Phenomenology of $J^{PC} = 3^{--}$ tensor mesons

We study the strong and radiative decays of the anti-quark-quark ground state $J^{PC} = 3^{--}$ ($n^{2 S + 1} L_J = 1^3 D_3$) nonet {$\rho_{3} (1690)$, $K_{3}^{\ast} (1780)$, $\phi_{3} (1850)$, $\omega_{3} (1670)$} in the framework of an effective quantum field theory approach, based on the $SU_\mathrm{V}(3)$-flavor-symmetry. The effective model is fitted to experimental data listed by the Particle Data Group. We predict numerous experimentally unknown decay widths and branching ratios. An overall agreement of theory (fit and predictions) with experimental data confirms the $\bar{q} q$ nature of the states and qualitatively validates the effective approach. Naturally, experimental clarification as well as advanced theoretical description is needed for trustworthy quantitative predictions, which is observed from some of the decay channels. Besides conventional spin-$3$ mesons, theoretical predictions for ratios of strong and radiative decays of a hypothetical glueball state $G_3 (4200)$ with $J^{PC} = 3^{--}$ are also presented.


I. INTRODUCTION
The spectroscopy and the phenomenological description of conventional mesons is important for many reasons. It allows to test quantum chromodynamics (QCD) in the nonperturbative regime. Furthermore, it aims to the correct and systematic understanding and assignment of experimentally measured states, resonances and mesons [1]. It is also fundamental for the search of mesons that go beyond the antiquark-quark (qq) picture, such as glueballs, hybrids, and multiquark states. Namely, both in the light-quark and heavy-quark sectors, it is possible to search for "exotics" only, if the conventionalqq mesons are fully under control [2].
Consequently, our work is just an incremental contribution to this extensive project in high-energy physics. In our work, we provide a (qualitative) analysis of the phenomenology of the spin-3 mesons ρ 3 (1690), K * 3 (1780), φ 3 (1850), ω 3 (1670) with J P C = 3 −− (n 2S+1 L J = 1 3 D 3 ) as well as the potential spin-3 tensor glueball G 3 (4200) [3][4][5]. This involves a fit of our effective model to experimental data as well as predictions for decay widths and ratios of the decays of the spin-3 mesons. Although we provide explicit values for the decay widths and branching ratios, these should be considered only as first estimates and mainly interpreted as qualitative results. Hence, this study may for example help in identifying dominant decay channels etc..

A. Contextualization
In the low-energy sector of QCD, conventional mesons contain the light up (u), down (d), and strange (s) quarks, hence they can be grouped into nonets:ūd,ūs, etc.. Nonets are classified by the quantum numbers J P C , where J is the absolute value of the total spin of the meson (|L − S| ≤ J ≤ |L + S|, where L is the angular momentum and S the spin of the system), P is the signchange under parity transformations, and C the signchange under charge conjugation. Forqq states, besides J P C , also the older spectroscopic notation n 2S+1 L J , with L = S, P, D, . . . and with n = 1, 2, . . ., which is the radial quantum number, is used. In the following, we shall restrict to the radial ground state nonets with n = 1. For aqq system, where P = (−1) L+1 and C = (−1) L+S , notice that L and S are, strictly speaking, not well defined quantum numbers in a relativistic setup, hence mixing of states with the same J P C but with different L and S is possible -even if it is oftentimes a subordinate effect. However, this nomenclature for the states is still well upheld also in presence of mixing for practical purposes.
In the low-energy regime of QCD the "classical" symmetries of QCD at high-energy scales still play a crucial role -more precisely the spontaneous, anomalous or explicit breaking of these symmetries. Quickly recapitulated, the situation is as follows: gluons are "democratic": they interact equally strongly with right-and left-handed quarks implying that in the so-called chiral limit (i.e. assuming negligibly small bare quark masses m u , m d , and m s ) the QCD Lagrangian is invariant under U L (3) × U R (3) transformations. However, this symmetry of the classical action is broken due to quantum effects, which is called the U A (1)-anomaly [6][7][8][9]. Still, the baryon number is conserved, which corresponds to the unaffected U V (1) symmetry. The remaining SU L (3)×SU R (3) symmetry is spontaneously broken at low energies into SU V (3) [10,11]. The SU V (3) amounts to a rotation in the flavor space spanned by u, d and s and is consequently called flavor symmetry or vector symmetry. Hence, at the composite mesonic level, chiral partner-mesons, which are linked through a chiral transformation, are not degenerate anymore. After all, the residual realization of an approximate SU V (3), which is only explicitly broken by the bare quark masses, is the reason, why nonets of qq states can be used to classify the low-energy spectrum of QCD [12][13][14].
2. Many decay channels are known [1], which allows for further theoretical and experimental tests of the assignment.
This nonet is therefore tailor made for an effective QFT study of decays. In particular, we shall answer whether and to what extend theqq assignment works, we can test validity of flavor symmetry for various decay channels, make various "postdictions" and -most interestinglypredictions for many decay channels. In view of the ongoing experimental efforts in hadron physics at GlueX [24][25][26] and CLAS12 [27] at Jefferson Lab, at COMPASS and LHCb at CERN [28,29], at BESIII in Beijing [30,31], and at the future PANDA experiment [32] at the GSI facility, we consider a revival of interest on such resonances valuable.
As an additional application of our approach, we also study some decays of the J P C = 3 −− glueball state. This can be done by a simple modification of the action of spin-3 mesons and by using the mass of approximately 4.2 GeV found in lattice simulations [3][4][5] (in the quenched approximation) as an input. Due to the flavorsinglet nature of the glueball, only few decay channels are possible.

B. Method
From a technical point of view, we construct SU V (3)invariant effective actions/Lagrangians that involve the mesonic nonet (6) as well as its various decay products consisting in the well-establishedqq nonets that were previously introduced. From a (Functional) Renormalization Group (FRG) perspective, the effective actions/Lagrangians can be interpreted as residual infrared (IR) effective actions with coupling constants that already involve all quantum effects from higher energy scales, the ultraviolet (UV). These coupling constants are determined via fits to experimental data, rather than by ab initio QCD-calculations. All calculations for the decays are therefore performed at tree-level. This approach was already implemented in earlier works for studies of various light mesonic nonets, such as tensor mesons [33], axial-vector, and pseudovector mesons [34], the scalar mesons [35,36], the pseudotensor mesons [37], and the orbitally and radially excited vector mesons [38].

C. Organization and structure
We organize the paper as follows: in Chap. II, after a discussion of our low-energy effective model, we present antiquark-quark nonets, their transformation rules, and the corresponding Lagrangian of the model. It consists of seven interaction terms. In Chap. III we show the results for the decay widths and branching ratios of spin-3 mesons, while in Chap. IV we list some branching ratios for an hypothetical heavy 3 −− glueball. A discussion of the validity of the employed effective interaction terms is reported in Chap. V and conclusions can be found in Chap. VI. Many technical aspects concerning the QFT treatment of J = 3 fields can be found in the numerous and detailed appendices.
As mentioned in the introduction, in the chiral limit (m u,d,s = 0) the SU L (3) × SU R (3) × U V (1) symmetry of QCD is spontaneously broken into flavor symmetry SU V (3) and baryon-number conservation -the U V (1) symmetry. One of the first successful models that describes this process using four-Fermi interactions is given by Refs. [39,40]. At the level of confined light hadrons, this symmetry is evident: quark-antiquark mesons are clearly grouped into nonets, some of which were already listed in the introduction [12][13][14]. One possibility to describe low-energy QCD makes use of effective low-energy hadronic models. If the degrees of freedoms are only hadrons, then confinement and color-neutrality is automatically built in. Typically, these models are defined by a proper action that mimics the chiral symmetry of QCD and its spontaneous and explicit breaking.
Hadronic models in which only the residual flavor symmetry SU V (3) is explicitly conserved have also been constructed for a variety of nonets in a series of publications by one of the authors and collaborators [33][34][35][36][37][38]. These models can be interpreted as the effective emerging terms of chiral models after spontaneous symmetry breaking is worked out. When the effective action is written down, only flavor symmetry is retained and an expansion in TABLE I. Transformation properties of the pseudoscalar P (7), the vector V1 (9), the pseudovector B1 (11), the axialvector A1 (15), the rank-2 tensor A2 (13), and rank-3 tensor W3 (20) nonets under parity transformations P , charge conjugation C, and SUV(3) flavor transformations U . Notice the position of the Lorentz indices for parity transformations, since spatial and time-like indices do not transform identically. We use the Minkowski metric in the (ηµν ) = diag(+1, −1, −1, −1) convention. nonet parity charge conjugation flavor dominant and subdominant terms in 1/N c is carried out. Moreover, terms that break explicitly flavor symmetry, either because of the underlying breaking due to nonzero and unequal quark masses (m u ≈ 2 MeV, m d ≈ 5 MeV and m s ≈ 93 MeV) or due to the chiral anomaly can be included.
In this work, we study the decays of theqq groundstate J P C = 3 −− nonet by constructing such a model. We shall consider only the dominant terms in the large-N c expansion [76][77][78] and neglect flavor symmetry breaking corrections, since the present level of data accuracy does not allow for their investigations.
A. Particle content

Symmetries
Mesonic nonets that transform under the adjoint representation of the approximate flavor symmetry SU V (3) are the main ingredients of the current work. Transformation rules under parity, charge conjugation and flavor transformations of the mesonic nonets of our model are summarized 1 in the Table I. On the other hand, the mesonic nonets are in direct correspondence to the physical states, which is listed in Table II. Eachqq nonet can be assigned to a certain microscopic antiquark-quark current N ij ≡ (q j Γ q i )/ √ 2, where Γ is a combination of Dirac matrices and derivatives, whichin the nonrelativistic limit -reduces to the corresponding nonrelativistic configuration with the desired L and S. 1 Other mesonic nonets constructed in the same way can be found for instance in Refs. [34,37,53,79].

Pseudoscalar mesons
The first nonet of Table I describes the matrix of pseudoscalar mesons P with quantum numbers L = S = 0, leading to J P C = 0 −+ (n 2S+1 L J = 1 1 S 0 ). The elements are P ij ≡ (q j iγ 5 q i )/ √ 2 and the mesons are {π, K, η (958), η}, where the first entry represents the isospin I = 1 triplet (the three pions), the second entry the two I = 1/2 isodoublets (the four kaons), and the last two entries the two isoscalar states (the etas). Pseudoscalars form the basis of almost all low-energy effective hadronic models/limits or theories of QCD, e.g. ChPT [41][42][43][44][45][46][47][48][49][50] and σ-models [52][53][54][55]57] (for an link between ChPT and σmodels, see Ref. [68,69,71,80]). Explicitly, the matrix P reads: where η N ≡ (ū iγ 5 u +d iγ 5 d)/ √ 2 stands for the the purely nonstrange state and η S ≡s iγ 5 s stands for the pure strange state. For the pions and kaons, the physical states are directly assigned to the fields in the model. In the isoscalar sector, physical and model fields are related by mixing 2 , Here we shall use β p = −43.4 • obtained in Ref. [81]. Note, the rather large mixing angle and the unexpectedly high mass of the η (958) result from the chiral (or axial) anomaly U A (1) [82,83]. According to a classifications introduced by two of the authors of the present work in Ref. [84], pseudoscalar mesons together with their chiral partners (the scalars), belong to what we call a "heterochiral multiplet", which allows for the constructions of chirally anomalous mass and interaction terms. Vector mesons Next, the second entry in Table I refers to the J P C = 1 −− (n 2S+1 L J = 1 3 S 1 ) nonet with L = 0 and S = 1. These are the very well-known vector states {ρ(770), K * (892), ω(782), φ(1020)}, which are for example also included in certain extensions of ChPT [47][48][49][50] as well as in enlarged (and realistic) versions of hadronic models [52][53][54][55][56]. The matrix V µ 1 with elements 2 We refer to App. A for the description of the mass terms of the corresponding Lagrangian(s) and for the derivation of the PDGmixing formula, which was applied in the PDG [1] in the case of ground-state vector and pseudoscalar mesons as well as for the ground-state tensor mesons with J P C = 2 ++ and J P C = 3 −− . We shall also present the link between the singlet-octet basis used in the PDG and the strange-nonstrange one employed in this work.
where ω N and ω S are purely nonstrange and strange states, respectively. Similarly as before, the physical fields arise upon mixing where the very small isoscalar-vector mixing angle β v1 = −3.9 • is taken from the PDG [1]. Hence, the physical states ω(782) and φ(1020) are dominated by nonstrange and strange components, respectively. This is in agreement with the "homochiral" nature of these states [84].
In fact vector and axial-vector mesons form a "homochiral multiplet", for which the effect of the chiral anomaly is of subleading order and can safely be ignored.
Pseudovector mesons Next, we consider L = 1. The choice L = 1 and S = 0, leading to J P C = 1 +− (n 2S+1 L J = 1 1 P 1 ), contains the established pseudovector mesons {b 1 (1235), The mixing angle is not known. Quite interestingly, since this nonet belongs to a heterochiral multiplet (the chiral partners are the orbitally excited vector mesons) the mixing angle could be nonnegligible, just as for pseudoscalar mesons.
Here, we shall consider two scenarios for our model calculations: in the first, the mixing angle β b1 is set to zero, in the second we consider a large mixing similar to the pseudoscalar sector, β b1 ≈ −40 • and similar to what two of the authors found for the pseudotensor meson nonet in Ref. [37] and elaborated on in Ref. [84].
Tensor mesons For L = S = 1 three nonets are possible. The well-known J P C = 2 ++ (n 2S+1 L J = 1 3 P 2 ) tensor states {a 2 (1320), 2, represent an almost ideal nonet of quark-antiquark states The physical isoscalar-tensor states are where β a2 = 5.7 • is the small mixing angle reported in the PDG [1], in agreement with the fact that tensor mesons belong to a homochiral multiplet. The decays of tensor mesons were studied in great detail in Refs. [33,85] and fit very well into theqq picture [15].
Axial-vector mesons The choice L = S = 1 allows also for the J P C = 1 ++ (n 2S+1 L J = 1 3 P 1 ) axial-vector nonet A 1 , which contains the resonances {a 1 (1260), K 1,A , f 1 (1285), f 1 (1420)} which are linked to the vector mesons mentioned above by chiral transformations, see e.g. Refs. [86,87] (building a homochiral multiplet). The nonet matrix, whose microscopic currents are A µ 1,ij ≡q j γ 5 γ µ q i / √ 2, reads: For the isoscalar sector we find, The mixing angle β a1 is expected to be small, as the homochiral nature of the multiplet [84] and the decay properties [34] suggest. We set β a1 = 0 for simplicity (an anyhow small mixing angle would not affect our results very much). It is important to note at this point, that the kaonic states K 1,A and K 1,B mix. For a study of the axialvector and pseudovector nonets, with focus on the K 1 (1270)/K 1 (1400) system, by using an approach similar to the one used in this work, see Ref. [34]. The main result concerning the kaonic mixing reads: Equivalently, in terms of kets and the angle θ K = ϕ K + 90 • typically used in the literature [86][87][88]: The final result reads ϕ K = (56.4 ± 4.3) • , or equivalently θ K = (−33.6 ± 4.3) • , hence K 1 (1270) is predominantly K 1,B and K 1 (1400) predominantly K 1,A , yet the mixing is still large. Scalar mesons For completeness, we recall that L = S = 1 yields also the scalar states J P C = 0 ++ (n 2S+1 L J = 1 3 P 0 ), corresponding to the chiral partners of the pseudoscalar mesons. The scalars are subject to an ongoing controversial debate on the correct assignment of measured states. The identification is still uncertain, but the set seems to be favored (for the description of mixing with e.g. the long-searched scalar glueball, see e.g. Refs. [35,36,[89][90][91][92][93][94] and Refs. therein). The reason why this nonet is so peculiar is due to the fact that it has the quantum numbers of the QCD vacuum, hence condensation of the scalar-isoscalar fields should take place. The corresponding nonet S, whose microscopic currents are S ij ≡ (q j q i )/ √ 2, is not included in this work, since no decays of spin-3 states into scalars can be realized in our model approach, see Sub.Sec. II B 3.
Moreover, light scalar four-quark states (in the form and combination of tetraquark/molecular/companion poles) below 1 GeV also exist [1]. Nevertheless, they are also not included our work.
Higher-spin mesons Besides the peculiar case of the scalar nonet, all other nonets mentioned above are wellestablished. Yet, what about nonets with L = 2? For L = 2 four nonets can be constructed, one of which is the main subject of this work. Thus, even if, besides spin-3 fields, they do not enter into the Lagrangian of this work, since they are too heavy, they allow us to put the nonet of mesons with J = 3 into the correct physical framework. Hence, we briefly discuss them.
The latter two nonets are omitted in this work, because their members are too heavy.
Spin-3 mesons Finally and most importantly, the choice L = 2 and S = 1 can also lead the nonet of In this case the mixing angle is β w3 = 3.5 • [1]. The small mixing is also in agreement with the homochiral nature of the corresponding chiral multiplet in which this nonet is embedded [84]. In the next section we couple this nonet to the other nonets listed in Table I.

B. Effective mesonic interactions
In this section, we present the effective mesonic interactions and discuss the derivation of the tree level decay widths from our model.

The effective action
Using the nonets introduced in the previous subsection, we construct the effective Lagrangian describing the strong decays of spin-3 tensor mesons as follows, where L kin = 1 2 tr (∂ µ P ) 2 + . . . contains the usual kinetic terms and L mass contains all the quadratic mass terms describing the masses of all relevant nonets. In this work, all masses are taken from the PDG [1] and assumed to be exact, while some mixing angles are derived from the masses following the quark model review of the PDG [1], see also App. A. The other terms describing the decays are listed explicitly in Table III. The quantities g w3•• are the coupling constants, which are fitted to experimental results, ε µνρσ is the antisymmetric Levi-Civita pseudotensor, [•, •] − stands for the commutator, and {•, •} + for the anticommutator.
All the interaction Lagrangians in Table III are invariant under CP T -, Poincaré-and flavor transformations, listed in Table I. We considered only couplings involving a minimal number of derivatives within each single interaction term. This strategy turned to be successful in φ(1020) −ω1,N sin βv 1 + ω1,S cos βv 1

Decay width
The tree level decay widths have the following general form [1,10], where m w3 is the mass of a (decaying) spin-3 particle, while m a and m b are the masses of the decay products "A" and "B", Θ(x) denotes the Heaviside step function, and the modulus of the outgoing particles momentum has the following analytic expression, We obtain the factors κ i in Eq. (23) for the i-th decay channel from the explicit forms of the Lagrangians in the tables in App. B by considering the square of the coefficients for a given decay channel as well as eventual sum over members of the same isospin multiplet. The κ i for all interaction Lagrangians and all relevant channels are presented in App. C. The decay amplitudes |M| 2 are derived via Feynman rules under the use of the polarization vectors and tensors as well as their corresponding completeness relations in App. D. The results for 1 7 |M| 2 are listed in 4 Table IV. 3. Remarks on the exclusion of scalar mesons As a final side remark, we note that we cannot couple the spin-3 nonet (20) to scalar mesons (19). In fact, if we try to couple them to scalars and pseudoscalars, the only possible CPT-and flavor-invariant interaction term, vanishes identically since it involves the contraction of antisymmetric and symmetric tensors. Correspondingly, scalar mesons do not enter our study, which agrees with the experimental results reported in the PDG [1]. This is also of direct advantage for our work, since we do not have to deal with the identification of the scalar states, which, as already mentioned, is a long-standing and yet unsolved issue of low-energy QCD [89][90][91][92][93][94].
The Lagrangian terms considered in Table III are large-N c (and thus Okubo-Zweig-Iizuka) dominant and symmetric under U V (3). As we shall discuss later on, this approximation is expected to be acceptable in view of the precision of the experimental data presently available. Yet, by considering the W 3 B 1 P interaction as an example, the Lagrangian L w3b1p can be regarded as the first term of an expansion in large-N c and/or symmetry breaking terms that takes the form: w3b1p tr W µνρ tr B 1µ tr ∂ µ ∂ ρ P + + g (4) w3b1p tr δ W µνρ B 1µ , (∂ µ ∂ ρ P ) + + + g (5) w3b1p tr δ W µνρ tr B 1µ (∂ µ ∂ ρ P ) + + g (6) w3b1p tr W µνρ tr δ B 1µ (∂ µ ∂ ρ P ) + + . . . Table III, is a flavor symmetric term and scales as g w3b1p ∝ N −1/2 c , hence it is the dominant term in a large-N c expansion. The second and the third term are also flavor symmetric, but scale as g (2) w3b1p ∝ N

The first term, reported in
Isospin violation is proportional to δ d ∝ m d − m u and is expected to be very small, yet the breaking due to the s quark δ s ∝ m s − m u can be nonnegligible following Ref. [89,90], it could be along the order of g (4) w3b1p g w3b1p diag{0, 0, 0.1 − 0.2}, but the actual value for J = 3 mesons should be determined by an independent fit to data. It is however expected to be sufficiently small to be neglected in this work. At this stage, the first large-N c correction g (2) w3b1p and the first flavor symmetric correction g (4) w3b1p are expected to be of the same intensity and should be the first to be included. Further terms g (n≥5) w3b1p are both large-N c subdominant and flavor suppressed, thus are regarded to be very small.
The same analysis can be carried out for all the interaction terms, yet it is interesting to observe that the second and the third term would vanish whenever the commutator is present. (Flavor-symmetry violation is expected to be the main next-to-leading-order contribution for those interaction terms). In this section we present our results for decay rates and branching ratios of the J P C = 3 −− mesons. In each subsection and for each interaction Lagrangian term, we compare the experimental data to our theoretical results.
We recall that the total decay widths of the J P C = 3 −− mesons under consideration are [1], We shall verify that the sum of all the single decay channels from our theoretical calculations never overshoots these values.
For what concerns the accuracy of our results, whenever possible we determine the coupling and its error via a simple fit of experimental decay width and ratios. We ignore the experimental uncertainties for all masses of the particles of the model and assume the masses to be exact, because their errors are small and they are of minor importance for the overall errors. The experimental errors for the decay widths and branching ratios as well as the systematic errors in the model are much larger, which justifies this approximation. Yet, the quoted errors represents a lower(!) limit of the actual error of this work, since other indeterminacy features are inherent in the approximations, that lead us to the effective action, where only those terms are included, which are expected to be dominant in terms of large-N c , flavorand momentum-space expansions. Moreover, the decay widths are calculated at tree level. Since the width/mass ratio for the decaying resonances is rather small, contributions due to loops are expected to be negligible [98,99], if seen from a perturbative perspective. Otherwise, if the interaction terms are interpreted as effective couplings in a full quantum effective IR action, calculations have to be performed at tree level anyhow.
A. Decay process W3 → P + P The effective interaction term describing the decay of spin-3 tensor mesons into two pseudoscalar mesons has the following form Its extended version is listed in App. B in Eq. (B1). Correspondingly, the tree level decay rate is compare Eq. (D11). The factors κ i are reported in Table  XIX. In order to determine the coupling constant g w3pp , we use the above formula and the following experimental data: 1. For ρ 3 (1690) → π π one finds κ 1 = 1. Using the experimental result one obtains a first determination of the coupling constant squared and its error. We shall denote them asg 2 1 and ∆g 2 1 .

The experimental value
together with κ 2 = 2 1 TABLE V. Decays of J P C = 3 −− mesons into two pseudoscalars. Experimental data is taken from Ref. [1].
decay process theory experiment The comparison of theoretical and experimental results, which is obtained by using this value for the coupling constant, is reported in Table V. A good overall agreement is obtained, but there is also a sizable mismatch: the experimental value for K * 3 (1780) →K η is much larger than our theoretical prediction. Still, the experimental error is large and a better experimental determination would be interesting. Moreover, a noteworthy prediction concerning φ 3 (1850) →K K is obtained. From the theoretical large prediction, we conclude that an experimental determination should be feasible.
B. Decay process W3 → V1 + P The interaction Lagrangian for the vector and pseudoscalar decay mode reads An extended version is given by Eq. (B2) in App. B. In this case, the tree level decay rate formula has the form  Table XXX. In order to define the coupling constant we proceed as in the pseudoscalar-pseudoscalar case. We use Γ exp ρ3(1690)→ω(782) π = (25.8 ± 9.8) MeV , .0) MeV . The coupling constant and its error are hence This value leads to the results listed in Table VI. We observe that an acceptable agreement is reached, although the Γ K * 3 (1780)→ρ(770) K mode is theoretically underestimated (the experimental error is nevertheless large). Quite remarkably, the two theoretically sizable and dominant decays ω 3 (1670) → ρ(770) K and φ 3 (1850) → K * (892) K have been indeed seen in experiments, though they could not be quantified. Their future determination would represent a test of our approach -at least on a qualitative level.
In addition, we present the following ratio which does not totally contradict the PDG [1] average taken from Ref. [100], Interestingly Ref. [101] even reports a slightly larger ratio of Furthermore, we would like to mention a recent lattice QCD study [22], which also confirms our overall predictions on dominant and less dominant vector-pseudoscalar decay channels. Reference [22] predicts (without providing explicit errors due to large uncertainties), It is quite remarkable that our (quite simple) model is able to predict at least qualitatively rather similar results to such an advanced and comprehensive lattice QCD study. This leads us to the conclusion, that even for high-spin mesons with masses above 1 GeV like the conventional 3 −− -nonet chiral symmetry (breaking) is the decisive guiding principle for their phenomenology.
C. Decay process W3 → γ + P As a next step, we also present the results for the radioactive decays W 3 → γ P , where γ represents the photon. These can be obtained by using vector meson dominance [102][103][104], which takes into account of the photonvector-meson mixing through the shift where is the charge quark matrix, which includes the charges of the up, down, and strange quark. The electromagnetic field tensor is denoted as F µν = ∂ µ a ν − ∂ ν a µ , with a µ being the photon field, while e = √ 4π α is the electric coupling constant, and g ρ 5.5 parametrizes the photonvector-meson transition. By applying the shift of Eq. (54) into Eq. (40) we obtain the Lagrangian for radioactive decays The extended version of this Lagrangian is provided by Eq. (B3) in App. B. For the tree level decay rate formula, we obtain via Eq. (D17), where one of the masses m a and m b in Eq. (24) is set to zero, because the photon is massless. The factors κ γ i are listed in Table XX. (The Heaviside function is not needed, because all pseudoscalars are lighter than the conventional spin-3 mesons and the photons can be arbitrarily soft.) Various predictions for the radiative decays W 3 → γ P are calculated and presented in Table VII. Because of the rather large errors in the coupling constant (43), we round all results to integers in units of keV. Quite large radiative decay channels are ω 3 (1670) → γ π 0 as well as In general, these processes imply that a photoproduction of mesons with spin J P C = 3 −− can take place at the ongoing GlueX [24][25][26] and CLAS12 [27] experiments at Jefferson Lab. D. Decay process W3 → A2 + P The interaction Lagrangian describing the decay of the spin-3 tensor mesons into a spin-2 tensor and a pseudoscalar mesons has the following form The extended version is provided in App. B in Eq. (B4). Using Eq. (D20) the corresponding decay formula is where the κ i are listed in Table XXI. For this channel, there are no experimental reported branching ratios that allow for a direct determination of the coupling constant. Yet, we can use the experimental ratio listed by the PDG [1] and taken from Ref. [105], together with the previously determined theoretical value Γ ρ3(1690)→ρ(770) η = (3.8±0.8) MeV reported in Table VI. The corresponding value for the coupling constant is Once the coupling constant is fixed, we get the results for the decay rates reported in Table VIII. The decay width Γ K * 3 (1780)→K * 2 (1430) π is safely smaller the experimental upper limit [1,106]. Moreover, the quite large mode ρ 3 (1690) → a 2 (1320) π is seen in experiments, but also in this case no branching ratio is listed in the PDG [1].
The interaction Lagrangian describing the decay into a pseudovector and pseudoscalar meson is given by where the extended form is given in Eq. (B5) in App. B. The tree level decay rate takes the form, Here, we use Eq. (D25) and the κ i can be found in Table  XXII. Also in this case, a direct determination of the coupling constant g w3b1p is not possible due to the lack of experimental information. In order to estimate it, we proceed as it follows.
1. According to the PDG [1], there exists a lower limit for the ratio The PDG [1] extracts this lower bound for the ratio from Ref. [107] and does not list this bound as confirmed data. In Ref. [107] the authors in fact report They further argue that the decay mode ω 3 (1670) → b 1 (1235) π is the dominant contribution to the ω(782) π π final state via the subsequent decay b 1 (1235) → ω(782) π. They claim that one can assume that and deviations are expected to be smaller than 10%.
2. Based on this assumption we use the following approximation Here, one should mention that the PDG [1] cites Ref. [108] for the second ratio, but still excludes it from their confirmed data. Additionally, Ref. [108] even lists a different branching ratio, as well as Γ ω3 (1670) It is not clear to the authors of this work, how the PDG [1] extracted their ratio from Ref. [108]. Nevertheless, we will stick to the data reported in the PDG [1] and assume that there was some reasonable reanalysis of the data of Ref. [108]. In this context, it might be worth mentioning, that all experimental data concerning the ω 3 (1670) is rather old and its mass and total width could not be determined up to high accuracy at the time when Refs. [107,108] were published. Definitely, there is a need for future experimental investigations of this state.
For what concerns our work, we use the result Γ ω3(1670)→ρ(770) π = (97 ± 20) MeV presented in the previous section as well as the ratio (66). We obtain the following estimate for the coupling constant We are aware that this value and the corresponding results, which are listed in Table IX, are only first rough estimates. Still, they might again help identifying dominant decay channels. Additionally, we consider the following aspects for the determination and interpretation of our results.
1. We consider K 1,B ≈ K 1 (1270), since this is the dominant contribution; then, the decay rate Γ φ3(1850)→K 1,B K is rather small. A disclaimer is in order: due to the large mixing between K 1,A and K 1,B the results involving the identification K 1,B ≈ K 1 (1270) can be only considered as a first approximation. The more correct procedure should be to consider the full mixing in Eq. (17), hence K 1,B should be expressed as a superposition of K 1 (1270) and K 1 (1400). Yet, this is not an easy task, since the interaction in Eq. (61) alone is not enough. One should also take into account the interaction terms for the decays W 3 → A 1 P in Eq. (70), which includes K 1,A , see next section. Then, the decays into K 1 (1270) and K 1 (1400) should be calculated by the joint Lagrangians (61) and (70) together with the mixing (17). Furthermore, interference between the two Lagrangians is expected. This calculation is however not possible, since the coupling constant g w3a1p cannot be determined by the present experimental data (see next subsection). Hence, we must limit our study to the dominant assignment K 1,B ≈ K 1 (1270) (in this subsection) and K 1,A ≈ K 1 (1400) (in the next subsection). Decays of J P C = 3 −− mesons into a pseudoscalar-pseudovector pair obtained by using the mixing angle β b 1 = 0 and β b 1 = −40 • , respectively. No experimental value is listed in the PDG [1]. The decay channel ω3(1670) → b1(1235) π is possibly seen. decay process theory for theory for We present the results in Table IX for two values of the unknown mixing angle in the isoscalar sector. In one case, we assume that the mixing angle vanishes, β b1 ≈ 0, in the second we study β b1 ≈ −40 • , which is a quite large and negative value similar to the mixing angle for the pseudoscalar-isoscalars. Note, the mixing angle β b1 appears in the decay channels Table  XXII. The only experimentally possibly seen decay ω 3 (1670) → b 1 (1235) π, see Ref. [108], corresponds to a quite large theoretical partial decay widths and is independent of any assumption on β b1 .
3. The decay ρ 3 (1690) → h 1 (1170) π is also quite large for both choices of the mixing angle, hence it is a potentially interesting channel for future search. Moreover, it is also an interesting decay channel in order to determine the value of the mixing angle β b1 . The remaining decay channels are pretty small, which might be an explanation why they could not be observed in experiment.
F. Decay process W3 → A1 + P The interaction Lagrangian and the decay formula for the decay into an axial-vector and a pseudoscalar meson are given by branching ratio theory The extended version of the Lagrangian is given by Eq.
(B6) in App. B. The κ i -values for each channel are taken from Table XXIII and the decay formula was derived by using Eq. (D26). As explained previously, for a first rough estimate, we assume here that K 1,A ≈ K 1 (1400).
Since we do not have enough experimental information for obtaining the coupling constant g w3a1p , we can only get some theoretical predictions for ratios among different decay channels, that are reported in Table X. Note, the mixing angle β a1 does not appear in any of the nonzero reported decay channel, as it can be read from Table XXIII. Anyway, the mixing is expected to be small [84].
In the end, it is interesting to mention that there might be an option to size the coupling constant by linking the present model to an underlying chiral model, e.g. the eLSM. This approximately predicts 2 MeV for ρ 3 (1690) → a 1 (1260) π, see Chap. V for an estimate of g w3a1p . It seems therefore that the decays of the type The interaction Lagrangian describing the decay into two vector mesons is given by see Eq. (B7) in App. B for the extended version . The corresponding decay rate reads  branching ratio theory where the κ i 's are listed in Table XXIV. For the derivation of the decay formula, we use Eq. (D29). Also in this case, we cannot determine the coupling constant, but we can present the branching ratios reported in Table XI. Moreover, we can find an upper limit for the coupling constant by considering which follows from the fact that the ρ 3 (1690) → ρ(770) ρ(770) mode is part of the π ± π + π − π 0 decay mode (and eventually it is one of its dominating contributors). Since the experimental value for Γ ρ3(1690)→π π = (38.0 ± 3.2) MeV is known, we find Based on this information we can predict upper limits for the decay rates in Table XII. The theoretically largest decay is ρ 3 (1690) → ρ(770) ρ(770), which has been experimentally seen. Yet, the decay width of about 108 MeV is surely too large, since using a value close to this upper limit would imply that the the sum of all the decays of the state ρ 3 (1690) would overshoot the experimental total width of (161 ± 10) MeV. This is in agreement with the ratio which is smaller than 1 for all experimental measurements [109][110][111][112], which are listed by the PDG [1]. However, we can then obtain a second, more realistic estimate of these decay channels by summing up the largest decay channels: = Γ ρ3 (1690) from which follows, 148 .
The corresponding estimates for the other channels can be found in Table XII. Following this estimate, we perform the sum of the predicted and sizable decay channels for the other spin-3 mesons, obtaining: which are in agreement with the experimental values from the PDG [1], Γ tot ω3(1670) = (168 ± 10) MeV , These results demonstrate that the model is internally consistent.
IV. PHENOMENOLOGY OF THE J P C = 3 −− GLUEBALL In this chapter, we study the branching ratios of the decays of a hypothetical glueball with J P C = 3 −− . Similar to all other bound states of gluons, this glueball is not yet experimentally detected. However, lattice QCD calculations in the quenched approximation predict its mass of approximately 4.13 GeV [3], 4.33 GeV [4], or 4.20 GeV [5]. For general works on glueballs, see e.g. Refs. [113][114][115][116]. For model approaches to the phenomenology of glueballs, which are similar to this one, see for example previous publications by some of the authors [33,37,56,79,117].
Before we start our discussion, we explicitly state that all branching ratios should be considered as first indicative results.

A. The effective model for glueball decays
The Lagrangian describing the decays of this glueball can be obtained by making use of the fact that each glueball is a flavor singlet -an object, which is invariant under SU V (3) transformations. Hence, we can perform the simple replacement in the effective interaction terms, where 1 1 3×3 is the flavor-space-identity matrix and G µνρ 3 is the glueball field [33,117]. As a consequence, the interaction terms that come with a commutator vanish and the only nonzero contributions to a residual IR-effective action for the 3 −− glueball can be derived from the former interaction terms involving the anticommutator. Thus, The corresponding Lagrangian terms are reported in Table XIII. In addition, there are also decays of the type G 3 (4200) → V 1 + A 1 and G 3 (4200) → B 1 + A 1 that were not included in for the lightest conventional mesons with J P C = 3 −− because they were kinematically forbidden. They are however possible for the glueball G 3 (4200) since it is expected to be much heavier than the 3 −− nonet (6). Below we present the theoretical predictions for ratios of decays, which can be easily calculated following the same steps of the previous sections. The only ingredients that are needed for a determination of the branching ratios are the masses of the particles, which are for the sake of simplicity again assumed to be exact. For the 3 −− glueball mass, we use m g3 = 4.2 GeV [5].
B. Decay process G3 → V1 + P The interaction Lagrangian for decaying spin-3 tensor glueball into a vector and a pseudoscalar meson reads An extended version is given by Eq. (B8) in App. B. The corresponding decay formula is analogous to Eq. (41), Here, the κ i 's can be taken from Table XXV. We use again Eq. (24) for the momenta of the outgoing particles throughout this chapter, while replacing the mass of the conventional spin-3 mesons m w3 with the glueball mass m g3 = 4200 MeV. Since the coupling constant c g3v1p is unknown, we calculate the nonzero ratios in Table XIV. This is an interesting decay channel to search for this glueball. Since the vector mesons further decay in two (or three) pseudoscalar mesons, one should search for final channels with three (or four) pseudoscalar mesons.
C. Decay process G3 → γ + P Using vector meson dominance via the shift of Eq. (54) and analogous replacements, we can also construct a radioactive decay term for the spin-3 glueball, = c g3γp e gρ G 3,µαβ (∂ ν a ρ ) ε µνρσ tr Q, (∂ α ∂ β ∂ σ P ) + .  where the corresponding factors κ γ i are listed in Table  XXVI. Similar to Eq. (56) one of the masses m a and m b in Eq. (24) has to be set to zero, due to the masslessness of the photon.
From these formula, we calculate ratios for the radioactive decays Γ G3→γ P w.r.t. the strong decay channel Γ G3(4200)→ρ(770) π . The results are listed in Table XV. D. Decay process G3 → V1 + A1 We also present the glueball decays into vectors and axial-vectors by considering the Lagrangian in which G 3 The extended version is provided in App. B in Eq. (B10). The tree level decay formula is given by where we used Eq. (D30). The coefficients κ i are listed in Table XXVII. Results are listed in Table XVI. E. Decay process G3 → B1 + P The second dominant decay of the J P C = 3 −− glueball is the one into a pseudovector and a pseudoscalar meson. The interaction Lagrangian is given by The factors κ i are listed in Table XXVIII. Again, we study two possible choices for the mixing angle (β b1 ≈ 0 • and β b1 ≈ −40 • ) and upon using m K 1,B ≈ m K1(1270) , we present the results in Table XVII.
Moreover, we present the results for the glueball into a pseudovector and an axial-vector, G 3 → B 1 + A 1 in Table XVIII for β a1 ≈ 0 • . Namely, the Lagrangian for this decay can be obtained from Eq. (93) as The extended form is presented in Eq. (B12) in App. B. The corresponding decay formula is given by where the κ i 's are listed in Table XXIX.

V. DISCUSSION OF THE CONSTRUCTION OF THE MODEL
In this section we discuss the interpretation of Eq. (22), the interaction Lagrangian L W,total , which was  constructed by keeping only the lowest possible number of derivatives for a given interaction term. We justify this choice by following two different lines of argumentation. First, our effective model naturally emerges, if the Lagrangian L W,total is interpreted as being part of a more general and complete chiral hadronic model in the vacuum, such as the (extended) linear sigma model [57], which is aforementioned in the introduction. Second, a (Functional) Renormalization Group perspective also indicates that terms with the lowest derivatives might be retained as the dominant contributions within our effective hadronic model.
The eLSM is built under the assumption of chiral symmetry, thus and the assumption of scale invariance. Besides the small explicit breaking of both of these symmetries, because of the nonzero bare quark masses, which derive from the electro-weak sector, the symmetries are also broken dynamically. The SU L (3) × SU R (3) breaks down to a SU V (3) symmetry spontaneously, which was already discussed before. Also the scale invariance breaks down via gluonic quantum fluctuations and the emergence of a gluon condensate, which is also called the conformal/trace anomaly [10,118,119]. Additionally, a second anomaly -the chiral anomaly -breaks U A (1) and explains the heavy mass of η (958) as well as the large pseudoscalar mixing angle [83]. As discussed in Ref. [84] and commented in various part of this work, this anomaly may affect also the masses and the decays of other nonets (those being part of homochiral multiplets). In the chiral limit and ignoring the chiral anomaly all the interaction terms of the eLSM are invariant under scale/dilatation transformations. Consequently, they are parametrized by dimensionless coupling constants.
The Lagrangian L w3,total (22), that we use in this work, contains various decay constants, see Table III. With the exception of the last entry, all of them are dimensionful. At a first sight, this feature seems in disagreement with the requirement that should descend from a chiral model such as the eLSM [53,56]. Yet, this is not the case and a closer inspection also shows why the lowest number of derivatives should be kept. First, let us recall that the spontaneous symmetry breaking implies a peculiar scalar-axial-vector mixing, that is removed by the a shift of the axial-vector nonet A 1 . In the simplified case with vanishing bare quark masses, where U V (3) is exact (this is enough for our illustrative purposes), it takes the form [53,120]: where Z ≈ 1.6 and w ≈ g1φ N m 2 a 1 where g 1 ≈ g ρ ≈ 5.5, m a1 1.4 GeV, and φ N ≈ Zf π (f π = 92.4 MeV is the pion decay constant). Now, the eLSM was studied in a variety of frameworks: Besides (pseudo)scalar and (axial-)vector mesons [53] and the aforementioned calculation of glueballs [56,79,117], it was also applied to study hybrid mesons [120] and excited scalar mesons [121]. Even if various mesonic nonets were not included in the eLSM yet, it is clear, which features such terms should have.
Within this framework, the W 3 P P -Lagrangian, where g w3pp has dimension E −2 and the W 3 A 1 P -Lagrangian, where g w3a1p has dimension E −1 , can be seen as the result of the interaction Lagrangian involving solely the dimensionless coupling g w3a1a1 via the shift of Eq. (97) applied once/twice. In other words, only the Lagrangian L w3a1a1 is part of a generalized eLSM Lagrangian that includes fields with J = 3, in agreement with dilatation invariance. Then, upon spontaneous symmetry breaking and mixing/shifts, the Lagrangians L w3a1p and L w3pp are a consequence of L w3a1a1 , thus explaining how dimensional couplings appear in the model, even if one starts solely with dimensionless ones. While the decay W 3 → A 1 A 1 is not kinematically allowed, the corresponding term explains how the first and the fifth entries of Table III emerge. The simple relations and follow. Moreover, the ratio g w3a1p /g w3pp ≈ (Z w) −1 , out of which we may estimate that is obtained (it should be however stressed that the present heuristic discussion cannot provide a precise determination of ratios of couplings, but it solely offers a guide to understand the origin of the model). This value leads to Γ ρ3(1690)→a1(1260) π ≈ 2 MeV, as previously mentioned.
Next, let us consider the W 3 B 1 P -Lagrangian, with g w3b1p , which can be seen as emerging from with a dimensionless coupling g w3b1a1 . The A 1 -shift introduced above implies that Next, let us study the remaining terms. The W 3 V 1 V 1 -Lagrangian is already dilatation invariant, while the two other Lagrangians L w3v1p and L w3a2p are not, since they involve the coupling g w3v1p and g w3a2p , having dimensions E −3 and E −2 . In this case, one could build analogous Lagrangians L w3v1a1 and L w3a2a1 , but their coupling coupling constants g w3v1a1 and g w3a2a1 still have dimension E −2 and E −1 . This is however understandable, since these terms involve the Levi-Civita pseudotensor ε µνρσ and hence are a manifestation of the the chiral anomaly.
In conclusion, terms being part of the model are (i) dilatation invariant or (ii) can be obtained from dilatation invariant terms of a more general underlying chiral model via the shift of the axial-vector nonet or (iii) they are linked to the chiral anomaly. Now, one also understand why only terms with the lowest number of derivatives appear.
For the effective interaction terms of the 3 −− glueball, the argument is completely analogous.
Still, we also provide a brief alternative motivation for the construction of the model from a renormalization group perspective. In this framework, we know that the UV limit of all hadronic physics is QCD, where the degrees of freedom are quarks and gluons. However, we want to describe low-energy QCD and the effective hadronic degrees of freedom. By integrating out quantum fluctuations of gluons and quarks via renormalization group transformations from the UV scales to the IR scales including dynamical hadronization, one finally ends up with an effective hadronic theory, were hadrons are the effective degrees of freedom. While integrating out quantum fluctuations momentum shell after momentum shell, all kinds of effective couplings/vertices, which are in accordance with the UV symmetries of the theory -the symmetries of QCD -will be generated. This process is effectively described as the RG flow of the coupling constants of the theory, where effective couplings of hadrons are initialy zero and only generated dynamically during the RG flow. The same applies to the effective fermionic and hadronic quantum fields.
An efficient and modern framework to describe this process, which effectively is an implementation of Wilsons idea of the RG [122][123][124], is the Functional Renormalization Group (FRG) in its formulation via the Exact Renormalization Group Equation [60,61], see for example Refs. [58,59,63,[125][126][127] and Ref. [62] for an comprehensive up-to-date review. For our purpose, it is solely important to note that the FRG is most efficiently formulated on the level of the quantum effective IR action Γ[Φ], which is the generating functional of all 1PIn-point-correlation functions. The FRG allows to calculate Γ[Φ] -at least within certain truncations -from the UV theory, which here is QCD. The interesting part is that the quantum effective IR action contains all kinds of effective vertices, that were generated during the RG flow and contain all information about the higher energy scales. For the calculation of decays etc., however, only these effective couplings are relevant and calculations are performed at tree level, because all loop contributions are already contained in the couplings and because we are working with the 1PI-n-point-correlation functions. Dynamical symmetry breaking is realized in this framework as a nontrivial vacuum/ground-state, which realizes as a minimum of the effective action Γ[Φ]. All 1PI-n-point functions must be extracted at this physical point, which yields shifts like in Eq. (97).
In practice, it is almost impossible to really do this calculation from first principles, especially concerning couplings for mesons with higher spin and large masses. Nevertheless, we know from the previous argumentation that a low IR-effective action Γ[Φ] must contain all possible interaction terms, which are in accordance with the symmetry of the theory and if shifted to the physical point, then it must contain all symmetries of the vacuum of the theory. This also applies to the effective couplings of our spin-3 mesons. Thus, we can use the following argument: instead of calculating all interaction terms and couplings, which are in accordance with the symmetries of QCD, via an RG flow from first principles (which is anyhow impossible), we just write down all possible effective interaction terms, that are invariant under the residual IR vacuumsymmetries of QCD and simply fit the effective coupling constants to experimental data. This approach additionally justifies the use of tree level-calculations, because experimental measurements for decay widths etc. already contain all information about the high energy scales of QCD and so does an effective IR action Γ[Φ], from which information is extracted at tree level. We therefore interpret all effective Lagrangians L w3•• as being part of an effective IR action Γ[Φ] for QCD.
Last, we argue, that as long as the momentum exchange in processes like three-point-interactions (decays) is small, we can expand an effective IR action in powers of momenta of the effective hadronic fields. In the context of the FRG, this approach is denoted as a derivative expansion and turned out as a decent approximation in a lot of low-energy models of QCD [64][65][66][67][68][69][70][71][72][73][74][75].
All in all, both approaches complement each other and explain that we only retain terms that are of the lowest orders in the derivatives of the fields, that are of leading order in a large-N c expansion and that are flavor invariant.

VI. CONCLUSIONS
In this paper, we have studied the decays of the lightest mesonicqq nonet with quantum numbers J P C = 3 −− using an effective QFT model based on flavor symmetry. Our model retained only the dominant terms in an large-N c expansion and the lowest possible number of derivatives. By comparing the theoretical results with the current status of experimental data for decay width and some known branching ratios, which are all reported by PDG [1], we conclude that theqq assignment works quite well. Still, we remark that there are also other rather successful approaches different from theqq picture towards a coherent description of the nature of the J P C = 3 −− states, see e.g. via multi-ρ(770) resonances [128,129]. However, also in our work some of the decay channels deserve deeper theoretical and experimental investigation. Additionally, we presented decay ratios of a putative and not yet detected J P C = 3 −− glueball by considering decays into the vector-pseudoscalar and pseudovector-pseudoscalar mesonic pairs. In summary, we provide many qualitative predictions of strong and radiative decays of conventional J P C = 3 −− mesons and the J P C = 3 −− glueball for future experimental tests. For a certain nonet, we denote as a the triplet of fields describing the isospin I = 1 elementsdu,ūd, 1 √ 2 (ūu −dd), with K ± the complex kaonic fields representing the doubletsu andūs, andK 0 the analogous fieldsds andsd for the neutral kaonic elements, and as f 8 and f 0 the octet and the singlet elements, whose quark-antiquark configurations are 1 √ 6 (ūu +dd − 2ss) and 1 √ 3 (ūu +dd +ss), respectively. The corresponding SU V (3)-invariant mass term reads (omitting possible Lorentz indices for simplicity), where also a singlet-octet mixing parameter m 08 has been introduced. The masses of the fields a,K 0/± , and f 8 are expressed as where m nonet is the mass of the nonet in the chiral limit as a consequence of spontaneous symmetry breaking (see e.g. Ref. [53]), and δ 2 n represents the (almost negligible) contribution of the bare nonstrange quarks u and d and δ 2 s the (on hadronic scales) small but nonnegligible contribution of the bare quark s. (Besides the pseudoscalar nonet, one simply approximates δ n ∝ m u m d and δ s ∝ m s .) From the expressions above it follows that The mass of the singlet member of the nonet may contain an additional unknown contribution (due to the anomaly and/or conversion to gluons), parametrized by: Next, we diagonalize the system in the isoscalar sector. We then introduce the physical fields f and f as where β PDG is the corresponding mixing angle, which is obtained via diagonalization of the mass matrix together with the relation, In Eq. (A7) the extra minus sign in front of f is introduced for later convenience. Using Eq. (A9) one gets but on the other hand using the above expressions for the masses the l.h.s. can be written as leading to expressions used in the PDG [1] to calculate the mixing angle in the singlet-octet basis: In the last step we relate this to the (for our purposes) more convenient nonstrange-strange basis f N and f S that corresponds to the configurations 1 √ 2 (ūu +dd) andss. Via we introduce the strange-nonstrange mixing β as: or, as reported in the main text, as: The strange-nonstrange mixing angle β and the angle β PDG can be obtained by Eq. (A15) as: The mixing angle β PDG (and hence β) has been calculated in the PDG for certain well-known mesonic nonets (pseudoscalar, vector, as well as J P C = 2 ++ and J P C = 3 −− ). While for the pseudoscalar case we employ the result of Ref. [81] (where not only masses but also decays are used to get β), in the remaining three cases we used the value β PDG to determine the strange-nonstrange mixing angle.
With the exception of the peculiar pseudoscalar nonet, where the axial anomaly is large, the other values of β calculated from β PDG are quite small -in agreement with the theoretical expectations presented in Ref. [84]. As a consequence, the parameter α of Eq. (A6) is negligible. In particular, for J = 3 the numerical value β = β w3 = 3.5 • has been used throughout our calculations. When future experimental data will be more accurate, one could also include β w3 as a fit parameter in an improved model. In this respect, better experimental data for radiative decays would be useful.

Appendix B: Extended form of the Lagrangians of the model
In this appendix, we provide the explicit form of the interaction Lagrangians in Table III and in Table XIII.
Appendix C: Coefficients for the decay channels In this appendix we present the explicit forms of the coefficients κ i and κ γ i in Eq. (23) based on the extended form of Lagrangians presented in the previous section in Tables XIX -XXX.

Appendix D: Unpolarized invariant decay amplitudes
This appendix is devoted to the calculation of the unpolarized invariant decay amplitudes. These form the normal decay amplitudes by multiplication with factors κ i depending on the specific decay channels. We apply the following Feynman rules for each interaction vertex in order to derive the expressions for the decay amplitudes,               = ig w3pp µνρ (λ w3 , k w3 ) k µ p (2) k ν p (2) k ρ p (2) .
The square of the amplitude reads 2. The unpolarized invariant decay amplitude for the interaction of massive spin-3 fields, massive vector fields and massive pseudoscalar fields The square of the amplitude is 3. For the decays of massive spin-3 fields into massive pseudoscalar fields and photons, the generic interaction Lagrangian is of the general type The corresponding decay amplitude reads The averaged square of the amplitude is 4. Decays of massive spin-3 fields into massive spin-2 tensor and massive pseudoscalar fields is described by the following interaction which leads to the amplitude Squaring the amplitude, we find, 5. The amplitudes for the decays of massive spin-3 fields into massive pseudoscalar and massive axial-/pseudovector fields, which are described via the interaction Lagrangians and have the same shape, Their squares have the following form, 6. The interaction Lagrangian which describes the decay of massive spin-3 fields into two massive vector fields leads to the following amplitude The square of this amplitude is given by 7. For the decays of massive spin-3 fields into massive vector and axial-vector fields, we use the following Lagrangian to obtain the amplitude The square of the amplitude is We remark that the form of the amplitudes M of most of the decay channels was already derived in Ref. [16]. In contrast to Ref. [16] we neglect higher order derivative couplings and purely rely on the lowest order contribution in derivatives of the fields. Furthermore, we do not include form factors.
In order to obtain the results of this appendix one has to average over all incoming spin polarizations, sum up all possible outgoing polarizations and consider the following completeness relations for massive particles, which are discussed in the subsequent appendix, and (η µν ) = diag(+1, −1, −1, −1). For the square of the amplitudes of the radioactive decays in Eq. (D17) the following completeness relation for photons is used [130], Here, n = (1, 0, 0, 0) T is the unit vector in temporal direction and k is the four momentum of the outgoing photon.
The summations over Minkwoski spacetime indices in the squares of the amplitudes were performed on a symbolic level with Wolfram Mathematica 12.1 [131] using Sum and ParallelSum. We work in the center of mass frame of the decaying particle. In order to reduce the number of addends in the sums drastically, w.l.o.g. we choose the three momenta of the outgoing particles in z direction.

Appendix E: Completeness relations
In this appendix, we present how the completeness relations for the polarization vectors and tensors of massive higher-spin fields, like Eqs. (D33), (D34) and (D35), are constructed. These completeness relations are used for the calculation of the unpolarized decay amplitudes in the previous App. D.
We begin by deriving the completeness relation for massive spin-1 fields explicitly. This derivation is based on the discussion in Ref. [130] and presented here for reasons of consistency. Starting from this result for spin-1 fields, we generalize the procedure and show, how the completeness relations for massive higher-spin fields can be constructed. We then provide an explicit derivation of the completeness relations for massive spin-3 tensor fields. We close the discussion, by elaborating on the degrees of freedom of massive fields of arbitrary integer spin-J and their description via rank-J tensor fields.
None of the results in this appendix is original to our work. However, we think that the discussion might still be useful for a nonexpert reader.

Spin-1 fields
The construction of the completeness relation for the polarization vectors µ (λ, k) of the massive (real) vector fields V µ (x) is based on the field equations that de-scribe free massive vector particles. 7 These field equations are given by the Proca equations, which are the Klein-Gordon (KG) equation and a transversality condition (which is explicitly not a gauge), For neutral massive vector mesons, the solutions to the KG equation are plane waves, where a(λ, k) = a * (λ, − k) are the Fourier amplitudes and E 2 ( k) = k 2 + m 2 is the square of the energy. The transversality condition further implies that the polarization vectors (λ, k) have to be transversal to the four momentum k = (E( k), k) T of the the fields, From this equation, we can construct the completeness relation for the polarization vectors. We start with the most general ansatz for a dimensionless rank-two tensor that only depends on the four momen- where a and b are dimensionless constants. The transversality condition (E3) implies which simplifies our ansatz to +1 λ=−1 The coefficient a can be fixed by choosing an appropriate normalization condition. This is done by demanding that in the rest frame k = (m, 0) T for the spatial components the completeness relation should reduce to the Kronecker delta, to have a three-dimensional orthonormal basis of polarization vectors.
7 For free massive axial-and pseudovector fields the construction is identical.
We obtain the completeness relation and define the projector G µν , which projects on the subspace of the Minkowski space that is transversal to the four momentum vector k = (E( k), k) T of the massive spin-1 field.

Higher-spin fields
For all kind of higher-spin tensor fields, we proceed analogously. The starting point are the so called Fierz-Pauli equations for a massive field with integer spin J. The field is described by a rank-J tensor field that fulfills the following equations (see also Refs. [130,[132][133][134][135][136][137] for more details on wave equations and degrees of freedom of higher-spin fields), and ∀ 1 ≤ m = n ≤ J, and ∀ 1 ≤ m ≤ J, These equations reduce the 4 J degrees of freedom of a general rank-J tensor field in 4-dimensional Minkowski space to the (2J +1) degrees of freedom that are necessary to describe a massive spin-J field. (We will come back to this point at the end of this appendix.) Note that the Proca equations (E1) for spin-1 fields are merely a special case of the Fierz-Pauli equations (E9) -(E12). Completely analogous to the vector field, we can thus find a solution to the KG equation and derive the following constraints for the corresponding polarization tensors, ∀ 1 ≤ m = n ≤ J , Next, to find the corresponding completeness relation, one has to start with an appropriate ansatz for Consequently, one should write down the most general rank-2J tensor, which only depends on the four momentum k = (E( k), k) T and the metric tensor. This means that all possible combinations of η µν and kµkν k 2 , with all possible permutations of the indices α 1 , . . . , α J , β 1 , . . . , β J of the polarization tensors have to be included, = + a 1 η α1α2 · · · η β J−1 β J + all permutations + Here, a i , b i , . . ., z i stand for the various dimensionless coefficients of the ansatz. An alternative approach is to write down the ansatz for the rank-2J tensor on the r.h.s. of the above equation in terms of projection operators which also form a complete basis.
What follows is identical for both approaches: To determine the various coefficients, one has to use Eqs. (E13), (E14) and (E15) (or combinations of these constraints) successively until one reaches the point, where only one overall coefficient/factor is left, see for example Ref. [132] for massive spin-2 fields. The leftover constant can only be fixed by appropriate normalization. For most problems, a normalization, which produces an orthonormal basis of polarization tensors in the rest frame, like for the vectors in Eq. (E7), is suitable.
Following these procedures, one eventually arrives at the completeness relations (D34) and (D35) for the polarization tensors of spin-2 and spin-3 fields.
We remark, that an expression for arbitrary spin fields was already presented in Refs. [19,138]. The expressions for spin-2 and spin-3 were already presented among others in Refs. [16-18, 132, 139].

Spin-3 fields revisited
Here, we explicitly present the derivation of the completeness relation for massive spin-3 fields.
Next, we have to determine the coefficients a i , b i , c i and d by applying the constraints (E19), (E20) and (E21). Note that for convenience only, the coefficients a i are dimensionless, while b i , c i and d contain factors of 1 k 2 , in order to have the r.h.s. of the equation dimensionless. The last symmetry, which includes all other possible index interchanges, is µ ν ρ ↔ α β γ, from which follows that a 1 = a 13 , a 2 = a 6 , a 4 = a 10 , = + a G µν G ρα G βγ + a G µν G ρβ G αγ + + a G µν G ργ G αβ + a G µρ G να G βγ + + a G µρ G νβ G αγ + a G µρ G νγ G αβ + + a G µα G νρ G βγ + a G µα G νβ G ργ + + a G µα G νγ G ρβ + a G µβ G νρ G αγ + + a G µβ G να G ργ + a G µβ G νγ G ρα + + a G µγ G νρ G αβ + a G µγ G να G ρβ + + a G µγ G νβ G ρα + + c G µν k ρ k α k β k γ + c G µρ k ν k α k β k γ + + c G µα k ν k ρ k β k γ + c G µβ k ν k ρ k α k γ + + c G µγ k ν k ρ k α k β + c G νρ k µ k α k β k γ + + c G να k µ k ρ k β k γ + c G νβ k µ k ρ k α k γ + + c G νγ k µ k ρ k α k β + c G ρα k µ k ν k β k γ + + c G ρβ k µ k ν k α k γ + c G ργ k µ k ν k α k β + + c G αβ k µ k ν k ρ k γ + c G αγ k µ k ν k ρ k β + + c G βγ k µ k ν k ρ k α + + d k µ k ν k ρ k α k β k γ .
To determine the remaining eight constants, we use Eqs.
Thus, we conclude b =b = 0. Before we calculate the relation between the coefficients a and a , we provide the remainder expression for the completeness relation. = + a G µν G ρα G βγ + a G µν G ρβ G αγ + + a G µν G ργ G αβ + a G µρ G να G βγ + + a G µρ G νβ G αγ + a G µρ G νγ G αβ + + a G µα G νρ G βγ + a G µα G νβ G ργ + + a G µα G νγ G ρβ + a G µβ G νρ G αγ + + a G µβ G να G ργ + a G µβ G νγ G ρα + + a G µγ G νρ G αβ + a G µγ G να G ρβ + + a G µγ G νβ G ρα .
The last constant a can be chosen arbitrarily. Nevertheless it might be convenient to orthonormalize the polarization tensors, µνρ (λ, k) µνρ (λ , k) = −δ λλ , which is the higher-spin equivalent to the orthonormality condition Eq. (E7). Then it directly follows that The final result is.