Interaction of the vector-meson octet with the baryon octet in effective field theory
MMITP/15-040
Interaction of the vector-meson octet with the baryon octet ineffective field theory
Y. ¨Unal,
1, 2
A. K¨u¸c¨ukarslan, and S. Scherer PRISMA Cluster of Excellence, Institut f¨ur Kernphysik,Johannes Gutenberg-Universit¨at Mainz, D-55099 Mainz, Germany Physics Department, C¸ anakkale Onsekiz Mart University, 17100 C¸ anakkale, Turkey (Dated: October 8, 2015)
Abstract
We analyze the constraint structure of the interaction of vector mesons with baryons using theclassical Dirac constraint analysis. We show that the standard interaction in terms of two inde-pendent SU(3) structures is consistent at the classical level. We then require the self-consistencycondition of the interacting system in terms of perturbative renormalizability to obtain relationsfor the renormalized coupling constants at the one-loop level. As a result we find a universal in-teraction with one coupling constant which is the same as in the massive Yang-Mills Lagrangianof the vector-meson sector.
PACS numbers: 11.10.Ef, 11.10.Gh, 12.40.Vv, 14.40.Be, 14.40.Df a r X i v : . [ nu c l - t h ] O c t . INTRODUCTION The ground-state baryon octet as well as the vector-meson octet played a vital role inshaping our understanding of the symmetries of the strong interactions (for an overview,see, e.g., Ref. [1]). According to Coleman’s theorem [2], the multiplet structure of the lighthadrons is related to an approximate SU(3) symmetry of the ground state of QCD. In fact,in the limit of massless up, down, and strange quarks, the QCD Lagrangian exhibits a chiralSU(3) L × SU(3) R symmetry, which is assumed to be dynamically broken down to SU(3) V in the ground state. As a result of this mechanism one expects the appearance of eightmassless Goldstone bosons [3–5], which are identified with the members of the pseudoscalarmeson octet. The masses of the pseudoscalars in the real world are attributed to an explicitchiral symmetry breaking due to the finite quark masses. The masses of hadrons other thanthe Goldstone bosons stay finite in the chiral limit.Symmetry considerations not only affect the spectrum of QCD but also put constraintson the interaction among hadrons. The dynamics of hadrons may be described in termsof an effective field theory [6]. To that end, one considers the most general Lagrangiancompatible with the symmetries of the underlying theory. Given a power-counting scheme,one may then calculate observables in terms of perturbation theory or, alternatively, by ap-plying non-perturbative methods such as solving integral equations. While the interactionof the pseudoscalar octet ( π, K, η ) with the baryon octet is largely constrained by sponta-neous symmetry breaking (see, e.g., Ref. [7]), this is not the case for the coupling of thevector-meson octet to the baryon octet. Moreover, when describing the dynamics of vectormesons in a Lagrangian framework, one inevitably faces the following challenge. EffectiveLagrangians for vector particles (spin S = 1, parity P = −
1) are constructed with Lorentzfour-vector fields V µ or anti-symmetric tensor fields W µν = − W νµ with four and six in-dependent fields, respectively (see, e.g., Refs. [8, 9]). Therefore, one imposes constraintswhich, for an interacting theory, may lead to relations among the coupling constants of theLagrangian. For example, by applying a Dirac constraint analysis [10] to the interaction ofthe pion triplet with the Delta quadruplet, it was shown in Ref. [11] that the number of inde-pendent coupling constants reduces from three at the Lagrangian level to a single coupling.Additional constraints beyond the consistency at the classical level may be obtained if werequire the theory to be perturbatively renormalizable in the sense of effective field theory[12]. An investigation of this type for the pure vector-meson sector results in a massiveYang-Mills theory [13–15]. All case studies found a reduction in the number of parameterswhich seemed to be independent from the point of view of constructing the most generalLagrangian. This is of particular importance when working with purely phenomenologicalLagrangians, because one is likely to introduce more structures, and thus seemingly freeparameters, than allowed by a self-consistent treatment.In the present article, we want to focus on the lowest-order effective Lagrangian for theinteraction of the vector-meson octet with the ground-state baryon octet. For that purpose,in Sec. II, we will summarize the idea of the Dirac constraint analysis. In Sec. III, we definethe relevant Lagrangians and then apply the Dirac constraint analysis in Sec. IV. In Sec. V,we investigate the constraints resulting from renormalizability in the sense of effective fieldtheory at the one-loop level. Our results are summarized in Sec. VI. Some technical detailsare relegated to the appendices. 2
I. REVIEW OF THE DIRAC CONSTRAINT ANALYSIS
A common procedure for the quantization of a classical system with given symmetries isto first construct the Lagrangian of the system, which is assumed to be invariant under thecorresponding transformation of the dynamical variables, and then to perform the transitionto the Hamiltonian in terms of a Legendre transformation. On the one hand, the Lagrangianformalism is suitable for satisfying Lorentz invariance and other symmetries, on the otherhand, the Hamiltonian formalism is needed to calculate the S matrix [12]. For a systemincluding constraints, we perform the transition from the Lagrangian to the Hamiltonian byapplying Dirac’s constraint analysis to be discussed below [10, 16, 17]. The quantization ofthe constrained system is performed using path-integral methods [13, 16, 17].In the following, we will summarize Dirac’s constraint analysis in terms of a classicalsystem with a finite number N of degrees of freedom (DOF). To start with, we considera Lagrange function L ( q, ˙ q ) which depends on N coordinates q i and the corresponding ve-locities ˙ q i = dq i dt , collectively denoted by q and ˙ q , respectively. We assume that L does notexplicitly depend on time and that the ˙ q i appear in monomials of maximal degree 2 in L : L ( q, ˙ q ) = 12 A ij ( q ) ˙ q i ˙ q j + b i ( q ) ˙ q i + c ( q ) , (1)where A ij = A ji , i.e., A = ( A ij ) is a symmetric N × N matrix, A = A T . To perform thetransition to the Hamilton formalism, one needs to introduce the momenta p i conjugate tothe coordinates q i , p i = ∂L ( q, ˙ q ) ∂ ˙ q i = A ij ( q ) ˙ q j + b i ( q ) , (2)or p ( q, ˙ q ) = A ( q ) ˙ q + b ( q ) . (3)Because the Hamiltonian is a function of ( q, p ), one needs to be able to invert Eq. (2) to goover from the set of dynamical variables ( q, ˙ q ) to ( q, p ). To uniquely solve Eq. (3) for thevelocities, the existence of the inverse matrix A − is required,˙ q = A − ( p − b ) , (4)where A ij = ∂∂ ˙ q j p i = ∂∂ ˙ q j ∂L∂ ˙ q i = ∂ L∂ ˙ q j ∂ ˙ q i = ∂ L∂ ˙ q i ∂ ˙ q j = A ji . (5)In other words, for a unique description of the velocities in terms of the momenta, theJacobian matrix ∂ ( q, p ) /∂ ( q, ˙ q ) cannot be singular, i.e.,det (cid:18) ∂ L∂ ˙ q i ∂ ˙ q j (cid:19) (6)cannot vanish. However, in the case that the determinant vanishes, the theory is singularand one cannot pass from the Lagrange function to the Hamiltonian formulation in thestandard manner. In this case, we make use of a method originally proposed by Dirac [10].In a singular system, we are not able to determine all velocities as functions of thecoordinates and the independent momenta. Let the unsolvable ˙ q i be the first M velocities3 q , . . . , ˙ q M . The so-called primary constraints occur as follows. The Lagrange function L can be written as L ( q, ˙ q ) = M (cid:88) i =1 F i ( q ) ˙ q i + G ( q, ˙ q M +1 , . . . , ˙ q N ) , (7)from which we obtain as the canonical momenta p i = (cid:26) F i ( q ) for i = 1 , . . . , M, ∂G ( q, ˙ q M +1 ,..., ˙ q N ) ∂ ˙ q i for i = M + 1 , . . . , N. (8)The first part of Eq. (8) can be reexpressed in terms of the relations φ i ( q, p ) = p i − F i ( q ) ≈ , i = 1 , . . . , M, (9)which are referred to as the primary constraints. Here, φ i ≈ H ( q, p ) = N (cid:88) i =1 p i ˙ q i − L ( q, ˙ q ) , (10)we consider the so-called total or extended Hamilton function [10] H T ( q, p ) = N (cid:88) j = M +1 p j ˙ q j ( p, q ) − G ( q, ˙ q M +1 ( p, q ) , . . . , ˙ q N ( p, q )) + M (cid:88) i =1 λ i φ i ( q, p )= H ( q, p ) + M (cid:88) i =1 λ i φ i ( q, p ) , (11)where the λ i , i = 1 , . . . M , are Lagrange multipliers taking care of the primary constraintsand the ˙ q i ( p, q ) are the solutions to Eq. (8) for i = M + 1 , . . . , N .The constraints φ i , i = 1 , . . . , M , have to be zero throughout all time. For consistency, ˙ φ i must also be zero. According to this statement, the time evolution of the primary constraints φ i is given by the Poisson bracket with the total Hamilton function, leading to the consistencyconditions { φ i , H T } = { φ i , H } + M (cid:88) j =1 λ j { φ i , φ j } ≈ , i = 1 , . . . , M. (12)The “weak” equality sign refers to the fact that the conditions hold only after the evaluationof Poisson brackets. Either all the λ i can be determined from these equations, or newconstraints arise. The number of these secondary constraints corresponds to the number of λ ’s (or linear combinations thereof) which could not be determined. Again one demandsthe conservation in time of these (new) constraints and tries to solve the remaining λ ’sfrom these equations, etc. The number of physical degrees of freedom is given by the initialnumber of degrees of freedom (coordinates plus momenta) minus the number of constraints.In order for a theory to be consistent, the chain of new constraints has to terminate such thatat the end of the procedure the correct number of degrees of freedom has been generated.Using this consistency condition, we could have some restrictions on the possible interactionsterms. 4 II. LAGRANGIAN
The vector mesons are described by eight real vector fields V µa , and the spin- baryonsby eight complex Dirac fields Ψ a (and adjoint fields Ψ † a ). The behavior of the fields underinfinitesimal global SU(3) transformations is given by V µa (cid:55)→ V µa + f abc (cid:15) b V µc , (13a)Ψ a (cid:55)→ Ψ a + f abc (cid:15) b Ψ c , (13b)Ψ † a (cid:55)→ Ψ † a + f abc (cid:15) b Ψ † c , (13c)where f abc denotes the structure constants of SU(3). Equations (13a)–(13c) express the factthat the corresponding fields in each case transform according to the adjoint representationas SU(3) octets.The most general effective Lagrangian for a system of a massive vector-meson octetinteracting with a massive baryon octet can be written as L = L + L / + L int + . . . . (14)Let us first comment on the terms which are not explicitly shown in Eq. (14). The ellipsesstand for an infinite string of “nonrenormalizable” higher-order interactions as well as forinteractions with other hadrons. We make the assumption that the “nonrenormalizable”interactions are suppressed by powers of some large scale and concentrate, at the presenttime, on the leading-order Lagrangians L , L / , and L int given by L = − V aµν V µνa + M V V aµ V µa − gf abc ( ∂ µ V aν ) V µb V νc − g f abc f ade V bµ V cν V µd V νe , (15a) L / = i a γ µ ∂ µ Ψ a − i ∂ µ ¯Ψ a ) γ µ Ψ a − M B ¯Ψ a Ψ a , (15b) L int = − i G F f abc ¯Ψ a γ µ Ψ b V cµ + G D d abc ¯Ψ a γ µ Ψ b V cµ . (15c)We have taken these Lagrangians to be invariant under the infinitesimal global SU(3) trans-formations of Eqs. (13a)–(13c). As a result, the members of the vector-meson octet have acommon mass M V , and the mass of the baryon octet is denoted by M B . In Eq. (15a), thefield-strength tensor is defined as V aµν = ∂ µ V aν − ∂ ν V aµ . Moreover, for the vector-meson selfinteraction, the constraint analysis of Refs. [14, 15] has already been incorporated, leadingto a reduction from originally five independent couplings to one single coupling g . The La-grangian L is hence nothing else but the massive Yang-Mills model. Owing to the assumedSU(3) symmetry, the interaction between the vector-meson octet and the baryon octet,Eq. (15c), can be parametrized in terms of two couplings G F and G D , where d abc denotesthe d symbols of SU(3). Note that in SU(2) a structure proportional to d symbols does notexist. The interaction Lagrangian of Eq. (15c) represents the analog of the D and F termsin the interaction of the Goldstone-boson octet with the baryon octet [18]. To summarize, atthe Lagrangian level we start with a massive Yang-Mills Lagrangian for the vector mesonsinvolving one dimensionless coupling g as justified in Refs. [13–15]. The interaction betweenthe vector-meson octet and the baryon octet contains two SU(3) structures with couplingsG F and G D . For the sake of simplicity, we suppress subscripts 0 denoting the bare parameters and the bare fields. V. CLASSICAL CONSTRAINT ANALYSIS
The Lagrangian description of spin-1 particles in terms of vector fields V µ contains toomany degrees of freedom, namely, four instead of three fields. In other words, we needconstraints to eliminate the redundant degrees of freedom. We perform the transition tothe Hamiltonian formulation and investigate whether the Lagrangians of Eqs. (15a)–(15c)lead to a consistent interaction with the correct number of degrees of freedom. This isthe case as soon as one has obtained the appropriate number of constraint equations andsimultaneously can solve for all the Lagrange multipliers. Moreover, to include the fermionicdegrees of freedom at a “classical” level, we treat the fields Ψ αa and Ψ ∗ αa as independentGrassmann fields related by formal complex conjugation [19]. The indices α and a refer tothe Dirac-spinor components and the SU(3)-flavor components, respectively. We will alsoneed the corresponding generalization of the Poisson bracket which is given in Appendix A.Before performing the Dirac constraint analysis, let us count the number of DOF in theHamiltonian framework, where the fields and the momentum fields are regarded as indepen-dent variables. Starting from the vector fields V µa together with the conjugate momentumfields π µa , we have 8 · · · · / fields we start with 8 · · αa and Ψ ∗ αa and 64 conjugate momentum fieldsΠ Ψ αa and Π Ψ ∗ αa . Indeed, we expect 8 · · · · π aµ = ∂ L ∂ ˙ V µa = ∂ L ∂ ˙ V µa , (16a)Π Ψ αa = ∂ L L ∂ ˙Ψ αa = ∂ L L / ∂ ˙Ψ αa = − i ∗ αa , (16b)Π Ψ ∗ αa = ∂ L L ∂ ˙Ψ ∗ αa = ∂ L L / ∂ ˙Ψ ∗ αa = − i αa . (16c)Here, we follow the convention of Ref. [17] and define both conjugate momentum fields Π Ψ αa and Π Ψ ∗ αa in terms of left derivatives. As a result, Π Ψ ∗ αa = − Π ∗ Ψ αa . Using these relations,we immediately see that the “velocities” cannot be expressed in terms of the “momenta.” Inthis case, we cannot immediately pass over from the Lagrangian description in terms of fieldsand velocity fields to the Hamiltonian description in terms of fields and momentum fields.To define the Hamiltonian of the system, we introduce 3 equations for so-called primaryconstraints [10], θ V a = π a + gf abc V b V c ≈ , (17a) χ αa = Π Ψ αa + i ∗ αa ≈ , (17b) χ αa = Π Ψ ∗ αa + i αa ≈ , (17c)where a = 1 , . . . , α = 0 , , ,
3. In these equations, a relation such as θ V a ≈ · · { λ αa , λ αa , λ V a } , for each constraint oneLagrange multiplier, and define a constraint Hamiltonian (density) H c through H c = λ αa χ αa + λ αa χ αa + λ V a θ V a . (18)We make use of χ ∗ αa = − χ αa and θ ∗ V a = θ V a , and require H c to be real. Noting that λ αa , λ αa , χ αa , and χ αa are all odd functions (see Appendix A), this implies for the Lagrangemultipliers λ ∗ αa = λ αa and λ ∗ V a = λ V a . The so-called total or extended Hamiltonian (den-sity) is constructed in terms of a Legendre transformation and the constraint Hamiltonian(density) H c as H T = H + H / + H int + H c . (19)The explicit expressions for the Hamiltonian densities are given in Appendix B. TABLE I: Counting the DOF for the free vector, Dirac, and interacting theories, respectively.Case Total DOF Constraints Physical DOFFree vector fields 64 16 48Free Dirac fields 128 64 64Interacting theory 192 80 112
The requirement that Eqs. (17a)–(17c) have to be zero throughout all time results in { θ V a , H T } = ∂ i π ia + M V V a − gf abc π ib V ci + . . . ≡ ϑ V a ≈ , (20a) { χ αa , H T } = i ( ∂ i Ψ ∗ βa )( γ γ i ) βα + M B Ψ ∗ βa γ βα + . . . + iλ αa = 0 , (20b) { χ αa , H T } = i ( γ γ i ) αβ ∂ i Ψ βa − M B γ αβ Ψ βa + . . . + iλ αa = 0 , (20c)where H T = (cid:82) d x H T is the total Hamilton function. The full expressions for the Poissonbrackets are displayed in Appendix C.From Eqs. (20b) and (20c) we can solve for the Lagrange multipliers λ αa and λ αa ,respectively. In other words, in the fermionic sector, we have produced the correct numberof constraints, namely 64, and have also determined the 64 Lagrange multipliers, without anyconditions for the coupling constants G F and G D . Equation (20a) is a so-called secondaryconstraint, and, therefore, we obtain 8 additional constraints. Also these constraints have toremain conserved with time. In fact, evaluating the Poisson bracket of ϑ V a and H T resultsin an equation for the Lagrange multiplier λ V a [see Eq. (C2)]. By inserting the results forthe fermionic Lagrange multipliers λ αa and λ αa , at this stage, we have solved for all theLagrange multipliers and have generated the correct number of constraints. The results forthe number of DOF are summarized in Table I.As a result of Dirac’s constraint analysis, at the classical level we have a self-consistenttheory with the correct number of constraints and thus the correct number of physicalDOF without any relation among the couplings. In other words, at the classical level, theconstants g , G F , and G D may be regarded as independent parameters.7 . RENORMALIZABILITY We have seen in Sec. IV that, at the classical level, the leading-order Lagrangians ofEqs. (15a)–(15c) provide consistent interactions with the correct number of DOF. In par-ticular, at this stage, the coupling constants g , G F , and G D are independent parametersof the theory. When using these Lagrangians in perturbative calculations beyond the treelevel, we will encounter ultraviolet divergences which need to be compensated in the processof renormalization [20]. At the one-loop level, the perturbative renormalizability conditionstates that all the divergent parts of the one-loop diagrams must be canceled by the tree-level diagrams originating from the corresponding counter-term Lagrangian. Since we areworking with the most general effective Lagrangian satisfying the underlying symmetries,perturbative renormalizability in the sense of EFT requires that the ultraviolet divergencesof loop diagrams can be absorbed in the redefinition of the masses, coupling constants, andfields of the effective Lagrangian [6, 12]. However, it may turn out that this is only possibleif certain additional relations exist among the coupling constants. A. Counter-term Lagrangian
In order to see whether the couplings g , G F , and G D are related through renormalizability,we investigate the vector-meson self energy as well as the V V V and
V V V V vertex functionsat the one-loop level. To identify the counter-term Lagrangian, we relate the bare fields Ψ and V µ to the renormalized fields Ψ and V µ ,Ψ = (cid:112) Z Ψ Ψ , V µ = (cid:112) Z V V µ , (21)and express the bare parameters and the wave-function renormalization constants in termsof the renormalized parameters, g = g + δg, (22a)G F0 = G F + δ G F , (22b)G D0 = G D + δ G D , (22c) M B = M B + δM B , (22d) M V = M V + δM V , (22e) Z Ψ = 1 + δZ Ψ , (22f) Z V = 1 + δZ V . (22g)The functions δg etc. depend on all the renormalized parameters and on the renormalizationcondition. The counter-term Lagrangian is then given by L ct = − δZ V V aµν V µνa + 12 δ { M V } V aµ V µa − δ { g } f abc f ade V bµ V cν V µd V νe − δ { g } f abc ( ∂ µ V aν ) V µb V νc + i δZ Ψ (cid:0) ¯Ψ a γ µ ∂ µ Ψ a − ( ∂ µ ¯Ψ a ) γ µ Ψ a (cid:1) − δ { M B } ¯Ψ a Ψ a − iδ { G F } f abc ¯Ψ a γ µ Ψ b V cµ + δ { G D } d abc ¯Ψ a γ µ Ψ b V cµ , (23)where we display only those terms generated from the Lagrangians in Eqs. (15a)–(15c). Theexpressions for the counter-term functions δ { M V } etc. are given in Appendix D.8 . Derivation of the conditions We now investigate the divergent parts of all one-loop contributions to the self energiesand the vertex functions shown in Fig. 1. Omitting for simplicity both flavor and Lorentzindices, the relation between the unrenormalized (or bare) and renormalized proper vertexfunctions involving three and four vector fields, respectively, reads [21]Γ R V = Z V Γ V , (24a)Γ R V = Z V Γ V , (24b)where Γ V and Γ V are unrenormalized vertex functions and Z V is the wave-function renor-malization constant of the vector field. The vertex functions and the wave-function renor-malization constant may be expanded in powers of ¯ h ,Γ = Γ tree + ¯ h Γ + O (¯ h ) , (25a) Z V = 1 + ¯ h δZ V + O (¯ h ) . (25b)Substituting Eqs. (25a) and (25b) into Eqs. (24a) and (24b), we obtain the expansionsΓ R V = Γ tree3 V + ¯ h (cid:18) Γ V + 32 δZ V Γ tree3 V (cid:19) + O (¯ h ) , (26a)Γ R V = Γ tree4 V + ¯ h (cid:16) Γ V + 2 δZ V Γ tree4 V (cid:17) + O (¯ h ) . (26b)The tree-level diagrams have the following form,Γ tree3 V = g S V , (27a)Γ tree4 V = g S V , (27b)where S V and S V denote both Lorentz and flavor structures. The corresponding divergentparts of the loop diagrams in Fig. 1 contain the same Lorentz structures. In terms of therenormalized coupling g , the bare coupling g can be written as g = g + ¯ hδg + O (¯ h ) , (28)where δg is the one-loop counter term. Using Eq. (28) in Eqs. (27a) and (27b), weobtain from Eqs. (26a) and (26b) the expressionsΓ R V = gS V + ¯ hδg S V + ¯ h (cid:18) Γ V + 32 δZ V gS V (cid:19) + O (¯ h ) , (29a)Γ R V = g S V + 2¯ hgδg S V + ¯ h (cid:16) Γ V + 2 δZ V g S V (cid:17) + O (¯ h ) . (29b)The left-hand sides of Eqs. (29a) and (29b), i.e., Γ R V and Γ R V , are finite. On the right-handsides, the tree contributions, i.e., gS V and g S V , are also finite. In Eq. (29a), δg The one-loop contributions involving internal vector-meson lines, generate expressions of orders g and g which need to be canceled by separate counter-term contributions. δg has tocancel the divergences inside the parentheses of Eq. (29b), which also depend on the couplingconstants, but with a different functional form. These two conditions for δg ultimatelylead to relations among the coupling constants. From Eqs. (29a) and (29b) we obtain forthe terms linear in ¯ h the conditions δg S V + (cid:18) Γ V + 32 δZ V gS V (cid:19) = 0 , (30a)2 gδg S V + (cid:16) Γ V + 2 δZ V g S V (cid:17) = 0 . (30b) C. SU(2)
Before addressing the universality principle in SU(3), we first want to reproduce the caseof SU(2) [22]. To that end, we consider the diagrams of Fig. 1. Introducing Lorentz- andisospin indices, the vector-meson self energy may be parameterized as [23]Π µνij ( p ) = δ ij (cid:2) g µν Π ( p ) + p µ p ν Π ( p ) (cid:3) . (31)Using dimensional regularization, the result for the divergent part of the self-energy diagramreads Π div1 ( p ) = − λ π g V NN p , (32a)Π div2 ( p ) = λ π g V NN , (32b)where λ is given by λ = 116 π (cid:26) D − −
12 [ln(4 π ) + Γ (cid:48) (1) + 1] (cid:27) , (33)with D the number of spacetime dimensions. The wave-function renormalization constantis related to the residue of the propagator at the pole, p = M V . In terms of the self-energyfunction Π ( p ) it is given by Z V = 11 − Π (cid:48) ( M V ) . (34)Since we are working at one-loop order, Z V can be written as Z V = 1 + Π (cid:48) ( M V ) + O (¯ h ) , (35)where O (¯ h ) stands for two-loop corrections. Inserting Eq. (32a) into Eq. (35), we have forthe part proportional to λ , δZ λV = − λ π g V NN . (36)10 + + + + (b)(a)(c) FIG. 1: (a) Nucleon-loop contribution to the vector-meson self-energy diagram, (b) one-loop contri-butions to the three-vector vertex function, and (c) four-vector vertex function. Single and doublelines correspond to fermions and bosons, respectively.
The divergent parts of the one-loop contributions to the three- and four-vector vertex func-tions read, respectively,Γ µνρ div ijk ( p , p , p ) = (cid:15) ijk λ π g V NN [ g µν ( p − p ) ρ + g µρ ( p − p ) ν + g νρ ( p − p ) µ ] , (37a)Γ µνρσ div ijkl ( p , p , p , p ) = − iλ π g V NN [(2 δ ij δ kl − δ ik δ jl − δ jk δ il ) g µν g ρσ + (2 δ ik δ jl − δ il δ jk − δ ij δ kl ) g µρ g νσ + (2 δ il δ jk − δ ik δ jl − δ ij δ kl ) g µσ g νρ ] . (37b)Substituting Eq. (36) and Eqs. (37a) and (37b) in Eqs. (30a) and (30b), we obtain thefollowing two expressions for δg λ , δg λ = λ π gg V NN − λ π g V NN , (38a) δg λ = λ π gg V NN − λ π g V NN g . (38b)In a self-consistent theory the two expressions for δg λ must coincide. This is true for thetrivial case g V NN = 0, i.e., for a theory without lowest-order interaction between the vectormesons and the nucleon. The non-trivial solution is given by g V NN = g, (39)which corresponds to the universality principle in SU(2). Consequently, from the EFT per-spective, the universal coupling g V NN = g is a result of the consistency conditions imposedby the requirement of perturbative renormalizability (see also Ref. [22]). D. SU(3)
In this section, we look for relations among the renormalized coupling constants G F , G D ,and g of the SU(3) Lagrangian of Eq. (14). The method is similar to the case of SU(2), but11his time we need to consider different SU(3) flavor combinations in order to disentangle theconditions for G F and G D .The divergent part of the self-energy diagram in Fig. 1 is given byΠ µν div ab ( p ) = − λ π (5G + 9G ) δ ab ( g µν p − p µ p ν ) , (40)from which we obtain by using Eq. (35) δZ λV = − λ π (5G + 9G ) . (41)In contrast to the SU(2) case, we will calculate the four-vector vertex function for twodifferent combinations of flavor indices to obtain two expressions for δg λ . To be specific, weconsider the combinations ( a, b, c, d ) = (1 , , ,
3) and ( a, b, c, d ) = (1 , , , µνρσ div1313 = iλ π (11G + 90G G + 27G )( g µρ g νσ + g ρσ g µν − g νρ g µσ ) , (42a)Γ µνρσ div1616 = iλ π (35G + 90G G + 27G )( g µρ g νσ + g ρσ g µν − g νρ g µσ ) . (42b)We now consider Eq. (30b) for the combinations ( a, b, c, d ) = (1 , , ,
3) and ( a, b, c, d ) =(1 , , , δg λ , namely, δg λ = − λ π g (11G + 90G G + 27G ) + λ π (5G + 9G ) g, (43a) δg λ = − λ π g (35G + 90G G + 27G ) + λ π (5G + 9G ) g. (43b)In a self-consistent theory, the two expressions for δg λ must be equal. This impliesG D = 0 , (44)because otherwise the interacting theory would not be renormalizable in the perturbativesense of effective field theory. Using G D = 0, the universality G F = g is obtained in analogyto the SU(2) case by comparing the expressions for δg λ obtained from the three-vector andfour-vector vertex functions, respectively. We obtain for the three-vector vertex function δg λ = 3 λ π g G + λ π G , (45)which needs to be compared with δg λ = − λ π g G + 3 λ π G g (46)from the four-vector vertex function. As solutions we either obtain G F = 0 or G F = g .Consequently, our final result is G D = 0 and G F = g . (47)12he renormalizability analysis thus generates relations among the dimensionless couplingconstants of the most general Lagrangian with a global SU(3) symmetry. We end up with auniversality principle in SU(3), in which the leading-order Lagrangian is that of a massive Yang-Mills theory with a universal coupling g . Of course, Lagrangians of such type wereoften used in phenomenological applications (see, e.g., Refs. [24–27]). Indeed, by using therequirement of renormalizability in the sense of EFT, the present analysis provides a furthermotivation for the universal coupling of vector mesons as originally discussed in Ref. [28] forisospin, baryon number, and hypercharge (see also Ref. [29]). VI. CONCLUSIONS
At the classical level, the lowest-order SU(3)-invariant Lagrangians of Eqs. (15a)–(15c),involving three independent coupling constants g , G F , and G D , define a self-consistent start-ing point for the self interaction of the vector-meson octet as well as the interaction ofthe vector-meson octet with the baryon octet. This was explicitly shown by using Dirac’smethod.However, the requirement of renormalizability in the sense of effective field theory impliesadditional constraints among the renormalized couplings. By comparing the expression for δg obtained from the V V V vertex function on the one hand and the
V V V V -vertex functionon the other hand, we were able to show that there are relations among the renormalizedcoupling constants g , G F , and G D . We found a universal interaction with g = G F andG D = 0. In other words, starting from the most general leading-order Lagrangian invariantunder a global SU(3) transformation, we have seen that, after obtaining a universal coupling,the interaction Lagrangian is that of a (massive) SU(3) Yang-Mills theory. VII. ACKNOWLEDGMENTS
When calculating the loop diagrams, we made use of the packages FeynCalc [30] andLoopTools [31]. This work was supported by the Deutsche Forschungsgemeinschaft (SFB443). The work of Y. ¨U was supported by the Scientific and Technological Research Councilof Turkey (T ¨UB˙ITAK). The authors would like to thank D. Djukanovic and A. Neiser foruseful discussions and comments during the work. The authors are also greatly indebted toJ. Gegelia for his very valuable contributions to the work.13 ppendix A: Generalized Poisson brackets
With the inclusion of Grassmann fields, we need a generalization of the standard Poissonbrackets. Here, we only collect the results needed for the present purposes and refer thereader to chapter 6 of Ref. [17] for more details. Let F denote a function of the dynamicalvariables Ψ αa , Π Ψ αa , Ψ ∗ αa , Π Ψ ∗ αa , V aµ , and π aµ . The Grassmann parity (cid:15) F is defined tobe equal to 0 if the function consists of monomials of Grassmann variables of even degree,and the function is then said to be even. An odd function has Grassmann parity (cid:15) F = 1and consists of monomials of Grassmann variables of odd degree. Any function F can bedecomposed into its even and odd components, respectively, F = F E + F O . The Poissonbracket of two functionals (or functions) is defined as { F, G } = (cid:90) d x (cid:18) δFδV aµ ( (cid:126)x ) δGδπ µa ( (cid:126)x ) − δFδπ µa ( (cid:126)x ) δGδV aµ ( (cid:126)x ) (cid:19) + ( − ) (cid:15) F (cid:90) d x (cid:18) δ L Fδ Ψ αa ( (cid:126)x ) δ L Gδ Π Ψ αa ( (cid:126)x ) + δ L Fδ Π Ψ αa ( (cid:126)x ) δ L Gδ Ψ αa ( (cid:126)x )+ δ L Fδ Ψ ∗ αa ( (cid:126)x ) δ L Gδ Π Ψ ∗ αa ( (cid:126)x ) + δ L Fδ Π Ψ ∗ αa ( (cid:126)x ) δ L Gδ Ψ ∗ αa ( (cid:126)x ) (cid:19) , (A1)where a summation over repeated indices is implied, and F is assumed to have a definiteGrassmann parity (cid:15) F . We suppress time as an argument of the fields, as they are to beevaluated at the same time. The symbol L in the functional derivative indicates that therelevant function entering the functional has to be moved to the left with an appropriatesign factor resulting from the necessary permutations. The fundamental Poisson bracketsare given by { V aµ ( (cid:126)x ) , π bν ( (cid:126)y ) } = δ ab δ µν δ ( (cid:126)x − (cid:126)y ) , (A2a) { Ψ αa ( (cid:126)x ) , Π Ψ βb ( (cid:126)y ) } = − δ αβ δ ab δ ( (cid:126)x − (cid:126)y ) , (A2b) { Ψ ∗ αa ( (cid:126)x ) , Π Ψ ∗ βb ( (cid:126)y ) } = − δ αβ δ ab δ ( (cid:126)x − (cid:126)y ) . (A2c)In addition, the following properties are useful: { F, G } = − ( − ) (cid:15) F (cid:15) G { G, F } , (A3) { F, GH } = { F, G } H + ( − ) (cid:15) F (cid:15) G G { F, H } , (A4) { F, G } ∗ = −{ G ∗ , F ∗ } . (A5) Appendix B: Hamiltonian densities
The Hamiltonian densities relevant for the evaluation of the Poisson brackets read H = − i (cid:2) ¯Ψ a γ i ∂ i Ψ a − ( ∂ i ¯Ψ a ) γ i Ψ a (cid:3) + M B ¯Ψ a Ψ a , H = − π ai π ia + ( ∂ i V a ) π ia + 14 V aij V ija − M V V aµ V µa − gf abc π ia V b V ci + gf abc ( ∂ i V ja ) V bi V cj + g f abc f ade V bi V cj V id V je , H int = i G F f abc ¯Ψ a γ µ Ψ b V cµ − G D d abc ¯Ψ a γ µ Ψ b V cµ . ppendix C: Results of the constraint analysis { θ V a , H T } = ∂ i π ia + M V V a − gf abc π ib V ci + ( − i G F f abc + G D d abc )Ψ † b Ψ c ≡ ϑ V a ≈ , (C1) { ϑ V a , H T } = − M V ∂ i V ia + gf abc V b ∂ i π ic + g f abe f ecd V c ( V bi π id − π ib V di ) − g f abe f ecd V bi [( ∂ j V cj ) V id + ( ∂ i V jc ) V dj ] + g f abe f ecd ∂ i ( V ic V dj V jb ) − g f abc f dbe f dfg V ci V ej V if V jg + gf abc ( i G F f cde − G D d cde ) V bi ¯Ψ d γ i Ψ e + ( i G F f abc − G D d abc ) ∂ i ( ¯Ψ b γ i Ψ c )+ ( − i G F f abc − G D d abc ) λ αb Ψ ∗ αc + ( i G F f abc − G D d abc ) λ αb Ψ αc + M V λ V a , (C2) { χ αa , H T } = i ( ∂ i Ψ ∗ βa )( γ γ i ) βα + M B Ψ ∗ βa γ βα − ( i G F f abc + G D d abc )Ψ ∗ βb ( γ γ µ ) βα V cµ + iλ αa ≈ , (C3) { χ αa , H T } = i ( γ γ i ) αβ ∂ i Ψ βa − M B γ αβ Ψ βa + ( − i G F f abc + G D d abc )( γ γ µ ) αβ Ψ βb V cu + iλ αa ≈ . (C4) Appendix D: Counter-term functions
The expressions for the counter-term functions in Eq. (23) are given by δ { M B } = δM B + δZ Ψ M B ,δ { M V } = δM V + δZ V M V ,δ { g } = δg + 32 δZ V g,δ { g } = 2 δgg + 2 δZ V g ,δ { G F } = δ G F + (cid:18) δZ Ψ + 12 δZ V (cid:19) G F ,δ { G D } = δ G D + (cid:18) δZ Ψ + 12 δZ V (cid:19) G D . We only displayed the terms relevant at leading order in an expansion in ¯ h , i.e., we omittedterms of the type δM V δZ V etc. 15 ppendix E: Feynman rules TABLE II: Propagators and vertices of Feynman diagrams in SU(2) and SU(3). Single and doublelines correspond to fermions and bosons, respectively. a, b, c, d correspond to SU(3) octet indices, i, j, k, l and r, s correspond isospin triplet and doublet indices, respectively.SU(2) SU(3) kµ, i ν, j kµ, a ν, b
Propagators − i g µν − kµkνM V k − M V + i(cid:15) δ ij − i g µν − kµkνM V k − M V + i(cid:15) δ ab pr s pa b i/p − m + i(cid:15) δ rs i/p − M B + i(cid:15) δ ab a bµ, i a bµ, cigγ µ ( τ i ) ba iγ µ [ G D d abc + iG F f abc ] ν, j, p ρ, k, p µ, i, p ν, b, p ρ, c, p µ, a, p Vertices g(cid:15) ijk [ g µν ( p − p ) ρ + g µρ ( p − p ) ν + g νρ ( p − p ) µ ] gf abc [ g µν ( p − p ) ρ + g µρ ( p − p ) ν + g νρ ( p − p ) µ ] µ, i ν, jρ, k σ, l µ, a ν, bρ, c σ, d − ig [ g µν g ρσ (2 δ ij δ kl − δ ik δ jl − δ jk δ il )+ g µρ g νσ (2 δ ik δ jl − δ il δ jk − δ ij δ kl )+ g µσ g νρ (2 δ il δ jk − δ ik δ jl − δ ij δ kl )] − ig [ f abe f cde ( g µρ g νσ − g µσ g νρ )+ f ace f bde ( g µν g ρσ − g µσ g νρ )+ f ade f bce ( g µν g ρσ − g µρ g νσ )] ppendix F: Loop integrals The scalar loop integrals of the two-, three-, and four-point functions which are used forthe calculation of the self energy and the vertex diagrams are given by A ( m ) = (2 πµ ) − D iπ (cid:90) d D k k − m ,B ( p , m , m ) = (2 πµ ) − D iπ (cid:90) d D k k − m ][( k + p ) − m ] ,C ( p , p , p , m , m , m ) =(2 πµ ) − D iπ (cid:90) d D k k − m ][( k + p ) − m ][( k + p + p ) − m ,D ( p , p , p , p , p , p , m , m , m , m ) =(2 πµ ) − D iπ (cid:90) d D k k − m ][( k + p ) − m ] , [( k + p + p ) − m ][( k + p ) − m ]with the abbreviation p ij = ( p i + p j ) for the momenta. For the sake of brevity, we havesuppressed the boundary conditions i(cid:15) in the individual factors of the denominators. [1] M. Gell-Mann and Y. Ne’emam, The Eightfold Way (Benjamin, New York, 1964).[2] S. Coleman, J. Math. Phys. , 787 (1966).[3] Y. Nambu and G. Jona-Lasinio, Phys. Rev. , 345 (1961); , 246 (1961).[4] J. Goldstone, Nuovo Cim. , 154 (1961).[5] J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. , 965 (1962).[6] S. Weinberg, Physica A , 327 (1979).[7] S. Scherer and M. R. Schindler, Lect. Notes Phys. , 1 (2012).[8] G. Ecker, J. Gasser, H. Leutwyler, A. Pich, and E. de Rafael, Phys. Lett. B , 425 (1989).[9] M. C. Birse, Z. Phys. A , 231 (1996).[10] P. A. M. Dirac, Lectures on Quantum Mechanics (Dover, Mineola, N.Y., 2001).[11] N. Wies, J. Gegelia, and S. Scherer, Phys. Rev. D , 094012 (2006).[12] S. Weinberg, The Quantum Theory Of Fields. Vol. 1: Foundations (Cambridge UniversityPress, Cambridge, 1995), Chap. 12.[13] D. Djukanovic, J. Gegelia, and S. Scherer, Int. J. Mod. Phys. A , 3603 (2010).[14] A. Neiser, Effective Field Theories for Vector Particles and Constraint Analysis , DiplomaThesis, Johannes Gutenberg University Mainz, 2011.[15] A. Neiser and S. Scherer,
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