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Quark and pion effective couplings from polarization 6 effects 1 0 2 Fa´bio L. Braghin r p A Instituto de F´ısica, Federal University of Goias, 9 Av. Esperan¸ca, s/n, 74690-900, Goiânia, GO, Brazil ] h April 12, 2016 p - p e Abstract h [ A flavor SU(2) effective model for pions and quarks is derived by considering polariza- 2 tion effects departing from the usual quark-quark effective interaction induced by dressed v 6 gluonexchange, i.e. aglobalcolor modelforQCD.For that, thequarkfieldisdecomposed 1 into a component that yields light mesons and the quark-antiquark condensate, being in- 9 tegrated out by means of the auxiliary field method, and another component which yields 4 0 constituent quarks, which is basically a background quark field. Within a longwavelength . 1 and weak quark field expansion (or large quark effective mass expansion) of a quark de- 0 terminant, the leading terms are found up to the second order in a zero order derivative 6 expansion, by neglecting vector mesons that are considerably heavier than the pion. Pi- 1 : ons are considered in the structureless limit and, besides the chiral invariant terms that v i reproduce previously derived expressions, symmetry breaking terms are also presented. X The leading chiral quark-quark effective couplings are also found corresponding to a NJL r a and a vector-NJL couplings. All the resultingeffective coupling constants and parameters are expressed in terms of the current and constituent quark masses and of the coupling g. PACS: 12.40.Yx, 12.38.Lg, 12.39.Fe, 14.40.Be 1 Introduction Hadron and nuclear structure and dynamics are ultimately ruled by Quantum Chromody- namics (QCD) which, due to its intrincated structure, is not exactly solved with currently known analytical methods [1]. There is a large amount of works dedicated to establish con- nections between (low and intermediary energies) QCD and observable hadrons which rely on the derivation and/or elaboration of effective models, either formally or pheonomenologically [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Hadron effective field models and theories must then be compatible with the more fundamental QCD symmetries and structure. These effective models and theories are expected to describe 1 observations already in the tree level or with first order corrections and the corresponding effective parameters and coupling constants might be expected to be calculable from QCD grounds. Several effective models for low energy hadron structure and interactions are formulated in terms of pions and constituent quarks and gluons, which have shown to be a powerful way to describe many hadron properties [24, 25]. Currently there is a large effort by many groups to describe nuclear properties by considering hadron effective models or theories, few examples are given in [5, 11, 16, 22, 23, 24, 25, 28, 29, 30], whose QCD content is more transparent. Dy- namical chiral symmetry breaking (DχSB) is an extremely important QCD effect to be taken into account since it yields broad and well known consequences in the hadron level and a direct relation to confinement is expected, for example in [1, 31, 32]. The relevance of the corresponding chiral condensate for hadron structure and dynamics is widely recognized although recently its location became a controversial subject [33, 34]. When establishing relations between QCD and hadron phenomenology it is of utmost importance to understand each piece of the mechanisms and effects that provides the measurable quantities. The fundamental effects that yield hadron effective interactions/couplings might help to select the most realistic and relevant couplings in a hadron model. One of the most investigated sources of quark effective couplings is instanton mediation [35, 36, 37, 12] and polarization effects have also been shown to produce quark effective interactions [26, 27]. DχSB is to be investigated and understood in the QCD quark sector andm although this program still needs theoretical developments and improvements, one might address particular known limits of the full theory. The following low energy quark effective global color model [6, 3, 4] will be considered: Z = N [ψ¯,ψ]exp i ψ¯ i∂/ m ψ D − Z Zx g2 (cid:2) (cid:0) (cid:1) jb(x)(R˜µν)(x y)jc(y)+ψ¯J +J∗ψ − 2 µ bc − ν Zy (cid:21) Where the color quark current is jµ = ψ¯λ γµψ, the sums in color, flavor and Dirac indices a a are implicit, stands for d4x, the kernel R˜µν is a dressed gluon propagator. Even if other x bc terms might arise from the non abelian structure of the gluon sector, the quark-quark induced R R interaction (1) should be part of the quark effective action for QCD. Although simple one gluon exchange is known to not produce enough strength of quark interactions such as to yield DχSB there are different effects that improve such picture and make the strength strong enough. For example by considering gluons self interactions in the polarization tensor, by modeling confinement or corrections in the corresponding Schwinger Dyson approach already in the rainbow ladder approximation, few examples are given in Refs. [7, 38, 39, 40]. The aim of this work is to derive a flavor SU(2) low energy effective model when internal structure of pions is not important by departing from the generating functional for the model (1). A quark field splitting into two components is performed: one component whose composite excitationscorrespondtolightquark-antiquarkmesonsandthescalarcondensate, andtheother that remains as (constituent) background field quarks. If these components may correspond or not to low or high energy quark components will not be discussed since this procedure correspondstotheone-loopbackgroundfieldmethod[41,42]. Vectormesonsdegrees offreedom 2 will be neglected since they are considerably heavier than the pion and are not expected to contribute in the low energy limit. The analysis of problems related to the light vector mesons is left for another work. The quark and pion effective couplings emerge from an expansion of a quark determinant which therefore generates a series of interactions with progressively higher powers of quark bilinears and of the pion field and their derivatives. Mesons will be considered in the punctual approximation within the zero order derivative expansion and this basically reproduces the corresponding terms found previously in Refs. [6, 4] in the chiral limit. In addition to that, chiral symmetry breaking terms due to the current quark mass are also found in the punctual pion limit. Besides that, leading quark-quark effective couplings, such as the NJL coupling, are also exhibited, being higher order in 1/N in agreement with c Ref. [43]. We also believe this approach might provide insights for the investigation of the longstanding problem of the convergence of the QCD effective action. An effective field theory (EFT) for low and intermediary energies QCD might be though as enough to provide a reliable understanding of the corresponding processes with enough predictive power to describe Nature atthislevel. However, thispicturebecomesmorecompleteifthemorefundamental mechanisms that generate all the terms of the corresponding EFT, with their effective parameters, arefound or derived. With the present work we hope to provide further insight into this program by considering the global color model (1). The paper is organized in the following way. In the next section, with a Fierz transformed version of the above non local current-current quark effective interaction, the quark field is separated into two components. One of these components will be integrated out by considering a set of auxiliary fields which generate quark-antiquark meson fields and the chiral condensate. The other component is dressed by polarization effects. Auxiliary fields that correspond to vector mesons will be neglected since they are considerably heavier than pions and do not contribute in the low energy regime. With a chiral rotation in Sect. 2.2, the non linear realization of chiral symmetry is introduced in terms of covariant derivatives. The expansion of the quark determinant is exhibited in Section 3, where the effective coefficients of the leading terms of quark couplings are presented up to the second order and the pion sector up to the fourth order. In the last Section there is a summary and discussion. 2 Flavor structure and auxiliary fields The departure point is therefore expression (1), and the kernel R˜µν can be written in terms of ab transversal and longitudinal components such that: ∂µ∂ν ∂µ∂ν R˜µν = δ R gµν +R . (1) ab ab T − ∂2 L ∂2 (cid:20) (cid:18) (cid:19) (cid:21) With a Fierz transformation [6, 3, 28] by selecting only the color singlet terms, the quark interaction above can be written in terms of bilocal quark bilinears, jq(x,y) = ψ¯(x)Γqψ(y) i where q = s,p,v,a and Γ = I .I (for the 2x2 flavor and 4x4 identities), Γ = iγ σ , Γµ = γµσ s 2 4 p 5 i v i and Γµ = iγ γµσ , where σ are the flavor SU(2) Pauli matrices. The resulting non local a 5 i i 3 interactions are the following: Ω g2ja(x)R˜µν(x y)jb(y) ≡ µ ab − ν αg2 j (x,y)j (y,x)+ji (x,y)ji (y,x) R(x y) → S S P P − 1 j(cid:8)i(cid:2)(x,y)ji(y,x)+ji (x,y)ji (y,x) (cid:3)R¯µν(x y) , (2) − 2 µ ν µA νA − (cid:27) h i where i,j,k = 0,...(N2 1) and α = 8/9 for SU(2) flavor. The kernels above can be written f − as: R(x y) R = 3R +R , T L − ≡ ∂µ∂ν R¯µν(x y) = gµν(R +R )+2 (R R ). (3) T L T L − ∂2 − The local limit yields the Nambu Jona Lasinio (NJL) and vector NJL couplings. The coupling constants are roughly, for massless gluons, G g2 [44] or G g2 for non zero effective ∼ Λ2QCD ∼ MG2 gluon mass being comparable in any case [26, 28, 45, 46, 47, 48]. These couplings are of the order of G g2 1/N . c ∼ ∼ The quark field will be splitted such as to preserve chiral symmetry into a component, (ψ) , 2 that yields the scalar condensate and whose (composite) excitations correspond to (quark- antiquark) light mesons, and another component, (ψ) , that will be associated to constituent 1 quark. If the usual shift were performed with a background fermion field, for ψ ψ + ψ 1 ¯ ¯ ¯ → and ψ ψ +ψ , additional contributions of higher order would emerge. This field separation 1 → by means of the bilinears yields rather the background field method in the one-loop level [41, 42]. The shift in the bilinears also produces automatically chiral invariant structures, being however that quarks that are integrated out also provide bound light meson states and the quark-antiquark condensate. Physically, this shift of bilinears can also be associated to the quark-antiquark states built with auxiliary fields. Quark bilinears will therefore be written as: ψ¯Γqψ t (ψ¯Γqψ) +t (ψ¯Γqψ) , (4) 2 2 1 1 → where t and t are constants that can be set to one at the end and that help to understand 1 2 the role of each of these components in the resulting model. According to this separation, the current-current quark interactions can be written in three parts: Ω t2Ω + t2Ω + t t Ω → 1 1 2 2 1 2 12 where Ω mixes both components and Ω ,Ω stand for the terms exclusive to each of the 12 1 2 components. Ω can be written as: 12 Ω αg2 jS (x,y)RjS (x,y)+jP (x,y)RjP (x,y) 12 ≡ 1 2 i 1 i 2 +jS (x,y)RjS (x,y)+jP (x,y)RjP (x,y) (cid:2)2 1 i 2 i 1 αg2 R¯ jµ (x,y)jν (x,y)+jµ (x,y)jν (cid:3)(x,y) − 2 µν i 1 i2 i A1 iA2 +jµ (x,(cid:2)y)jν (x,y)+jµ (x,y)jν (x,y) . i 2 i1 i A2 iA1 The resulting ambiguity inthis splitting will not besolved here. Howe(cid:3)ver it will beshown below that it yields the ambiguity of determining the relative contribution of constituent quarks and pions (or pion cloud) to describe hadron observables, in particular baryons [5, 49]. 4 2.1 Auxiliary fields and pions ¯ To integrate out the component (ψψ) , a set of bilocal auxiliary fields (a.f.) with the quantum 2 numbers of the bilinears defined above is introduced to linearize Ω . The generating functional 2 is multiplied by a collection of normalized unity Gaussian integrals of the a.f., and these fields are shifted by fermion bilineares, preserving an unit Jacobian, such that all the terms in Ω are 2 canceled out. These integrals, with the corresponding shifts, are given by: 1 = N D[S]D[Pi]e−2it22 x,yRα[(S−gj(S2))2+(Pi−gjiP,(2))2] R Z D[Vi]e−4it22 x,yR¯µνα (Vµi−gjµi,(2))(Vνi−gjνi,(2)) µ R h i Z D[A¯i]e−4it22 x,yR¯µνα(cid:20)(A¯iµ−gjµi,(2)A)(A¯iν−gjνi,(2)A)(cid:21). (5) µ R Z The bilocal a.f. are S(x,y),P (x,y),Vi(x,y) and A¯i(x,y) and the corresponding shifts (with i µ µ unity Jacobian) were given by the following bilocal bilinears: j = ψ¯(x)ψ(y), ji = ψ¯(x)iγ σ ψ(y), S P 5 i ji = ψ¯(x)σ γ ψ(y), ji,A = ψ¯(x)σ iγ γ ψ(y). (6) µ i µ µ i 5 µ The resulting effective Lagrangian is given by: t = t ψ¯ (iγ ∂ m)δ +t αgR(S +iγ σ P )+ 2αgR¯µν(Viγ +A¯iiγ γ ) L1 2 2 · − x,y 2 5 i i 2 µ ν µ 5 ν Zy (cid:20) +t αg2 Rq(x,y)Γ jq(x,y) ψ +t ψ¯ (iγ ∂ m)ψ 1 q 2 1 1 1 · − # q X g2t2 t2α 1 1 jb,(1)(x)R˜µν(x,y)jc,(1)(y) 2 R S2 +P2 + R¯µν ViVi +A¯iA¯i (7,) − 2 µ bc ν − 2 i 2 µ ν µ ν Zy Zy(cid:26) (cid:27) (cid:2) (cid:3) (cid:2) (cid:3) where terms of the form Rq(x,y)Γ jq(x,y) correspond to the terms from Ω , being Rq = q 12 (R,R¯µν) with the corresponding operators Γq from the bilinears. In this expression the inverse Fierztransformationwasperformedforthetermsint2Ω whichwaswrittenasacurrent-current 1 1 effective interaction again. By integrating out the quark field ψ , in the limit of zero quark field 2 ψ (or t = 0) and zero quark mass, the resulting model is the same as the model presented 1 1 and investigated in Refs. [2, 4, 6] in the flavor SU(3) version. Therefore the resulting pion sector has the same structure. It has also been shown that a chiral rotation in the measure of the generating functional yields a Wess Zumino term [4, 6]. Since the pion sector obtained in these works is the same as the one obtained in the present article the calculation will not be exhibited here. Fromthisnonlocaltheory, thelocalmesonfieldsaredefinedbymeansofaformalexpansion of the bilocal a.f. φ (x,y) on a local meson field basis M which is given by: q k,q x+y φ (x,y) = F˜ (x y)+ M F (x y), (8) q q k,q k,q − 2 − k (cid:18) (cid:19) X 5 where F are the form factors associated to the corresponding k = 0,1,2...-meson excitation of k,q the channel q. F˜ (for q = s,p,v,a) correspond to the translational invariant vacuum functions q for each of the channels q and M are the local meson fields, being that k > 0 correspond to all k,q meson excitations in the corresponding quark-antiquark channel q. The aimof thepresent work is to obtain a local meson-quark effective model for the low energy regime in which internal structure of mesons is not relevant. Therefore only the local limit of these form factors will be considered, and it will be written as: F (x y)Rq(y x) F δ(x y). This limit yields k,q k,q − − ≃ − punctual mesons. Besides that, only the lowest quark-antiquark scalar and pseudoscalar states (mesons) shouldcontribute, i.e. k = 0forthelocalmeson fieldsdenotedbyM = s,p ,vi,ai k=0,q i µ µ since higher quark-antiquark excited states are heavier and only contribute for (relatively) higher energy processes. Finally, the vector/axial (quark-antiquark) mesons do not contribute for low energy regime since their masses are considerably higher than the pion mass. Even if the auxiliary fields for vector mesons were considered, their structure and dynamics could receive contributions from constituent quarks and pion cloud, inducing an ambiguity in their description similar to the one that will be found for baryons. This problem however is outside the scope of this work. Moreover, vector mesons are known to give rise to corrections for the Skyrme terms (fourth order pion couplings c ,c found below) [50] and therefore to some 1 2 extent their contribution can be incorporated by redefining the fourth order pion terms. Chiral transformations mix scalar and pseudoscalar fields and therefore, in the limit of small current quark masses, one must have F = F = F. From here on, only the local punctual meson 0,s 0,ps fields leading terms will be considered. This will produce the correct punctual meson limit of the previously derived low energy pion effective couplings [6, 3, 4, 12]. Consider the following terms from the quark and auxiliary fields interaction, ¯ ψ (x)Ξ(x,y)ψ (y) where: 2 2 Ξ(x,y) = gα F (x y)R[S(z)+P (z)iγ σ ] 0,0 i 5 i { − γ σ ν iR¯µν Fv(x y)Vi(z)+iγ Fa(x y)A¯i(z) (9) − 2 0 − µ 5 0 − µ o (cid:2) (cid:3) where z = x+y that reduces to z = x due to the structureless mesons approximation, and, in 2 the absence of the heavier vector mesons, it reduces to: Ξ(x,y) Φ (x,y) F (s+p γ σ )δ(x y) L i 5 i → ≃ − Φ˜ δ(x y). (10) L ≡ − Therefore, the quark ψ and meson coupling in terms of Ξ Φ˜δ4(x y) can be written 2 as: t2 ψ¯ Φ (x y)ψ , which corresponds to the linear realiza∼tion of ch−iral symmetry. The 2 2 L − 2 ~ canonically normalized definition of the pion field becomes: ξ = F~p. 2.2 Chiral rotation The non linear representation for chiral symmetry can be obtained as described below. The scalar field is frozen and then, by performing a chiral rotation, only the pion field and its (covariant) derivative remain [51, 52, 42]. This can be done by constraining the scalar and 6 pseudoscalar fields to the chiral radius, 1 = s2 + p~2 which yields s = 1 p~2. Now we note − there is a freedom to define the pion and quark fields and derivatives related among each other p by chiral rotations. The quark free terms and its coupling to (scalar and pseudoscalar) mesons are given by: ¯ t ψ [iγ ∂ m+t Φ ]ψ 2 2 2 L 2 · − ¯ = t ψ [iγ ∂ m+t F(s+iγ ~σ p~)]ψ , (11) 2 2 2 5 2 · − · Quark and scalar and pseudoscalar fields can be redefined as [51, 52, 42]: 1 ~π2 2π (1 iγ ~σ ~π) s = − , p = i , ψ = − 5 · ψ′. (12) i 1+~π2 1+~π2 √1+~π2 In the resulting non linear realization of chiral symmetry the above Lagrangian terms (11) can be written as: ∂ ~π ~π ∂ ~π ψ¯′ iγ ∂ m∗ +γµ~σ µ iγ +i × µ 2 · − · 1+~π2 5 1+~π2 (cid:20) (cid:18) (cid:19) ~π2 ǫ σ π π +4m ijk k i j ψ′, (13) 1+~π2 − 1+~π2 2 (cid:18) (cid:19)(cid:21) where it was used that σ σ = δ +iǫ σ , and it has been set i j ij ijk k t = 1. 2 The last two terms in this expression correspond to the chiral symmetry breaking term fromthe current quark mass. These terms yield the terms proportional to the pion mass or to powers of ~π2 in the resulting effective model. To improve the notation two covariant derivatives are defined as: ∂ ~π µ ~π , µ D ≡ (1+~π2) ~π ∂ ~π ψ¯ ∂ ψ ψ¯′Dcψ′ ψ¯′ ∂ +i~σ × µ ψ′. (14) 2 µ 2 → 2 µ 2 ≡ 2 µ · 1+~π2 2 (cid:18) (cid:19) Thecanonicallynormalizedpionfieldcorrespondsto~π′ = ~πF. Fromhereon,thenewdefinitions of pion and quark field will be used by writing simply π and ψ respectively. This redefinition i 2 of the fields however induces a non trivial change in the functional measure with terms that do not depend on this pion covariant derivative. These terms are of higher order in the pion and quark fields, therefore they should be less important from a dynamical point of view. This subject will not be addressed further in the present work, and therefore the Jacobian will not be exhibited and discussed here. A different parameterization of the non linear realization can be used for the pseudoscalar fluctuations around the vacuum to rewrite expression (11), as discussed in Refs. [4, 6], by means of : Φ Φ = F P U +P U† , (15) L NL R L → where U = ei~σ·~π and PR,L = (1 γ5)/2 are the ch(cid:0)irality project(cid:1)ors. These expressions allow to ± rewrite the pion sector in the standard shape of Chiral Perturbation Theory. 7 2.3 Integrating out quarks ¯ By integrating out the component (ψψ) the following non linear (non local) effective action 2 ¯ for quarks (ψψ) and pions is obtained: 1 S = i Trlog (Sc)−1 +Φ αt g2R¯µνγ σ (ψ¯γ σ ψ) +iγ (ψ¯iγ γ σ ψ) eff − 0 N − 1 µ i ν i 1 5 5 ν i 1 (~π2 ǫ σ π π ) + 2αt g2R (cid:8)(ψ¯ψ) +iγ σ (ψ¯iγ σ ψ) +4m(cid:2) − ijk k i j , (cid:3) (16) 1 1 5 i 5 i 1 1+~π2 − L2 (cid:27) Z (cid:2) (cid:3) where the following relation was used: det(A) = eTrlnA and where Tr stands for traces of discrete internal indices and integration of spacetime or momentum coordinates for the quark component ψ . The following kernel has been defined: 2 (Sc)−1 (iγ D µ m∗). (17) 0 ≡ µ · c − The contribution of the pion covariant derivative was written as: Φ = iγ γµ~σ ~π (18) N 5 µ ·D and the remaining terms for the first component of the quark field given by: g2t2 = t ψ¯ (iγ ∂ m)ψ 1 jb,(1)(x)R˜µν(x,y)jc,(1)(y), (19) L2 1 1 · − 1 − 2 µ bc ν Zy The a.f. vacuum expected values can be found from their gap equations. However these equations must be found from the effective action in terms of the fields s,p , i.e. by integrating i out quarks ψ without doing the chiral rotrations of the last section. These saddle point 2 equations correspond to: ∂S eff = 0. (20) ∂φ i (cid:12)[φ(0)=s(0),p(0),..] j i (cid:12) (cid:12) This set of equations corresponds basica(cid:12)lly to the usual set of gap equations of the NJL model being that in the Nambu Goldstone mode only the scalar field has a non zero expected value in the vacuum. This provides the only contribution to the quark effective mass that constitutes pions and that condense into the chiral condensate, m∗ = m+gs(0). These gap equations were solved, for example, in a model with a very simplified gluon propagator in Ref. [27] in terms ¯ of an ultraviolet Euclidean cutoff. When a particular gauge is choosen for R and R, the gauge ∂S fixing parameter can be determined by a condition of gauge independence such as: eff = 0. ∂λ All the quantities in the effective action found below for quarks and pions, and also the gap equations above, depend basically on the original QCD Lagrangian parameters: u-d current quark masses, gauge coupling g, a gauge fixing parameter λ. By factorizing log((Sc)−1) in the determinant, with the quantity χ = γµ~σ ~π×∂µ~π, this term 0 · 1+~π2 can be written as: S′ = i Tr log[iγ ∂ m∗ χ] = C +S d2 − · − − 0 d2 = i Tr log[iγ ∂ m∗] i Tr log[1 S χ], (21) 0 − · − − − 8 where S = Sc(π = 0), being that the first term (C ) becomes a multiplicative constant factor 0 0 i 0 in the generating functional and the second one can be expanded for weak pion field or in a longwavelength expansion. The first order term is zero, and the expansion can be written as: 1 n S = i Tr( S χ) . (22) d2 0 n − n X The expansion of S yields terms of higher order in the pion field than the expansion of the d2 pion sector of the remaining part of the determinant. The quark determinant can then be written as: S = S +S , det d d2 where the main part can now written as: (~π2 ǫ σ π π ) S = i Trlog 1+Sc Φ˜ +4m − ijk k i j +g2αt R Γ ψ¯Γ ψ , (23) d − 0 N 1+~π2 1 q q q " !# q X The pion coupling in Dc also produces further interactions in the expansion as shown below, µ however the most relevant one is the first order term. This determinant will be expanded in the longwavelength limit (low pion momenta) and for weak ψ quark field (or for small g2). This 1 expansion is also equivalent to a large quark mass (m∗) zero order derivative expansion [53]. 3 Effective quark and pion couplings By neglecting vector meson fields the expanded determinant can be written as: ¯ ¯ S i Tr c S Φ +2K R(x y) (ψ(x)ψ(y))+γ σ (ψ(x)γ σ ψ(y)) d n 0 N 0 5 i 5 i ≃ − n X (cid:8) (cid:2) (cid:2) (cid:3) K R¯µν(x y)γ σ ψ¯(x)γ σ ψ(y)+iγ ψ¯(x)iγ γ σ ψ(y) 0 µ i ν i 5 5 ν i − − ~π2 ~σ ~π ~π n (cid:2) (cid:3) + 4m − · × (24) 1+~π2 (cid:21)(cid:27) where: c = (−1)n+1, and K = αg2t . All the terms of this expansion will be calculated in the n n 0 1 zero order derivative expansion. Besides that, only the leading terms in the pion derivative will be shown, i.e., terms of higher order in (∂n~π) (n 2) will be neglected. ≥ Many terms in this expansion are zero due to the traces of Dirac and Pauli matrices. The ¯ only non zero first order terms yield a correction to the quark (ψψ) mass and a pion mass 1 term. They are respectively the following: ~π2 = t ∆m∗ (ψ¯ψ) M2F2 , (25) L1 1 1 − π 1+~π2 and where these masses were defined as: ∆m∗ = i N αg2 Tr′ S R, (26) c 0 − 1 M2 = iN Tr′ S m, (27) π cF 0 9 where Tr′, from here on, corresponds to traces in all internal and spacetime (or momentum) indices except color. Withthehelp ofthegapequationforthescalar field (20), it canbenoticed that expression (27) corresponds to the Gell Mann Oakes Renner relation: M2F2 = < q¯q > π − m . In this expression it is seen that pion mass becomes zero in the chiral limit (m = 0), i.e. q q the Goldstone theorem is satisfied. i.e. in the limit of structureless pions. If pion structure had not been neglected, expression above with the corresponding gap equation for the quark (and eventually gluon) propagator yield a rainbow ladder Schwinger Dyson and Bethe Salpeter equations which had been solved previously [54, 55] and that is outside the scope of this work. 3.1 Manohar and Georgi expansion By neglecting any coupling to gluons the series above yields terms of the following general type [22] for C = 0: π A ψ¯Γψ B [A1,a] C p D I C,D G µ (28) ABCD ∼ CA,B f f2Λ Λ Λ (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17) which was considered by Manohar and Georgi where C,D are the coefficients to be calculated, CA,B ( [Aa,(1)])C are gauge invariant combinations of Aa which are not considered in this work G µ µ (C = 0), with the chiral invariant combinations of interacting mesons/pions or quarks (A,B) with momenta of order D. Basically the n th term in the expansion above corresponds to − A+B = n . Momentum dependence will be considered solely for the pion field in this work. 3.2 Second order terms There isafirst ordertermoftheexpansion abovethatyields a secondorder pion-quarkcoupling if the kernel Sc is also expanded for the pion coupling χ in the first order, i.e. similarly to the 0 expansion (22). The resulting term is the same as the one emerging from the chiral rotation for the component ψ by considering the zero order derivative expansion. It can be written as: 2 ~π ∂ν~π ¯ = i t g × (ψγ ~σψ), (29) 2dπ 1 πdπ ν L 1+~π2 · where it was defined the following coupling constant: g g δ = ig2αN Tr′ S2σ γ R¯ γµσ . (30) πdπ ρν ij c 0 i ρ µν j In expression (30) the following trace properties must be used: tr(σ σ ) = 2δ , i j ij tr(γ γ ) = 4g , µ ν µν tr(γ γ γ γ ) = 4(g g +g g g g ), (31) µ ν ρ σ µν ρσ µσ νρ µρ νσ − and, besides that, rotational invariance for the traces in spacetime or momentum coordinates. These properties must be used in the second order terms of the expansion, in particular those 10