Stationary phase corrections in the process of bosonization of multi-quark interactions

Stationary phase corrections in the process of bosonization of multi-quark interactions arXiv:hep-ph/0601074v1 10 Jan 2006 A. A. Osipov†1, B. Hiller†2 , J. Moreira†3 , A. H. Blin†4 † Centro de Fı́sica Teórica, Departamento de Fı́sica da Universidade de Coimbra, 3004-516 Coimbra, Portugal Abstract The functional integration over the auxiliary bosonic variables of cubic order related with the effective action of the Nambu – Jona-Lasinio model with ’t Hooft term has recently been obtained in the form of a loop expansion. Even numbers of loops contribute to the action, while odd numbers of loops are assigned to the measure. We consider the two-loop corrections and analyse their effect on the low-lying pseudoscalar and scalar mass spectra, quark condensates and weak decay constants. The results are compared to the leading order calculations and other approaches. PACS number(s): 12.39.Fe, 11.30.Rd, 11.30.Qc 1 On leave from Joint Institute for Nuclear Research, Laboratory of Nuclear Problems, 141980 Dubna, Moscow Region, Russia. Email address: osipov@nusun.jinr.ru 2 Email address: brigitte@teor.fis.uc.pt 3 Email address:jmoreira@teor.fis.uc.pt 4 Email address: alex@teor.fis.uc.pt 1 1. Introduction The large distance dynamics of QCD is dictated to a great extent by the spontaneous symmetry breaking of chiral symmetry [1, 2]. The Nambu – Jona-Lasinio (NJL) model of fermionic fields [3] suggests that the dynamical mechanism for such breaking be in analogy with the Ginsburg – Landau theory of superconductivity [4]. Numerous studies [5]-[7] have been performed since that time with the respective effective mesonic action derived from four-quark interactions of the NJL type. During these years the resolution of UA (1) problem has been found and, in particular, the relevance of the U(3)L × U(3)R chiral symmetric NJL model combined with the sixquark ’t Hooft flavour determinantal interaction (NJLH) [8] for low-energy phenomenology of mesons was noted [9]-[12]. The explicit breaking of the unwanted UA (1) axial symmetry by the ’t Hooft determinant is motivated by the instanton approach to low-energy QCD [8], [13]. Originally written in terms of fermionic degrees of freedom, the NJLH model has been widely explored at mean field level with Bethe-Salpeter and Hartree-Fock techniques applied to quark-antiquark scattering in its various channels of interaction [9], [14, 15]. In parallel, functional integral methods have been used to obtain the Lagrangian in bosonized form [10], [16]-[18]. The bosonization gives rise to a doubling of the mesonic auxiliary fields, of which one set has to be integrated out. This latter, in the presence of the ’t Hooft interaction, involves a term of cubic order, which cannot be integrated out exactly. In [10] the leading order stationary phase approximation (SPA) was calculated. At this order the effective potentials obtained with both methods coincide [17]. Given its success in describing a large bulk of empirical data, the question arises whether corrections to the leading order SPA result are small. By embarking in this task, a series of startling facts came across our investigations [18]: (i) The stationary phase equations which one obtains in the NJLH model have more than one root (critical point). Only one has a regular behaviour in the limit κ → 0 of the six-quark coupling, the others are singular. The rigorous SPA treatment requires taking into account all critical points, which give rise to an unstable vacuum for the theory. (ii) The result obtained in [10] corresponds to the regular root contribution. It is an approximation which leads to an effective potential with a well separated local minimum, which approaches smoothly the stable NJL 2 vacuum as κ → 0. Such a local minimum is probably a good ground for phenomenological estimates, at least all known calculations made in the NJLH model are based on this approximation. It has been shown recently [19] that eight-quark interactions stabilize this vacuum state, opening the way to justify this approach theoretically. (iii) Two expansions of the effective action have been considered: the perturbative series in κ and the loop expansion. Both of them were never studied beyond the leading order. It is tacitly assumed that next to the leading order corrections are small, although this fact has never been proven. In this paper we quantify the two-loop order contributions to the Lagrangian derived previously by studying their impact on the mass spectrum of low-lying mesons. We show that the effect is of the order of a few percent compared to the leading order masses, improving them slightly. We think it is rather safe to conclude that the loop expansion is rapidly converging, at least for the mass spectra. The paper is structured as follows. In the next section we collect the essential information needed to extract the linear and quadratic terms which contribute to the gap equations and mass terms respectively. In section 3 we write out the expressions for the gap equations, masses, weak decay constants and condensates. Section 4 contains the numerical results and discussion. Conclusions are presented in section 5. 2. The ”tandem” Lagrangian To be self-contained and define the notation we review the main ingredients of our model calculations. The underlying multi-quark Lagrangian is bosonized in a two step (tandem) process, in which a semi-bosonized functional, quadratic in the fermionic fields, and another functional, depending only on the auxiliary bosonic variables, can be dealt with separately. The integration over the quadratic fermionic degrees of freedom is formally exact and is calculated using a generalized heat kernel method. We start by presenting how these two types of functionals emerge and how we obtain and calculate the loop expansion we are after. 3 2.1 The stationary phase contribution We consider the fermionic Lagrangian LN JLH = q̄(iγ µ ∂µ − m)q + LN JL + LH , (1) LH = κ(det q̄PL q + det q̄PR q), (3) which contains the NJL four-quark vertices of the scalar and pseudoscalar types i Gh (2) (q̄λa q)2 + (q̄iγ5 λa q)2 LN JL = 2 and the six-quark ’t Hooft interaction [8] where m is the diagonal current quark matrix for quark fields with Nf = 3 flavours and Nc = 3 colours. In eq.(2) λa , a = 0, 1...8, are the normalized (trλa λb = 2δab ) matrices in flavour space. The explicit form of these U(3) q hermitian generators is λ0 = 2/3 and λa for a 6= 0 are the usual GellMann matrices. The positive coupling G, [G] = GeV−2 has order G ∼ 1/Nc . In (3) the negative coupling κ of dimension [κ] = GeV−5 has the large Nc N asymptotics κ ∼ 1/Nc f . Therefore LN JL dominates over LH at large Nc . The matrices PL,R = (1 ∓ γ5 )/2 are projectors on the chiral states and the determinant is over flavour indices. We are assuming that the quark fields transform like the fundamental representations of the global U(3)L × U(3)R chiral group, i.e., δq = i(α + γ5 β)q, δ q̄ = −iq̄(α − γ5 β), (4) where the parameters of the infinitesimal transformations are chosen as α = αa λa , β = βa λa . One now observes that √ δLN JLH = iq̄ ([α, m] − γ5 {β, m}) q + 2i 6β0 κ (det q̄PR q − det q̄PL q) . (5) The global chiral symmetry is broken explicitly by the current quark mass term and the U(1)A axial symmetry is broken too due to the ’t Hooft interaction. The functional integral in bosonized form is derived in [10], and has the form Z = × Z Y A +∞ Z Y −∞ A  Z DΠA DqD q̄ exp i  Z DRA exp i 4 4 d4 xLq (q̄, q, σ, φ)  d xLr (Π, ∆; R) ,  (6) where Lq = q̄(iγ µ ∂µ − M − σ − iγ5 φ)q, G 2 κ Lr = RA (ΠA + ∆A ) + RA + ΦABC RA RB RC . 2 3! (7) (8) with a cubic polynomial in the fields RA in the exponent. The notation is as follows [18]: RA = (Ra , Rȧ ) = (sa , pa ) and ΠA = (Πa , Πȧ ) = (σa , φa ) are a very compact way to represent two sets of auxiliary bosonic variables, each containing a scalar sa (σa ) and a pseudoscalar pa (φa ) nonet. The indices (a, ȧ) run from 0 to 8 independently in flavour space. We also define the related quantity ∆A = (∆a , 0) = (Ma − ma , 0). The external scalar fields σ = σa λa have been shifted σa → σa + Ma by the constituent quark mass Ma , so that the expectation value of the shifted fields in the vacuum corresponding to dynamically broken chiral symmetry vanish. The vacuum expectation value of the “unshifted” scalar field hσi = Ma λa = diag(Mu , Md , Ms ) (9) gives the point where the effective potential of the model V (hσi) achieves its local minimum. The corresponding condition is known as the “gap” equation. It eliminates tadpole graphs and determines the values of constituent quark masses as functions of the model parameters and of the cutoff Λ. The case mu 6= md 6= ms corresponds to the most general breakdown of the SU(3) flavour symmetry, giving Mu 6= Md 6= Ms . In this way the ground state of the system includes effects of the explicit symmetry breaking. We will assume in the following that mu = md . The variables σa and φa must be replaced by the physical scalar and pseudoscalar states σaph and φph a determined through the appropriate normalization of their kinetic terms. Note that these terms, as well as other important contributions to the meson masses and interactions of the effective mesonic Lagrangian, are obtained as a result of integration over the quark fields in Z. One has σa , φa = gσaph, gφph (10) a at leading order of the heat kernel expansion of the effective mesonic action (see next section). The quark-meson coupling g, being a function of parameters of the model, fulfills in addition the Goldberger – Treiman relation at the quark level: g = Mu /fπ . Combining this relation with (9) and (10) one 5 finds the well-known linear sigma model result [20] hσuph i = fπ . (11) Finally, the three index coefficients ΦABC are defined as Φabc = −Φaḃċ = 3 Aabc , 16 Φabċ = Φȧḃċ = 0, (12) obeying ΦABC δBC = 0. (13) The totally symmetric constants Aabc are related to the flavour determinant, and equal to 1 Aabc = ǫijk ǫmnl (λa )im (λb )jn (λc )kl . (14) 3! Now the functional integral over the auxiliary variables RA in (6) Z[Π, ∆] ≡ +∞ Z Y −∞ A  Z DRA exp i  4 d xLr (Π, ∆; R) , (15) can be written in the form [18]  Z Z[Π, ∆] ∼ exp i × +∞ Z Y −∞ A ∞ X 4 d xLst D R̄A  i exp 2  κ 1 i ΦABC × 3! n=0 n!  Z Z 4 d 4 xL′′AB R̄A R̄B d xR̄A R̄B R̄C n  . (16) Here Lst is the stationary value of the Lagrangian Lr (Π, ∆; R) associated with the regular critical point, around which the effective Lagrangian has been expanded. The barred fields indicate that they are shifted with respect to their stationary values R̄ = R − Rst . We denote by L′′AB the coefficient of the expansion to second order in the fields. The expansion stops exactly at the third order L′′′ ABC = κΦABC /3!, whose exponent is represented here as an infinite series. The terms with odd values of n do not contribute in (16), because the corresponding functional integrals over R̄A equal to zero. The first term, 6 n = 0, sums all tree diagrams of a perturbative series in powers of the coupling κ resulting in Lst . The n = 2 term represents the first non-leading correction to the effective Lagrangian (for details we refer to [18], where we show that this correction can be associated with a “two-loop” contribution of quantum auxiliary bosonic fields) Leff λ = Lst + 2π !8 3κ2 M . 32N(N + 2)(N + 4) (17) The stationary Lagrangian reads to cubic order in the fields [16, 17] 1 (1) 1 (2) Lst = ha σa + hab σa σb + hab φa φb 2 2   i 1 h (1) (2) (3) σa habc σb σc + habc + hbca φb φc + 3 + O(field4 ). (18) The two-loop corrections, contained in the second term of (17), consist of a rather intricate dependence  M = tr L′′  −1 3 + 6 tr L′′ −1 tr(L′′  −1 2 ) + 8 tr L′′  −1 3 (19) on flavour traces of powers of L′′ −1 , which is the inverse of the real and symmetric N × N matrix (N = 18) 3κ 3κ Gδab + Aabc scst − Aabc pcst   , 16 16 L′′AB (Rst ) =    3κ 3κ c c − Aabc pst Gδab − Aabc sst 16 16   (20) calculated at the stationary points sast and past . These are expressed in increasing powers of the external fields σa , φa (1) (1) (2) sast = ha + hab σb + habc σb σc + habc φb φc (1) (2) + habcd σb σc σd + habcd σb φc φd + . . . past = + (2) (3) hab φb + habc φb σc + (4) habcd φb φc φd + . . . (i) (21) (3) habcd σb σc φd (22) with hab... depending on ∆a and coupling constants (see Appendix). In particular the coefficients have nonvanishing components for a = (0, 3, 8) and are 7 obtained, with h = ha λa = diag(hu , hd , hs ), in the case of isotopic symmetry (hu = hd ) as [17]          Ghu + ∆u + κ hu hs = 0, 16 (23) κ 2 h = 0. Ghs + ∆s + 16 u In eq.(17) λ denotes an ultra-violet cutoff associated with the stationary phase corrections to the functional integral over auxiliary bosonic fields. It is a free parameter to be fixed by phenomenology. To handle the new contribution M, we expand L′′AB (Rst ), which we abbreviate from now on as L′′ , to second order in the external fields σa , φa , L′′ = L0 + L1 + L2 + O(field3 ), (24) L′′−1 = L̄0 + L̄1 + L̄2 + O(field3 ). (25) and its inverse This is all one needs to extract the relevant terms to the masses arising from the two-loop correction term. Here Li , (i = 0, 1, 2), denote the matrices which are constant, linear and quadratic in the fields respectively. The L̄i are constructed order by order, starting from the 0th order L′′ L′′−1 = L0 L̄0 = 1, (26) i.e. L̄0 = L−1 0 . The next terms are conditioned by this relation. Combining the first order Lagrangians and truncating at the linear fields (L0 + L1 )(L̄0 + L̄1 ) → L0 L̄0 + L1 L̄0 + L0 L̄1 = 1 (27) one gets the matrix L̄1 , after using (26), L̄1 = −L̄0 L1 L̄0 . (28) In a similar fashion one derives the matrix L̄2 as     L̄2 = − L̄0 L2 L̄0 + L̄0 L1 L̄1 = − L̄0 L2 L̄0 − L̄0 L1 L̄0 L1 L̄0 . 8 (29) Using L̄i in (25) and inserting in (19), one obtains M2 = 3  n trL̄0   h trL̄1  2  + trL̄0  2   + 6 tr L̄0 tr L̄21 + 2 tr L̄0 L̄2        trL̄2 + 8 tr L̄0 L̄21 + L̄2 L̄20  i  + 2 tr L̄1 tr L̄0 L̄1 + trL̄2 tr L̄20 o ,  (30) where M2 stands for the part of M which contains only the second order terms in the fields σa , φa . This expression is used in “Mathematica” [21]. Although the results, after evaluating of traces, are analytical they are very lengthy and not illuminating, and will not be presented here. However some structures are relevant for the low-energy theorems and the results will be encrypted in them, as shown in the section where we present the mass formulae. 2.2 The heat kernel contribution It still remains to evaluate the functional integral over the quark degrees of freedom in eq.(6). The Lagrangian Lq is invariant under the chiral transformations (4) and the transformations δσ = i[α, σ + M] + {β, φ}, δφ = i[α, φ] − {β, σ + M}, (31) induced by them for the external fields. All symmetry breaking terms have been absorbed in Lr . This fact is of importance, since one can use then the generalized asymptotic expansion of the quark determinant [22, 23]. This method preserves the above mentioned symmetry at any order, taking into account the effects of the flavour symmetry breaking contained in the mass matrix M. Thus the corresponding part of the effective action can be written ln | det D| = − 1 32π 2 Z d4 xE ∞ X Ii−1 tr(bi ), (32) i=0 where D = iγµ ∂µ − M − σ − iγ5 φ is the Dirac operator present in Lq , eq.(7), and the bi are generalized Seeley-DeWitt coefficients [22], of which we show the first four for the case of SU(2)I × U(1)Y flavour symmetry b0 = 1, b1 = −Y, 9 Y 2 ∆us + √ λ8 Y, 2 3 ∆2us Y3 ∆us 1 + √ λ8 Y − √ λ8 Y 2 − (∂Y )2 . b3 = − 3! 12 6 3 2 3 b2 = (33) In the present case the background dependent structure Y is given by Y = iγµ (∂µ σ + iγ5 ∂µ φ) + σ 2 + {M, σ} + φ2 + iγ5 [σ + M, φ]. (34) We use the definition ∆ij ≡ Mi2 − Mj2 . In eq.(32) the trace is to be taken over colour, flavour and Dirac 4-spinor indices and the regulator-dependent integrals Ii are the weighted sums Ii =  1 2Ji(Mu2 ) + Ji (Ms2 ) 3 with Ji (Mj2 ) = Z∞ 0 dt ρ(tΛ2 ) exp(−tMj2 ). t2−i (35) (36) They are regularized with the Pauli-Villars regularization scheme [24] with two subtractions and one ultra-violet cutoff Λ ρ(tΛ2 ) = 1 − (1 + tΛ2 )exp(−tΛ2 ). (37) We obtain, for instance, [25] Λ2 J0 (M 2 ) = Λ2 − M 2 ln 1 + 2 , M ! 2 Λ2 Λ . J1 (M 2 ) = ln 1 + 2 − 2 M Λ + M2 ! (38) (39) Both of them are divergent in the limiting case Λ → ∞. Note that Λ does not need to be the same cutoff as λ of eq.(17). In the following we restrict our study to the two nontrivial terms, b1 and b2 , in the asymptotic expansion of ln | det D|. In this case only I0 and I1 are involved, related to the quark one-loop integrals of one- and two-point functions respectively, at zero fourmomentum transfer. 10 3. Gap equations, condensates, meson spectra 3.1 Gap equations and condensates The complete effective bosonized Lagrangian Lb = LHK + Lst + Lc (40) comprises contributions from the heat kernel expansion to order b2 , LHK , and from (17), where Lc stands for the two-loop corrections. We restrict to the case SU(2)I × U(1)Y symmetry, e.g. Mu = Md 6= Ms . The first two contributions remain the same as in the leading order calculations [12]. Equating the coefficient of σi , i = (u, d, s) in eq.(40) to zero we obtain the gap equations h i Nc 2 2 M 3I + (M − M )I u 0 s u 1 + 2cu = 0, 6π 2 h i Nc hs + 2 Ms 3I0 − 2(Ms2 − Mu2 )I1 + 2cs = 0, 6π hu + (41) where cu and cs denote the corrections arising from Lc . They depend on hu , hs , λ, κ. These equations must be solved self-consistently and in conjunction with the stationary phase conditions (23). The solutions Mi of (41) allow to calculate the condensates hūui and hs̄si (see eq.(54)) hq̄i qi i = − i Nc h 2 2 M J (M ) − m J (m ) , i 0 i 0 i i 4π 2 (42) where we have subracted the contribution from the trivial vacuum [9]. Although they are structurally identical to the condensates calculated at leading order, they encode the information of the correction terms ci implicitly through Mi . 3.2 Meson masses The expressions for the leading order masses, i.e. with Lc put to zero, will not be repeated here. They were obtained in [12]. The correction mass terms can just be added to the leading order terms in their ”raw” form, that is, as they are directly extracted from LHK , depending on I0 , I1 integrals. To check the low energy theorems, one can then use the new gap equations, with 11 the correction terms ci included, to eliminate these integrals. For example for the pion, φj (j = 1, 2, 3), one has LHK (m2π ) = Nc (3I0 + ∆su I1 ) φ2j , 12π 2 (43) φ2j . 2G(1 + ωs ) (44) Lst (m2π ) = − With “Mathematica” we are able to identify Lc (m2π ) cu φ2j , =− (4G)2 ωu (1 + ωs ) (45) where ωi = κhi /(16G). This connection between the pion mass correction to the gap equation correction term cu is crucial to guarantee the Goldstone limit. Indeed we obtain, after eliminating I0 , I1 from (43) with help of the gap equations, o2  m2π = mπ 1 + 2 o2 mπ = cu , hu  g 2mu , Mu G(1 + ωs ) (46) o where mπ is structurally identical with the leading order pion mass; g 2 = 4π 2 /(Nc I1 ) renormalizes the pion fields to the physical fields (see eq.(10)). We used also that hu 1 − = (47) ∆u G(1 + ωs ) which is a simple consequence of the stationary phase conditions (23). In an analogous way we are able to get for the kaon mass cu + cs , hu + hs g 2 (mu + ms ) , = G(Mu + Ms )(1 + ωu ) o2  m2K = mK 1 + 2 o2 mK o2  (48) where mK has the form of the leading order kaon mass. We used again (23) to obtain hu + hs 1 − = . (49) ∆u + ∆s G(1 + ωu ) 12 Concerning the η, η ′ corrections, we show here only some relevant properties obtained with the help of “Mathematica” in the SU(3) limit (∆m2π )c = (∆m2K )c = (∆m288 )c , (∆m208 )c = 0, (∆m200 )c − (∆m288 )c 6= 0, (50) where, for instance, (∆m2π )c is the contribution to the pion mass obtained from the Lagrangian Lc . Therefore the correction to the flavour (0, 8) components follow the same patterns as in the leading order case [12], complying with the low-energy requirements: the octet member m288 remains degenerate with the pion and kaon, the mixing m208 vanishes and in the chiral limit the correction to the mass of the singlet m200 is also non-vanishing, and will therefore contribute to the singlet-octet splitting. For the scalars we obtain in the SU(3) limit also that the corrections to the masses behave as 2 (∆Ma20 )c = (∆MK2 0∗ )c = (∆M88 )c , 2 (∆M08 )c = 0, 2 2 (∆M00 )c − (∆M88 )c 6= 0. (51) 3.3 Weak decay constants We use PCAC and the Gell-Mann-Oakes-Renner (GOR) [26] relation to extract the condensates. From PCAC the weak decay constants are given as fπ = Mu , g fK = Mu + Ms . 2g (52) Using this and the mass relations for mπ and mK , eqs.(46) and (48), one obtains the GOR equations with some model corrections of higher order in the current quark masses and from which one identifies the condensates m2π fπ2 m2K fK2 mu mu = mu (hu + 2cu ) 1 + = −2mu h0|ūu|0i 1 + , ∆u ∆u   mu + ms 1 = (mu + ms )(hu + 2cu + hs + 2cs ) 1 + 4 ∆u + ∆s   1 mu + ms = − (mu + ms )h0|ūu + s̄s|0i 1 + . 2 ∆u + ∆s     (53) Finally, by using the gap equations (41) in (53) and expressing I0 , I1 through J0 (Mi2 ), J1 (Mi2 ) with (35) we obtain the condensates as h0|q̄i qi |0i = − Nc Mi J0 (Mi2 ) + O(J2 ), 4π 2 (54) where the O(J2 ) terms are neglected, to conform with the truncation of the heat kernel series. We recall that the O(J2 ) emerge from a property of the generalized heat kernel series in which differences of Jk (Mu2 ) − Jk (Ms2 ) are expressed as an infinite series involving Jk+l , l > 0 [22]. 4. Numerical results and discussion There are six parameters in the model, mu , ms , G, κ, Λ, λ. To see the effects of the new contribution, proportional to the cutoff λ, we compare pairwise in the sets (a,b), (c,d) and (e,f) of Tables 1-3 the results calculated with λ = 0 and λ 6= 0, keeping the remaining input unchanged. In this way, within each pair of sets, a running value of λ between the indicated ones, will interpolate smoothly between the calculated observables shown. In a-d we fix four parameters through the pseudoscalar sector, mπ , mK , fπ , fK , and adjust κ through the quark condensate hūui. In sets (e,f) five parameters are fixed through mπ , mK , fπ , mη′ , and the scalar a0 . Input is indicated through a *. Table 1. The main parameters of the model: current quark masses mu , ms , and corresponding constituent masses, Mu and Ms in MeV, couplings G (in GeV−2 ) and κ (in GeV−5 ), and two cutoffs Λ, λ in GeV. The values of condensates are given in MeV. mu a 6.3 b 6.3 c 6.3 d 6.3 e 2.8 f 2.1 ms 194 194 194 194 92 69 Mu 398 398 398 398 216 196 Ms 588 588 588 588 385 354 G 13.5 13.5 13.4 11.8 3.14 2.15 −κ 1300* 1300* 1370* 1370* 120 53 Table 2. 14 Λ λ −hūui1/3 0.82 0* 229 0.82 1.8* 229 0.82 0* 229 0.82 1.7* 229 1.37 0* 302 1.64 1.9* 333 −hs̄si1/3 172 172 172 172 314 363 The main characteristics of the light pseudoscalar mesons in MeV. The singlet-octet mixing angle θp is given in degrees. mπ a 138* b 138* c 138* d 138* e 138* f 138* mK 494* 494* 494* 494* 494* 494* fπ 92* 92* 92* 92* 92* 92* fK 114* 114* 114* 114* 129 129 θp mη mη′ 476 986 -14 487 958 -15 480 1020 -13 472 959 -15 533 1097* -1.2 540 1097* 0.5 Table 3. The characteristics of the light scalar nonet in MeV and the singlet-octet mixing angle θs in degrees. ma0 ∼ a0 (980) mK0∗ ∼ K0∗ (800) mσ∼ f0 (600) mσ′ ∼ f0 (980) a 1040 1267 806 1438 b 981 1219 781 1427 c 1056 1280 805 1447 d 967 1208 762 1426 e 980* 1029 413 1123 f 980* 992 346 1073 θs 24 24 23.7 23.5 19.5 18 It is clear that the large differences observed among different pairs of sets come from the leading order contribution. For example the condensates, fK and the η mass are strongly dependent on the value of κ, which is one order of magnitude larger in the sets (a-d) as compared to set (e,f). This observation applies also to the scalar spectrum, with large changes resulting at leading order. They are best described in set (f), but this implies rather large values for fK and the condensates. One observes however that the corrections, although small, have the correct trend, diminishing the splitting in the singlet-octet members of the pseudoscalar and scalar spectra. Comparing sets (a,b) with (c,d), one sees that by enlarging the magnitude of κ, the effect of the corrections get stronger, even for a smaller value of λ. 15 We also remark the interesting fact that λ cannot be arbitrarily increased, maintaining the remaining input on observables fixed. At values quite close to the ones indicated, solutions cease to exist. There is an intrinsic constraint on the size of corrections. One might be struck by the large variation in the values of κ and their relation to the convergence of the loop expansion (16). This can be understood by identifying the dimensionless expansion parameter of the series. Following [18], where standard methods are used to justify the stationary phase approach to functional integrals, one obtains the dimensionless parameter κ2 ζ= 32G3 λ 2π !4 . (55) Here the group structure factor 3/16 of Φabc in eq.(12) as well as the factor 1/3! appearing at each order in (16) have been taken into account. Note also that each term of the expansion carries a further suppression factor 1/n! In the present case n = 2 so that one get for sets (b), (d), (f) that ζ 2/2 is 0.0096, 0.021, 0.0023 respectively. This attests for a fast convergence of the series. For a comparison with empirical values, we take from [27]: mu = 1.5 ÷ 4 MeV, md = 4 ÷ 8 MeV, ms = 80 ÷ 130 MeV, mπ± = 139.57018 ± 0.00035 MeV, mK ± = 493.677 ± 0.016 MeV, mη = 547 ± 0.12 MeV, mη′ = 957.78 ± 0.14 MeV, (56) for the masses in the low lying pseudoscalar sector. The weak decay constants fπe = 130.7√± 0.1 ± 0.36 MeV, fKe = 159.8 ± 1.4 ± 0.44 MeV relate to ours through a 2 normalization factor, thus fπ ≃ 92.4 MeV and fK ≃ 113 MeV. The low-lying scalar masses are presently5 : 5 ma0 (980) = 984.7 ± 1.2 MeV, As several data sets are presented in [27], please consult it for details. Here we indicate the lowest and the highest values collected from all samples. 16 mf0 (600) = 400 ÷ 1200 MeV, mf0 (980) = 980 ± 10 MeV, mK0∗ (800) = 701 ÷ 970 MeV. (57) The recent update of the light-quark condensate is h(ūu+d̄d)/2i(1 GeV) = −(242 ± 15 MeV)3 , the flavour breaking ratio is known to be hs̄si/h(ūu + ¯ dd)/2i = 0.8 ± 0.3 [28]. Finally, as our numerical calculations do not differ significantly from the leading order values, we refer to [12] where a thorough discussion of our leading order results has been done in comparison with the ones obtained from other approaches [29]-[35]. 5. Conclusions The bosonization of the model combining the NJL and the ’t Hooft multiquark interactions leads to corrections associated with the stationary phase integration over auxiliary bosonic variables in the functional integral of the theory. The purpose of the present work has been to quantify NLO corrections, and to study their phenomenological effect on the mass spectrum of light pseudoscalar and scalar mesons. To this end the first correction to the tree level effective action has been considered. We have obtained the linear and quadratic terms (in the external mesonic fields) of the NLO Lagrangian. The group structure of the SU(2) × U(1) flavour symmetry considered leads to quite intricate expressions for the mass corrections. We have shown in a transparent way that they comply with the QCD low energy theorems. We have calculated the mass spectra of the low lying pseudoscalars and scalars, quark condensates and weak decay constants fπ , fK . The corrections are small and improve slightly the leading order results. We conclude from these calculations that the series considered is well convergent. It is an important conclusion, because it justifies the leading order estimates made before, on one hand, and reports on the selfconsistency of the stationary phase approach applied to the bosonization of effective multi-quark interactions, on the other hand. At the same time one may still expect some noticeable effects of the NLO terms which can show themselves in three and higher order mesonic amplitudes, especially in the cases where there is a strong cancellation between 17 the tree level contributions. Acknowledgements This work has been supported by grants provided by Fundação para a Ciência e a Tecnologia, POCTI/FNU/50336/2003, POCI/FP/63412/2005. This research is part of the EU integrated infrastructure initiative Hadron Physics project under contract No.RII3-CT-2004-506078. A. A. Osipov also gratefully acknowledges the Fundação Calouste Gulbenkian for financial support. 18 Appendix (i) The equations and algebra leading to the coefficients hab... of the series for sa and pa can be found in [17]. The explicit expressions for the case of one and two lower indices are also given there. In this appendix we collect (1) (2) explicit expressions for the coefficients habc and habc entering in the expansion of sa and needed for the evaluation of meson mass terms. Due to the trace structure of (30) and since L̄0 contributes only in the blockdiagonal, with nonvanishing entries in the diagonal and elements of 0, 8 mixing , only the elements of the diagonal and (0, 8) mixing of L2 will contribute to the mass terms. For those we need: κ 1 + ωs − 2ωu (1) h0aa = √ 3 for a ∈ 1, 2, 3. 16 6G µ+ (1 − ωs )2 κ 1 (1) for a ∈ 4, 5, 6, 7. h0aa = √ 3 16 6G µ+ (1 − ωu ) κ 1 + ωs + ωu (1) h8aa = − √ 3 for a ∈ 1, 2, 3. 16 3G µ+ (1 − ωs )2 1 + 2ωu κ (1) for a ∈ 4, 5, 6, 7. (58) h8aa = √ 3 32 3G µ+ (1 − ωu )2 κ (1) h000 = − √ 3 3 (1 + ωs − 2ωu )(1 − ωu )2 8 6G µ+ κ (1) h088 = √ 3 3 (1 + 2ωu )[1 + ωs (1 − 2ωu )] 16 6G µ+ κ (1) (1) (1) h008 = h080 = h800 = √ 3 3 ωu (ωu − ωs )(1 − ωu ) 8 3G µ+ κ (1) (1) h808 = h880 = √ 3 3 (1 + ωs + 2ωu − 4ωs ωu2 ) 16 6G µ+ κ (1) h888 = √ 3 3 (1 + ωu + ωs )(1 + 2ωu )2 16 3G µ+ (2) κ 1 + ωs − 2ωu √ for a ∈ 1, 2, 3. 3 16 6G µ+ (1 + ωs )2 κ 1 − ωu for a ∈ 4, 5, 6, 7. =− √ 3 16 6G µ+ (1 + ωu )2 κ 1 + ωs + ωu = √ 3 for a ∈ 1, 2, 3. 16 3G µ+ (1 + ωs )2 h0aa = − (2) h0aa (2) h8aa 19 (59) (2) h8aa = − (2) (2) (2) h008 (2) h800 (2) h808 (2) h888 for a ∈ 4, 5, 6, 7. 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