap (37) equations, which as mentioned above must besolved self-consistently. This is how the explicit symme-try breaking is intertwined with the dynamical symmetrybreaking and vice versa. We use (37) to determine ˆ h, h s through Λ , M s and ˆ M . The ratio M s / ˆ M is related to theratio of the weak decay constants of the pion, f π = 92MeV, and the kaon, f K = 113 MeV. Here we obtain M s ˆ M = 2 f K f π − . . (66)Furthermore, the two eqs. (21) can be used to find thevalues of Λ and ˆ M if the parameters ˆ m , m s , G , κ , κ , g , . . . , g are known. Thus, together with g we haveat this stage thirteen couplings to be fixed. Let us con-sider the current quark masses ˆ m and m s to be an input.Their values are known, from various analyses of the chi-ral treatment of the light pseudoscalars, to be aroundˆ m = 4 MeV and m s = 100 MeV [65]. Then the remainingeleven couplings can be found by comparing with empiri-cal data. One should stress the possibility (which did notexist before the inclusion of mass-dependent interactions)to fit the low lying pseudoscalar spectrum, m π = 138MeV, m K = 494 MeV, m η = 547 MeV, m η (cid:48) = 958MeV, the weak pion and kaon decay constants, f π = 92MeV, f K = 113 MeV, and the singlet-octet mixing angle θ p = − ◦ to perfect accuracy, see Table I.One can deduce that the couplings κ and g are essen-tial to improve the description in the pseudoscalar sector;in particular, g is responsible for fine tuning the η − η (cid:48) mass splitting, see also Table II, where the difference in g between set (b) and sets (a,c,d) is due to the input θ P = − ◦ versus θ P = − ◦ respectively.The remaining five conditions are taken from the scalarsector of the model. Unfortunately, the scalar channel inthe region about 1 GeV became a long-standing problemof QCD. The abundance of meson resonances with 0 ++ quantum numbers shows that one can expect the pres-ence of non- q ¯ q scalar objects, like glueballs, hybrids, mul-tiquark states and so forth [41]. This creates known diffi-culties in the interpretation and classification of scalars.For instance, the numerical attempts to organize the U (3)quark-antiquark nonet based on the light scalar mesons, σ or f (600) , a (980) , κ (800) , f (980), in the frameworkof NJL-type models have failed (see, e.g. [8–10, 58, 66–68]). The reason is the ordering of the calculated spec-trum which typically is m σ < m a < m κ < m f , asopposed to the empirical evidence: m κ < m a (cid:39) m f .On the other hand, it is known that a unitarized non-relativistic meson model can successfully describe thelight scalar meson nonet as ¯ qq states with a meson-mesonadmixture [33]. Another model which assumes the mix-ing of q ¯ q -states with others, consisting of two quarks andtwo antiquarks, q ¯ q [29], yields a possible description ofthe 0 ++ meson spectra as well [38, 39]. The well knownmodel of Close and T¨ornqvist [40] is also designed to describe two scalar nonets (above and below 1 GeV).The light scalar nonet below 1 GeV has a core madeof q ¯ q states with a small admixture of a ¯ qq compo-nent, rearranged asymptotically as meson-meson states.These successful solutions seemingly indicate on the im-portance of certain admixtures for the correct descriptionof the light scalars. Our model contains such admixturesin the form of the appropriate effective multi-quark ver-tices with the asymptotic meson states described by thebosonized ¯ qq fields. We have found, that the quark massdependent interactions can solve the problem of the lightscalar spectrum and these masses can be understood interms of spontaneous and explicit chiral symmetry break-ing only. Indeed, one can easily fit the data: m σ = 600MeV, m a = 980 MeV, m κ = 850 MeV, m f = 980 MeV.In this case we obtain for the singlet-octet mixing angle θ s roughly θ s = 19 ◦ [48]. Without changing the massspectra better fits for the strong radiative decays of thescalars are obtained with θ s = 25 ◦ ÷ ◦ , in the nextsubsection.We obtain and understand the empirical mass assign-ment inside the light scalar nonet as a consequence ofthe quark-mass dependent interactions, i.e. as the re-sult of some predominance of the explicit chiral symme-try breaking terms over the dynamical chiral symmetrybreaking ones for these states. Indeed, let us consider thedifference m a − m κ = 2 g (cid:18) H a − H κ (cid:19) − M s + 2 ˆ M )( M s − ˆ M ) . (67)The sign of this expression is a result of the competitionof two terms. In the chiral limit both of them are zero,since at ˆ µ, µ s = 0 we obtain ˆ M = M s and H a = H κ ,for H a and H κ being positive. The splitting H κ > H a is a necessary condition to get m a > m κ . The followingterms contribute to the difference H κ − H a = κ ( h s − ˆ h ) + 2 κ ( µ s − ˆ µ ) − g ( h s + ˆ hh s − h )+ g (cid:16) µ s h s + µ s ˆ h + ˆ µh s − µ ˆ h (cid:17) + g ˆ µ ( µ s − ˆ µ ) + g (cid:0) µ s − ˆ µ (cid:1) . (68)Accordingly, from this formula we deduce the “anatomy”of the numerical fit, e.g. for set (d) (see next subsection): m a − m κ = (cid:0) [0 . κ + [0 . κ + [6 × − ] g + [0 . g + [0 . g + [ − . g − [0 . M = 0 . 24) GeV , (69)where the contributions of terms with corresponding cou-pling (see eq. (68)) are indicated in square brackets. Thelast number, marked by M , is the value of the last termfrom (67). It is a contribution due to the dynamical chiralsymmetry breaking (in the presence of an explicit chiral2 TABLE I: The same values for the pseudoscalar and scalar masses (except for m σ ) and weak decay constans (all in MeV) areused as input (marked with *) for different sets of the model. Parameter sets (a),(b),(c),(d) of all following tables differ byvarying the mixing angles and m σ : sets (a), (b) and (d) with m σ = 550 MeV versus set (c) with m σ = 600 MeV, sets (a),(c)and (d) with θ P = − ◦ versus set (b) with θ P = − ◦ . The scalar mixing angle is kept constant, θ S = 25 ◦ , in (a),(b),(c) andincreased to θ S = 27 . ◦ in set (d). m π m K m η m η (cid:48) f π f K m κ m a m f m, m s , and Λ are given in MeV. The couplings have the following units: [ G ] = GeV − ,[ κ ] = GeV − , [ g ] = [ g ] = GeV − . We also show here the values of constituent quark masses ˆ M and M s in MeV. See alsocaption of Table I.Sets ˆ m m s ˆ M M s Λ G − κ g g a 4.0* 100* 372 541 830 9.74 121.1 3136 133b 4.0* 100* 372 542 829 9.83 118.5 3305 -158c 4.0* 100* 370 539 830 10.45 120.3 2081 102d 4.0* 100* 373 544 828 10.48 122.0 3284 173TABLE III: Explicit symmetry breaking interaction couplings. The couplings have the following units: [ κ ] = GeV − , [ κ ] =GeV − , [ g ] = [ g ] = GeV − , [ g ] = [ g ] = [ g ] = [ g ] = GeV − , [ g ] = [ g ] = GeV − . See also caption of Table I.Sets κ κ − g g g − g − g g g g a 0* 6.14 6338 657 210 1618 105 -65 0* 0*b 0* 5.61 6472 702 210 1668 100 -38 0* 0*c 0* 6.12 6214 464 207 1598 133 -66 0* 0*d 0* 6.17 6497 1235 213 1642 13.3 -64 0* 0*TABLE IV: Strong decays of the scalar mesons, m R is the resonance mass in MeV, Γ BW and Γ Fl are the Breit-Wigner widthand the Flatt´e distribution width in GeV, R S = ¯ g SK ¯ g β .Set Decays m R Γ BW Γ Fl ¯ g β ¯ g SK R S θ P θ S a σ → ππ 550 465 1.95 0.97 0.497 -12 25 f → ππ 980 108 60 0.23 0.32 1.397 κ → Kπ 850 310 1.2 0 a → ηπ 980 419 45 1.32 2.69 2.05Set Decays m R Γ BW Γ Fl ¯ g β ¯ g SK R S θ P θ S b σ → ππ 550 465 1.955 0.986 0.504 -15 25 f → ππ 980 108 60 0.230 0.312 1.356 κ → Kπ 850 310 1.2 0 a → ηπ 980 459 50 1.44 2.805 1.944Set Decays m R Γ BW Γ Fl ¯ g β ¯ g SK R S θ P θ S c σ → ππ 600 635 2.39 1.52 0.61 -12 25 f → ππ 980 108 61 0.23 0.30 1.32 κ → Kπ 850 310 1.2 0 a → ηπ 980 419 46 1.31 2.67 2.03Set Decays m R Γ BW Γ Fl ¯ g β ¯ g SK R S θ P θ S d σ → ππ 550 461 1.94 0.63 0.33 -12 27.5 f → ππ 980 62 30 0.23 0.30 3.90 κ → Kπ 850 310 1.2 0 a → ηπ 980 420 46 1.32 2.73 2.07 TABLE V: Radiative decays of the scalar mesons Γ Sγγ in KeV , m R is the resonance mass in MeV.Set a m R Γ Sγγ Set b m R Γ Sγγ Set c m R Γ Sγγ Set d m R Γ Sγγ σ → γγ 550 0.212 σ → γγ 550 0.212 σ → γγ 600 0.277 σ → γγ 550 0.210 f → γγ 980 0.055 f → γγ 980 0.055 f → γγ 980 0.055 f → γγ 980 0.080 a → γγ 980 0.389 a → γγ 980 0.386 a → γγ 980 0.392 a → γγ 980 0.383TABLE VI: Anomalous decays Γ Pγγ for sets (a) and (c) in KeV, corresponding to θ P = − ◦ , m R is the particle mass in MeV.[For set (b), corresponding to θ P = − ◦ , we have Γ ηγγ = 0 . η (cid:48) γγ = 4 . m R Γ Pγγ Γ expPγγ [65] π → γγ 136 0.00798 0 . ÷ . η → γγ 547 0.5239 (39 . ± . . ÷ . η (cid:48) → γγ 958 5.225 (2 . ± . . ÷ . symmetry breaking). One can see that the g -interactionis the main reason for the reverse ordering m a > m κ ,the coupling g being responsible for the fine tuning ofthe result.We now briefly comment on the role of parameters re-garding the successful fit of f π and f K as well as theordering m K < m η . For these cases many parametersare at work simultaneously. To illustrate this trend, wedeviate (arbitrarily) the values of f K and m η from theirempirical values, keeping the remaining observables fixed.Let’s consider first the weak decays. We take set (d)as reference and change in the input data only f K = 116MeV. As a result we obtain that the constituent quarkmasses both decrease to ˆ M = 351 MeV and M s = 533MeV, thus decreasing as well the normalization g in or-der to fulfill eq. (62). Regarding the interaction couplingstrengths, the largest deviation in absolute value is for g ,which increases by 50%, followed by g which decreasesby 40%. The parameters { g , κ , g , g , g , κ } decreasein the given order by { , , , , , } parts in hun-dred, and g increases by 28%. The remaining parame-ters have much less significant changes. We conclude thata very subtle interplay takes place involving parametersrelated with and without the explicit symmetry breakingin this case.As for m K < m η : we take again set (d) as referenceand change in the input only the η mass, lowering itto η = 490 MeV. In this case the largest changes areobserved in { g , g , g } , with an increase of { , , } per cent and a decrease in κ by 73%, while a lesserincrease in { g , g , κ } of { , , } and decrease of g by 16 per cent is registered. B. Strong decays Let us now show the result of our global fitting ofthe model parameters. We study the effect of havinga slightly different m σ mass, sets (a), (b) and (d) with m σ = 550 MeV versus set (c) with m σ = 600 MeV, aswell as having different pseudoscalar and scalar mixing angles, as described in the caption of Table I, with allother meson masses and weak decay constants remainingfixed to the values there indicated.Table II contains the standard set of parameters, whichare known from previous considerations. Their values arenot much affected by the quark mass effects. We havealready learned (as seen again in Table II) that highervalues of g lead to the lower σ mass [53]. This eight-quark interaction violates Zweig’s rule, since it involves q ¯ q annihilation.Table III contains the couplings which are responsiblefor the explicit chiral symmetry breaking effects in theinteractions. Largest variations are observed in the cou-plings g and g in set (d) as compared to sets (a-c) andin g between set (b) and the other sets. In the formercase it is related with the change of the scalar mixing an-gle and in the latter with the change in the pseudoscalarmixing angle. The coupling g is seen to occur only in( h (1) ab ) − , thus it probes the mass spectrum of the scalars,whereas g appears only in ( h (2) ab ) − , related to the massspectrum of the pseudoscalars. With all observables keptfixed, except the mixing angle, changes in these couplingsare obviously related to them. Regarding g it enters inboth mass spectra. Comparing sets (a) and (c) whereboth θ S and θ P are the same, but the σ mass different,show that that g responds also to the change in the σ mass.The calculated values of quark condensates are ap-proximately the same for all sets: −(cid:104) ¯ uu (cid:105) = 232 MeV,and −(cid:104) ¯ ss (cid:105) = 204 MeV. Our calculated values for theconstituent quark masses agree with the ones found in[8–10, 47], showing their insensitivity to the new mass-dependent corrections.In Table IV are shown the results for the strong decaywidths of the scalar mesons for the four different sets.The experimental status is as follows. The mass andwidth of the σ meson quoted until recently had a largeuncertainty, m σ = (400 ÷ σ = (600 ÷ m σ = (400 ÷ σ = (400 ÷ TABLE VII: The coefficients coef HK and coef SPA of the heat kernel and of the SPA contributions to the total value of thecoupling g SP P resulting from the interaction Lagrangian for the open decay channels. Values are for the neutral channels.Units are in GeV. g SP P coef HK /g coef SPA /g total /g σπ π -0.0450 0.0215 -0.0235 f π π -0.0061 -0.0047 -0.0109 κ ¯ K π a ηπ -0.0666 -0.0178 -0.0844TABLE VIII: The coefficients coef HK and coef SPA of the heat kernel and of the SPA contributions to the total value of thecoupling g SK ¯ K resulting from the interaction Lagrangian. Values are for the neutral channels. Units are in GeV. g SK ¯ KP coef HK /g coef SPA /g total /g σK ¯ K -0.041 0.0178 -0.0232 f K ¯ K a K ¯ K analysis of [69–72] leads even to a very sharp value forthe pole position M − i Γ / ± − (276 ± 5) MeV.The mass and full width of the f (980) meson are quotedas m f (980) = 990 ± 20 MeV and Γ f (980) = 40 ÷ 100 MeVand for the a (980) meson as m a (980) = 980 ± 20 MeVand Γ a (980) = 50 ÷ 100 MeV. The results for the κ (800)quoted in the PDG table from a Breit-Wigner fit havethe pole at (764 ± +71 − ) − i (306 ± +143 − ) MeV.We obtain that the σ mass and σ → ππ decay arewithin the recent limits for sets (a-b) and (d) while set(c) has a mass larger than the upper limit by ∼ 50 MeV.While in set (a-b) and (d) the calculated width is smallerthan the nominal mass of the resonance, the opposite be-havior is seen in set (c). The coupling strength ¯ g σππ in-creases comparing e.g. set(a) to (c) explaining the largerwidth, however the ratio R σ = ¯ g σK ¯ g σππ of the σ to kaonand to the pion couplings also increases by 20%. Theobtained ratios for R σ are in agreement with the exper-imental value R σexp = 0 . ± . R σ ∼ . ÷ . κ (800) → Kπ ∼ 310 MeV issmaller roughly by a factor two than the quoted cen-tral value but lies still within the limits. The ratio of thecouplings ¯ g κKπ ¯ g σππ m κ m σ = 1 . q ¯ q and q ¯ q model approaches considered in the samepaper.The widths of the a (980) → πη and f (980) → ππ decays are well accomodated within a Flatt´e description.We read the width at half maximum of the elastic crosssection in Figs. 1 and 2, respectively Note the huge re-duction in width in the case of the a (980) meson whenthe kaon channels are taken into account. This possi-bility was already noticed by Flatt´e in his analysis [60]. This is explained in our description by the ratio R a ∼ R S = ¯ g SK ¯ g β of the cou-plings is a relatively stable quantity in despite of the largefluctuations in the experimental values extracted for theindividual couplings. Our calculated R S are compatiblewith the indicated values in [74]. It should be emphasizedthat the ratio R f = ¯ g f K ¯ g f ππ is strongly dependent on themixing angle θ S of the scalar sector. As can be seen com-paring sets (a-c) with set (d) the increase in θ S is respon-sible for the larger ratio R f = 3 . R f exp = 4 . ± . 46 ofBES [75]. An often considered quantity is the crossedratio r = R f R a , usually assumed to be larger than unity.The a (980) does not depend on the θ S mixing angle(an eventual correlation with the f (980) meson throughisospin mixing is discarded here), but does depend on thepseudoscalar θ P angle through its decay into the πη . The θ P is fixed in the pseudoscalar sector to yield the correct η and η (cid:48) masses, as well as their radiative two photondecay widths. Therefore the ratio R a of the a cou-plings to kaons and to the πη channels remains approx-imately constant for all parameter sets ( R a ) − ∼ . R a exp ) − = 0 . ± . 11 [76]. Re-quiring the ratio r > θ S ∼ ◦ .On the other hand the ratio R f increases until θ S reaches ideal mixing. In the interval θ id < θ S ≤ π itdecreases but stays much larger than the experimentalaccepted ratio, e.g. at θ S = 44 ◦ one has R f ∼ 11. Thecombined requirement r > R f exp confines the mix-ing angle to the narrow window 27 ◦ < θ S < ◦ . Fromthe point of view of the calculated strong decay widthshowever the somewhat smaller angle θ S = 25 ◦ is alsoacceptable. Our interval of values for the mixing angle525 ◦ < θ S < ◦ , corresponding to − . ◦ < ¯ ψ < − . ◦ are within the values − ◦ < ¯ ψ < − ◦ estimated in [77],more specifically ¯ ψ ∼ − ◦ if a Flatt´e distribution is usedin a complementarity approach of Chiral PerturbationTheory and the Linear Sigma Model. C. Radiative decays The two photon decays of the pseudoscalars are in verygood agreement with data, (Table VI), the π and η intwo photons are within the experimental error bars, the η (cid:48) decay lies 10% above the upper limit for sets (a), (c)and (d), i.e. θ P = − ◦ . In the case of set (b), θ P = − ◦ , the result for the η (cid:48) decay is at the upper margin,and for the η about 10% above the upper boundary.For the radiative widths of the σ , see Table V, thereis a large spread in the experimental data from differentfacilities. Our results for σ → γγ only account for about20% of the value (1 . ± . 4) KeV [79] obtained from thenucleon electromagnetic polarizabilities, which is one ofthe lowest estimates for this width. For the f (980) → γγ the PDG average is quoted as (0 . +0 . − . ) KeV. Sets(a-c) yield approximately 20% and set (e) 30% of thisvalue. These results meet the current expectations thata direct coupling to the photons via a quark loop are notsufficient to account for the observed radiative widths ofthese mesons.A natural question arises then why in our approachthe strong widths can be described reasonably well in allchannels and the radiative ones fall short of the empiricalvalues for the σ, f decays. This can be understood: onlythe strong decays probe directly the multi-quark cou-plings g i contained in the stationary phase (SPA) piece(28) of the total interaction Lagrangian (49). Since thispart of the Lagrangian has no derivative terms only theheat kernel (HK) Lagrangian involves the electromag-netic interaction, after minimal coupling. The informa-tion of the SPA conditions which leaks through the gapequations to the electromagnetic sector is rather weak;it is contained only in the wave function normalizationwhich is the same for all mesons, and the quark con-stituent masses and scale Λ which remain approximatelyconstant in all parameter sets. Thus, effectively, thetwo photon decays of the scalars yield a clean signaturewhether the electromagnetic decay of the mesons pro-ceeds dominantly through a q ¯ q channel or not.This in turn ties up with the strength distribution inthe HK and SPA contributions to the coupling g SP P shown in Tables VII and VIII for set (d). The HK piecerelates directly to the meson- q ¯ q channel, the SPA part tothe higher order multiquark interactions.Consider first the a meson: the calculated a (980) → γγ ∼ . 39 KeV overestimates the present average PDGvalue 0 . +0 . − . and points within our approach to thedominance of the direct one quark loop coupling to pho-tons of this meson.This is corroborated by the fact that the large bare width that we obtain for the a → πη decay is shownto stem mainly from the HK coefficient represented with80% of the total strength, see Table VII. The a meson inthe q ¯ q picture is composed only of u and d quarks, thusits coupling to the K ¯ K mesons requires a flavor changeat the kaon vertices, as opposed to the ηπ case. As canbe seen from a similar decomposition in HK and SPAcontributions of the a K ¯ K coupling in Table VIII, it ismuch more favorable to couple to the kaons through themultiquark vertices, which now represent 80% of the to-tal strength instead. Therefore for the overall strong de-cay width it is important to take this mode into accountthrough the two-channel Flatt´e distribution. From thepoint of view of the two photon decay of a , we note thata πη loop does not couple directly to two photons [86]and the decay proceeds through the quark loop of u or dquarks with the large strength of the corresponding HKcomponent. To access the dominant SPA component thetwo photon decay would have to proceed through cou-pling to the K ¯ K loop, a sub-leading process in N c count-ing as compared to the direct q ¯ q loop. Furthermore, dueto the relatively large mass of the kaons, this loop is notexpected to contribute significantly.Now let us analyze the σ, f channels: there are sub-stantial contributions or cancellations from the SPA part.For the f ππ and f K ¯ K cases, one sees that the strengthin the SPA coefficient is in magnitude about of the HKcoefficient for both cases, but changes relative sign in thelatter. In the σππ and σK ¯ K cases, the cancellations oc-cur in both cases, with the SPA piece contributing abouthalf of the HK part. There is a subtle interplay aboutthe HK and SPA coefficients which finally add up to thecorrect description of the mass spectra and strong decaysof these mesons. The lack of a pronounced dominance ofthe HK has as consequence that the q ¯ q coupling of thesemesons to the photons represents only a fraction of thetotal width. The remaining strength must derive fromthe multiquark channels which should be included in anextra step, taking into account explicitly meson loop con-tributions.Regarding the strong decay of the f , one can furtherinfer that because of the stronger participation of themulti-quark interactions and because of cancellations inthe kaon channel as opposed to the pion channel, a cou-pling to the kaon channel through the Flatt´e approach isnot imperative to obtain a reasonable magnitude of thewidth, as seen from the Table IV.Rescattering effects have been shown in several ap-proaches to yield the main contribution, e.g. for the σ → γγ extracted from the dispersion analysis of γγ → π π [80]. Claims for a tetraquark structure [29] of the σ me-son were forwarded e.g. in [81], and in [82] interpreted aspion and kaon loop contributions. Our approach shedslight on these phenomena from a different angle.Finally we mention that the radiative decays of thescalar mesons have been calculated a long time ago ina variant of the NJL model, with and without mesonloop contributions, [83]. The amplitudes differ from ours6in two key aspects: we use the unified description for allnon-anomalous decays based on the generalized heat ker-nel approach which leads (i) to a common wave functionnormalization for all mesons that implies the reductionfactor of ∼ in the amplitude and in the case of theradiative decays to (ii) the regularized one loop integralscarrying the factors ( Λ Λ + M i ) , in despite of the integralsbeing finite. The latter reduces the amplitude by approx-imately half. The combined effect is a dramatic reductionby a factor ∼ 10 in the decay widths, as compared to [83]for the quark loop contribution. Thus caution must beused when it comes to interpret and comparing our nu-merical results with seemingly related model calculations,e.g. [84],[85].Summarizing the results of sections IV B. and C., thestrong decays calculated from our tree level meson cou-plings encode leading and higher order N c and multi-quark effects in combinations that account for the mainbulk of the empirical widths. The two photon decays ofthe scalars at leading order of the bosonized Lagrangianyield complementary information, testing whether the di-rect one quark loop coupling to photons is the dominantdecay process. We obtained that the a meson decayinto two photons proceeds mainly through the q ¯ q loop,whereas for the σ, f mesons we conclude that higher or-der multi-quark interactions are necessary to account forthe observed widths. This does not mean that the a meson is mainly a q ¯ q state, but that the multi-quarkcomponent with the large strength in the two kaon chan-nel, important for the reduction of the a πη strong decaywidth, is not the leading process in the two photon decayof this meson. (cid:45) (cid:48)(cid:46)(cid:49)(cid:53) (cid:45) (cid:48)(cid:46)(cid:49)(cid:48) (cid:45) (cid:48)(cid:46)(cid:48)(cid:53) (cid:48)(cid:46)(cid:48)(cid:48) (cid:48)(cid:46)(cid:48)(cid:53) (cid:48)(cid:46)(cid:49)(cid:48) (cid:48)(cid:46)(cid:49)(cid:53)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:48)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:50)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:52)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:54)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:56)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:49)(cid:48)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:49)(cid:50) (cid:69)(cid:32) (cid:91)(cid:71)(cid:101)(cid:86)(cid:93) (cid:45) (cid:50) (cid:93) (cid:115) (cid:32) (cid:91) (cid:77) (cid:101) (cid:86) (cid:104) (cid:112) FIG. 1: The πη cross section as function E = √ s − m K forthe a resonance channel from the Flatt´e distribution (solidline) with parameters of set (b), ¯ g a πη = 1 . 44, ¯ g a K = 2 . R a = 1 . Fl = 50MeV. Dashed line corresponds to the single πη channel. (cid:45) (cid:48)(cid:46)(cid:49)(cid:53) (cid:45) (cid:48)(cid:46)(cid:49)(cid:48) (cid:45) (cid:48)(cid:46)(cid:48)(cid:53) (cid:48)(cid:46)(cid:48)(cid:48) (cid:48)(cid:46)(cid:48)(cid:53) (cid:48)(cid:46)(cid:49)(cid:48) (cid:48)(cid:46)(cid:49)(cid:53)(cid:48)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:49)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:50)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:51)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:52)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:53)(cid:48)(cid:46)(cid:48)(cid:48)(cid:48)(cid:48)(cid:54) (cid:69)(cid:32) (cid:91)(cid:71)(cid:101)(cid:86)(cid:93) (cid:45) (cid:50) (cid:115) (cid:32) (cid:91) (cid:77) (cid:101) (cid:86) (cid:93) (cid:112)(cid:112) FIG. 2: The ππ cross section as function E = √ s − m K forthe f resonance channel from the Flatt´e distribution (solidline) with parameters of set (b), ¯ g f ππ = 0 . 23, ¯ g f K = 0 . R f = 1 . 36. The width read at half peak value is Γ Fl = 60MeV. Dashed line corresponds to just the two pion channel. V. CONCLUDING REMARKS In this paper we have generalized the effective multi-quark Lagrangians of the NJL type by including higherorder terms in the current quark-mass expansion. Theprocedure is based on the very general assumption thatthe scale of spontaneous chiral symmetry breaking deter-mines the hierarchy of local multi-quark interactions. Asa consequence, one can distinguish a finite subset of ver-tices which are responsible for the explicit chiral symme-try breaking at each order considered. We have classifiedthese vertices at next to leading order and studied thephenomenological consequences of their inclusion in theLagrangian.We are led to a subset of ten quark-mass dependentinteractions which enter the Lagrangian at the same or-der as the ’t Hooft determinant and eight quark termspreviously analyzed in the literature. From these, threeare related with the Manohar-Kaplan ambiguity, andthe remaining seven with genuinely new vertices. Thesenew terms carry either signatures of violation of theZweig-rule or of admixtures of q ¯ q states to the quark-antiquark ones and are thus potentially interesting can-didates in the quest of analyzing the structure and inter-action dynamics of the low lying mesons.We have derived the bosonized Lagrangian up to cu-bic order in the meson fields, from which we obtain themeson spectra and their two body strong, weak and elec-tromagmetic decays. Here are our main conclusions:(1) We fit the low lying pseudoscalar spectrum (thepseudo Goldstone 0 − + nonet) and weak decay constantsof the pion and the kaon to perfect accuracy. The fit-ting of the η − η (cid:48) mass splitting together with the overallsuccessful description of the whole set of low-energy pseu-doscalar characteristics is actually a solution for a long7standing problem of NJL-type models. We have foundthat the quark mass dependent interaction terms mainlyresponsible for the fit belong to the class of OZI-violatinginteractions. They represent additional corrections to the’t Hooft U A (1) breaking mechanism. In the interactionterms independent of the quark masses, we observe how-ever that the g coupling of the non OZI-violating 8 q interactions carrying the signature of the q ¯ q states arealso relevant in fitting the f π , f K values as well as for theordering m K < m η .(2) We are also capable to describe the spectrum ofthe light scalar nonet. In this case we identify the quark-mass interaction terms related with the four quark ad-mixtures to be the main source of the fit associated withthe a (980) and κ (800) meson masses. The primary termresponsible for the correct ordering carries interactionstrength g , and some fine tuning is due to the g term.(3) Regarding the mixing angle of the singlet-octetscalar states θ S we have found that its value is particu-larly sensitive to the interaction term proportional to g ,which is OZI-violating. Together with the result that thestrength g of the eight quark OZI-violating and quarkmass independent interaction term studied in earlier pa-pers dictates the mass of the σ (500) meson, we concludethat these states are strongly affected by OZI-violatingshort range forces.(4) The calculation of the strong decays of the scalarmesons has revealed that the present Lagrangian is capa- ble of accounting for the decay widths within the actualmargins of empirical data. We corroborate other modelcalculations in which the coupling of the f (980) and a (980) mesons to the K ¯ K channel is needed for the de-scription of the decays f (980) → ππ and a (980) → πη .We find that this coupling is most crucial for the latterprocess.(5) The radiative decays of the scalar mesons into twophotons show that the main channel for the a (980) decayproceeds through coupling to a quark-antiquark state,while the radiative decays of singlet-octet states σ, f must proceed through more complex strutures. We referto the full discussion given in sections IV B and IV C.(6) Finally, the radiative decays of the pseudoscalarsare in very good agreement with data. Acknowledgements This work has been supported by the Funda¸c˜ao para aCiˆencia e Tecnologia, project: CERN/FP/116334/2010,developed under the iniciative QREN, financed byUE/FEDER through COMPETE - Programa Opera-cional Factores de Competitividade. 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