Scalar-involved three-point Green functions and their phenomenology
Ling-Yun Dai Javier Fuentes-Martin Jorge Portoles






ZU-TH 07/19



1cm







Scalar-involved three-point Green functions 

 [13pt]and their phenomenology

 

Ling-Yun Dai^1,[Email: dailingyun@hnu.edu.cn]
, Javier Fuentes-Martín^2,[Email: fuentes@physik.uzh.ch] and
Jorge Portolés^3,[Email: Jorge.Portoles@ific.uv.es]

 

^1School of Physics and Electronics, Hunan University, Changsha 410082, China 



^2Physik-Institut, Universität Zürich, CH-8057 Zürich, Switzerland 

^3Instituto de Física Corpuscular, CSIC - Universitat de València,
Apt. Correus 22085, E-46071 València, Spain 
 [13pt]












We analyse within the framework of resonance chiral theory the ⟨ SA_μ A_ν⟩ and ⟨ SV_μ V_ν⟩ three-point Green functions, where S, A_μ and V_μ are short for scalar, axial-vector and vector SU(3) hadronic currents. We construct the necessary Lagrangian such that the Green functions fulfill the asymptotic constraints, at large momenta, imposed by QCD at leading order. We study the implications of our results on the spectrum of scalars in the large-N_C limit, and analyse their decays.










§ INTRODUCTION

Green functions of quantum fields convey all the dynamics of a quantum field theory describing a system of many interacting particles. Their consistent construction in the hadronic low-energy region (typically E ≪ 1), driven by non-perturbative Quantum Chromodynamics (QCD), can be thoroughly carried out within the model-independent framework of  Chiral Perturbation Theory (ChPT) <cit.>. The predictability of this theory is however spoiled at O(p^4) and higher due to our poor knowledge of the chiral low-energy constants. At higher energies, in the hadronic resonances populated domain
(1   E  2.5), the construction of the Green functions has been addressed only under several specific model-dependent assumptions, such as the Extended Nambu-Jona-Lasinio model <cit.> and related ones <cit.>. Different implementations of large-N_C <cit.>: minimal hadronic ansatz <cit.> and
resonance chiral theory (RChT)
<cit.>, have also been explored in the last decades. At even higher energies (2.5   E), except where very narrow hadronic resonances arise, perturbative QCD starts to provide a correct description.
It is clear that QCD should rule the dynamics of those Green functions. However, our lack of knowledge of non-perturbative QCD makes that task very difficult and the use of models of QCD becomes necessary. The construction of those models should include chiral symmetry as a feature to be fulfilled in its low-energy domain. The properties of the model at high-energies are more difficult to implement due to hadronization and hence they are not obvious from a Lagrangian point of view. Several works have addressed this problem within RChT <cit.>, which provides a framework for the evaluation of the Green functions in the intermediate energy region. This is a Lagrangian setting in terms of pseudo-Goldstone bosons and resonances (as matter fields) that, by construction, respect the chiral symmetry. As in ChPT, this symmetry provides the structure of the operators but gives no information on the coupling constants. However, due to the presence of resonance fields, the Lagrangian has no obvious counting that controls the number of operators and, consequently, some extra features are needed in its application. On one side Green functions are computed using large-N_C premises <cit.>; this translates, essentially, in a loop expansion generated by the Lagrangian. This is not enough to limit the number of operators and, in addition, gives no information on the coupling constants. The extra help comes from the assumption that the correlation functions, as given by RChT (Π_), can be matched, at large momenta, with the known asymptotic behaviour of Green functions and form factors on QCD grounds (Π_). This sounds feasible as the RChT result (at tree level) and the operator product expansion (OPE), at O(α_S^0), generate an expansion in inverse powers of momenta. The method was originally applied to two-point Green functions in Ref. <cit.> and later to three-point functions <cit.> as:

    lim_λ→∞ Π^α_S^0_ (λ q)  =  lim_λ→∞ Π_^ (λ q)   .

Short-distance constraints are also imposed on vertex functions (form factors) by considering their Brodsky-Lepage <cit.> asymptotic behaviour, using parton dynamics <cit.>. These approaches can provide valuable information
on the structure of the operators and their coupling constants. Moreover, as the later do not depend on the masses of the pseudo-Goldstone bosons, the procedure can be carried out in the chiral limit. The question of the feasibility of this matching was discussed in Ref. <cit.>.
The above-mentioned procedure is particularly transparent for Green functions that are order parameters of the spontaneous breaking of the chiral symmetry, i.e. those that do not receive contributions of perturbative QCD, in the chiral limit, at large momentum transfers and, therefore, show a rather smooth behaviour.  Several works along this line have been produced <cit.> with noticeable results. One of the key
issues in order to carry out the matching procedure in Eq. (<ref>) lies in the construction of the appropriate operators in the RChT Lagrangian that make
the matching possible. The procedure may not always be feasible <cit.>, but most of the time it is just a matter of looking for the suitable operators.
In Ref. <cit.> it was pointed out the difficulty involved in the matching for the ⟨ SV_μ V_ν⟩ Green function (where S and V_μ are short for scalar and vector QCD currents, respectively) using a Proca representation for the vector resonance fields in RChT. As expected, the authors satisfied the matching by including a higher order (in derivatives) RChT operator that was needed to enforce the QCD short-distance behaviour even though it was non-leading at low energies.
In this article we perform a systematic analysis of the ⟨ SV_μ V_ν⟩ and ⟨ S A_μ A_ν⟩ Green functions (A_μ is short for axial-vector QCD
current) using an antisymmetric representation for the spin-1 resonances in the RChT framework. We will fulfill the matching indicated by Eq. (<ref>) for both Green functions by constructing a minimal set of RChT operators that provide the correct short-distance behaviour. We consider tree-level diagrams only and, accordingly, work in the N_C →∞ limit. Moreover we restrict our large-N_C description to only one multiplet for each hadron type: scalars, vectors and axial-vectors. As a final result we obtain several relations between the relevant coupling constants of the Lagrangian.
The description, classification and dynamics of hadronic scalar meson resonances, with masses M_S   2, has a long story of successes and failures (see the corresponding note in Ref. <cit.>). The light-quark spectrum of meson resonances is populated by many scalar states whose identification as SU(3) octets/nonets is far from clear and that are, probably, an admixture of exotic states that involve tetraquarks or even glueballs. The unsolved non-perturbative dynamics does not allow us to identify the nature of the bound states generated by QCD. Experimentally one observes a number of
J^P=0^+ states that could fit into two U(3) nonets constituted by quarks. Our present knowledge points out to usual [qq] states but also
tetraquark ones [qq][qq] <cit.>. The existence of a glueball (with J^P=0^+ and of similar properties to the quark resonances) with mass in the upper part of our spectrum (∼ 2) was also pointed out some time ago by the lattice <cit.>. Hence it is expected that all the scalar resonances in this energy region could be an admixture of all these basic states. 
By construction, the leading multiplets of resonances described by RChT should correspond to those remaining in the N_C →∞ limit. However, while this identification does not create discussion for vector, axial-vector and pseudoscalar resonances, the scalar case is much more complex.
In Ref. <cit.> a study within RChT in the large-N_C framework identified the preferred lightest scalar nonet as the one constituted by S_ = { f_0(980), K^*_0(1430), a_0(1450), f_0(1500) }, assuming that the a_0(980) is dynamically generated and making an octet together with f_0(500) and K^*_0(700) as a subleading spectrum.
In Ref. <cit.>, a new method to study the large-N_C behavior
of the final states interactions (FSI) within the dispersive approach was proposed. The N_C trajectories of the poles suggest that f_0(980) and f_0(1370) should have the [qq] component. This is further confirmed in Ref. <cit.>, by studying the semi-local duality in the large-N_C limit. Finally, there is also a broad consensus that S_ corresponds to the [qq] structure while the lightest nonet of resonances is constituted by [qq][qq] <cit.> (and references therein), with a possible large mixing between them. Even though we basically agree with this description, we will modify it slightly in order to include the f_0(1370) and the f_0(1710), aiming to account for the glueball in our framework.
Although the experimental situation of the scalar decays is rather poor and uncertain <cit.>, we intend to analyse the two-pseudoscalar decays of the spectrum of scalars in RChT, i.e. the S → PP decays of the leading multiplet in the N_C →∞ limit. In doing so, we will use the minimal set of operators in this framework. We will conclude that meanwhile the short-distance matching procedure of the three-point Green functions requires higher derivative operators in some cases, we do not need to introduce subleading operators (in the large-N_C counting) to fulfill the matching. On the contrary, the experimental data on the S → PP decays will require to break manifestly that counting by introducing subleading operators. Hence we conclude that the scalar related couplings in the matching of the Green functions are not given by the N_C →∞ limit.
In Section 2 we recall the RChT framework within our large-N_C model, leaving for Section 3 the matching procedure for the ⟨ SA_μ
A_ν⟩ and ⟨ S V_μ V_ν⟩ three-point Green functions. Section 4 is devoted to explain the features of our scalar resonance sector and the results of their decays into two pseudoscalar mesons. We establish our conclusions in Section 5. The chiral notation and several analytical expressions on the decays of scalars into other final states are given in the Appendices.



§ THE LARGE-N_C SETTING: RESONANCE CHIRAL THEORY
 
RChT is a Lagrangian framework that includes the interaction between the chiral pseudoscalar octet of mesons, in ChPT, and the hadron resonances in the energy region up to ∼ 2. The symmetries driving the operators are both the chiral (SU(N)_L ⊗ SU(N)_R) and flavour (SU(N)) symmetries, for light flavours, N=2,3 <cit.>. By construction the RChT method matches the chiral symmetric results at low energies.
Here we only recall the content needed for our present work. We will only consider scalar, vector and axial-vector resonances, and the case with N=3 flavours. For a detailed account and notation we refer the reader to Refs. <cit.> and Appendix A.
The RChT framework starts with the leading chiral Lagrangian involving only the octet of pseudoscalar Goldstone bosons (GB) and external currents. It is given by:

    L_(2)^  =    L_(2)^  =  F^2/4 ⟨  u_μ  u^μ  +  χ_+ ⟩  ,

where F is the decay constant of the pion in the chiral limit, and the symbol ⟨·⟩ stands for the trace in flavour space. This term collects the information on the spontaneous symmetry breaking of the chiral symmetry and coincides with the same order Lagrangian of ChPT.
RChT has no defined parameter (in the Lagrangian) on which to build a qualified counting to establish a classification for the operators. As the integration of the resonances should provide, generically, the ChPT Lagrangian of O(p^n), for n>2, it has been customary to classify the RChT operators by the order in momenta of the ChPT operators that they were producing upon integration. Therefore the general structure of the operators is O∼⟨ R_1 R_2 ... R_p χ(p^n)  ⟩, with R_a a U(3) nonet of resonance fields, namely V_μν (vector), A_μν (axial-vector) and S (scalar). Notice that we will use the antisymmetric representation for the spin-1 fields
<cit.>, given its relevance in the chiral framework <cit.>. In addition, χ(p^n) is a tensor (constructed with chiral invariants in terms of the pseudoscalar Goldstone fields and external currents of ChPT)
of n chiral order (see Appendix <ref>). The operators giving the O(p^4) terms in the chiral Lagrangian are of the type ⟨ R_a  χ(p^2)  ⟩:

    L^V_(2)    =    F_V/2 √(2) ⟨ V^μν  f_+ μν ⟩  +   i G_V/√(2) ⟨  V^μν 
    u_μ  u_ν ⟩  , 
     L^A_(2)     =    F_A/2 √(2) ⟨  A^μν  f_- μν ⟩  , 
     L^S_(2)     =     c_d  ⟨  S   u^μ  u_μ ⟩  +   c_m  ⟨  S  χ_+  ⟩  ,

where the real couplings: F_V, G_V, F_A, c_d and c_m are, a priori, unknown.
Those generating the O(p^6) chiral Lagrangian have been studied in Ref. <cit.> and have the general structures:
⟨  R_a  χ(p^4)  ⟩, ⟨   R_a   R_b  χ(p^2)  ⟩ and ⟨  R_a   R_b   R_c  ⟩. We will collect those of interest for our study in the next section.
It would also be possible to classify the operators into sets that provide the correct asymptotic behaviour of definite n-point Green function of QCD currents, that is, the relation in Eq. (<ref>). As has been concluded in previous studies of these Green functions, one starts with the two-point Green function (and related form factors) and determines the
appropriate set of operators and relations between couplings. For instance the study of two-point Green functions, with only one multiplet of resonances (single resonance approximation), gives <cit.>:

    F_V   G_V      =    F^2   ,       F_V^2   -   F_A^2       =  F^2   ,    
    F_V^2   M_V^2     =   F_A^2   M_A^2   , 
    
    4   c_d   c_m       =  F^2   ,      c_d     = c_m   ,

for the couplings in Eqs. (<ref>,<ref>). Here M_V and M_A are the masses of the vector and axial-vector nonet, respectively.
When the study is extended to three-point Green functions one may determine an extended set of operators and the initial relations between couplings could be modified <cit.>, and so on.
A comment on the nature of the resonances described in the Lagrangian of RChT is needed. This framework is embedded in a large-N_C setting. Accordingly, the spectrum described in the Lagrangian corresponds to states that stay in the N_C →∞ limit. Thus our framework cannot contain resonances that are generated by the Lagrangian (for instance on accounts of unitarity) because these are subleading in the 1/N_C expansion. A clear case is the f_0(500), generated by (or coincident with) a strong ππ wide s-wave.
Together with L_(2)^ in Eq. (<ref>) and the Lagrangian involving resonances, RChT requires the addition of operators with the same structure as the ones in
the ChPT Lagrangian at O(p^4) <cit.>, O(p^6) <cit.>,
and so on, although with different couplings.
 It is well known that the low-energy couplings in ChPT are, at least at
O(p^4), mostly saturated by the contribution of the lightest multiplets of resonances <cit.>. At O(p^6) the situation is less clear. 

Since the couplings are different from their ChPT counterparts, we will denote them as L̂_̂î and Ĉ_̂î (for L_(4)^ and L_(6)^, respectively):

    L_(4)^  =  ∑_i  L̂_i   O_(4)^i   ,              L_(6)^  =  ∑_i  Ĉ_i   O_(6)^i   .

Notice that the dimension of the couplings are  [L̂_i] = E^0 and [Ĉ_i] = E^-2.
In this article we intend to analyse the three-point Green functions ⟨  S   V_μ  V_ν ⟩ and
⟨  S   A_μ  A_ν ⟩, imposing the asymptotic behavior in Eq. (<ref>), at leading order in the 1/N_C expansion. In practice this
means that we will evaluate the three-point Green functions in RChT with tree-level diagrams only. For consistency, we should include in our computations an infinite set of resonances. We do not know how
to do this in a model-independent way. However, there are good phenomenological reasons that indicate that the lowest mass states (surviving in the N_C →∞ limit) contribute dominantly, as has been shown for instance in the determinations of the O(p^4) low-energy couplings <cit.>. This is in agreement with the usual decoupling of effective field theories where the contributions from heavy mass states to the low-energy theory is suppressed by powers of E/M, with E the energy scale of the effective theory and M the mass of the decoupled state. Accordingly, we model our N_C →∞ setting by including only the lightest multiplet of resonances for each hadron type.
The identification of the nonets in Eq. (<ref>) is simple for vector states
<cit.>: V_μν(1^-) = {ρ(770), K^*(892),ω(782),ϕ(1020)}. For axial-vector mesons the situation
is slightly more complicated <cit.>:  A_μν(1^+) = { a_0(1260), K_1(1270), f_1(1285), f_1(1420) }, since the strange doublet could also be K_1(1400) or an admixture of both. The common feature of these two multiplets is that they correspond to the
lightest states (experimentally identified) with those quantum numbers. For the scalar resonance case (and the glueball) the identification of the lightest nonet, surviving at N_C →∞, seems not to concur with the lightest nonet but with one of higher mass. We delay this discussion to Section 4.




§ THREE-POINT GREEN FUNCTIONS FROM RCHT
 
Similarly to the relations in Eq. (<ref>), based on two-point Green functions, one can obtain additional constraints on the RChT couplings by analyzing the three-point Green functions. A lot of work has already been employed in their study <cit.>. Here we focus on the scalar-involved Green functions ⟨  S   A_μ  A_ν ⟩ and ⟨  S   V_μ  V_ν ⟩. Both of them are order parameters of the spontaneous chiral symmetry breaking and, consequently, vanish at O(α_S^0) in the chiral limit.
The definition of these Green functions is given by

    Π^ijk_123(p_1,p_2)   =   i^2 ∫  d^4x  d^4y    e^i(p_1 · x + p_2 · y)⟨ 0 | T {( ψΓ_1 λ^i/2ψ) (0)  ( ψΓ_2 λ^j/2ψ) (x)  ( ψΓ_3 λ^k/2ψ) (y) } |0 ⟩  ,

where Γ_i = 2 for the scalar current, Γ_i = γ_μ for the vector current and Γ_i = γ_μγ_5 for the axial-vector current. Our conventions for the momenta are defined in Figure <ref>.



We will proceed to determine the general structure of those Green functions as provided by their chiral Ward identities, SU(3)_V, parity and time reversal. Then we will obtain their short-distance behaviour at leading order in the momenta expansion. We also calculate their expressions using RChT and including the necessary operators such that we have a perfect matching in the momenta expansion, following the relation in Eq. (<ref>). A simplifying aspect of the procedure is that, since the couplings do not depend on the masses of the pseudoscalar mesons, we can perform this operation in the chiral limit. Since our Green functions are order parameters of the chiral symmetry breaking, this implies that there is no perturbative contribution in the parton calculation, at least at O(α_S^0).



 §.§ ⟨  S   A_μ  A_ν ⟩


The ⟨ SA_μ A_ν⟩ Green function is defined by:

    ( Π^ijk_SAA)_μν  =   i^2  ∫ d^4x   d^4y  
    e^i ( p_1 · x + p_2 · y ) ⟨ 0 | T { S^i(0) A_μ^j(x) A_ν^k(y) } |
    0 ⟩  ,

where

    S^i(x)   =  ( q̅λ^i q )(x)        ,        
    A^i_μ(x)   =  ( q̅γ_μγ_5 λ^i/2 q )(x)   ,

with q(x) = (u,d,s)^T the quark fields.
In SU(3) it satisfies the Ward identities:

    p_1^μ( Π^ijk_SAA)_μν    =     - 2   d^ijk B_0 F^2  (p_2)_ν/p_2^2  ,
    
    
     p_2^ν( Π^ijk_SAA)_μν    =     - 2   d^ijk B_0 F^2  (p_1)_μ/p_1^2  .

Here B_0 parameterizes the spontaneous chiral symmetry breaking and it has been defined in Eq. (<ref>).
The general structure of the Green function is given by:

    ( Π^ijk_SAA)_μν    =      d^ijk B_0 [ - 2   F^2 (p_1)_μ (p_2)_ν/p_1^2 p_2^2 
    +   F_A (p_1^2,p_2^2,q^2)   P_μν  +
      G_A (p_1^2,p_2^2,q^2)   Q_μν]   ,

with the generic scalar functions F_A(p_1^2,p_2^2,q^2) and G_A(p_1^2,p_2^2,q^2),
q^2 = ( p_1 + p_2 )^2, and where P_μν and Q_μν are the two Lorentz structures
that vanish upon projection with the (p_1)_μ and (p_2)_ν momenta:

    P_μν    =    (p_2)_μ(p_1 )_ν  -  p_1 · p_2   g_μν  , 
    
    Q_μν    =     p_1^2(p_2)_μ(p_2)_ν  +  
    p_2^2 (p_1)_μ(p_1)_ν  -  
    p_1 · p_2 (p_1)_μ(p_2)_ν  -  
    p_1^2  p_2^2  g_μν  .

The Ward identities in Eq. (<ref>) are also at the origin of the first term of the Green function in Eq. (<ref>). This term is recovered in RChT  by the O(p^2) ChPT Lagrangian in Eq. (<ref>) through the diagram in Figure <ref>.




The short-distance behaviour of the ⟨ SA_μA_ν⟩ function, at leading order in the momenta
expansion, is given by:

    lim_λ→∞( Π^ijk_SAA)_μν( λ p_1, λ p_2 )
        =     -2   d^ijk  B_0 F^2 1/λ^2 1/p_1^2   p_2^2   q^2 [ q^2 (p_1)_μ( p_2 )_ν  +   Q_μν  -   p_1 · p_2   P_μν] 
           +   O( 1/λ^3)   ,  
    lim_λ→∞( Π^ijk_SAA)_μν( λ p_1, p_2 )     =   
    -2   d^ijk  B_0 F^2 1/λ (p_1)_μ( p_2 )_ν/p_1^2 p_2^2  +
      O( 1/λ^2)   ,
    lim_λ→∞( Π^ijk_SAA)_μν( p_1, λ p_2 )     =   
    -2  d^ijk  B_0 F^2 1/λ (p_1)_μ( p_2 )_ν/p_1^2 p_2^2  +
      O( 1/λ^2)   ,  
    lim_λ→∞( Π^ijk_SAA)_μν( λ p_1, q-λ p_1 )     =    O( 1/λ^2)   .

Let us now compute F_A and G_A in RChT at tree level. The content of the Lagrangian, as explained in
Section <ref> presents two main parts: the operators with Goldstone boson fields only (and external currents) and those with interactions among them and resonance fields. We have:

    L_  =   L_(2)^  +   L_(4)^  +   L_(6)^  +   L_(2)^A   +
      L_(2)^S   +   L_A  ,

where the  Lagrangians have been defined in Eqs. (<ref>,<ref>). For the reader's convenience we list the relevant operators in Table <ref>.
Their contribution to the Green functions are given by the diagrams in Figure <ref>.



Next we consider the resonance contributions. The Lagrangians L_(2)^A and L_(2)^S are given in Eq. (<ref>) while in L_A we include those
operators with resonances, Goldstone fields and external currents that, upon integration of the resonances, originate the O(p^6) ChPT Lagrangian. They have been constructed in Ref. <cit.>. Those contributing to our Green function are also collected in Table <ref>. They contribute through the diagrams in Figure <ref>. Previous short-distance constraints already concluded that λ_17^S = λ_18^S = λ_17^A=0 <cit.>. We include these couplings in our analysis and we set them to zero at the very end.








The final result for the F_A and G_A functions defined in Eq. (<ref>) is:

    F_A(p_1^2,p_2^2,q^2)     =     32  (Ĉ_12  -  Ĉ_80  -  Ĉ_85)    -   32  λ_16^S   P_S   -   16  λ_6^AA  P_A(p_1^2)   P_A(p_2^2)   
           +   8  √(2) ( 2  λ_16^A   -  λ_6^A   +   (  λ_1^SA  +    2  λ_2^SA  )  P_S   ) ( P_A(p_1^2) + P_A(p_2^2) ) 
            -   16  λ^SAA  P_S   P_A(p_1^2)   P_A(p_2^2)  ,

and

    G_A(p_1^2,p_2^2,q^2)     =    8/p_1^2  p_2^2(   2  L̂_5   +   4  Ĉ_12 (  p_1^2   +   p_2^2   -   q^2   )   -   2  Ĉ_85   (  p_1^2   +   p_2^2   )   +   2   c_d   P_S  
                   -    2  λ_12^S   (p_1^2   +   p_2^2   ) P_S   -    2  λ_17^S   q^2   P_S    
                  -  √(2) ( λ_6^A   -  λ_1^SA  P_S ) ( p_1^2     P_A(p_1^2)   +   p_2^2    P_A(p_2^2) ) )   ,

where

    P_S    =    c_m   -  λ_18^S   q^2/M_S^2   -    q^2  ,                   
    P_A(p^2)    =   F_A   -   2  √(2) λ_17^A   p^2/M_A^2   -   p^2   ,

and M_S and M_A are the mases of the nonet of scalars and axial-vector mesons in the U(3) and chiral limits.
We can now expand our RChT results for the F_A and G_A functions and impose the constraints by Eqs. (<ref>,<ref>,<ref>) and  (<ref>). We get:

    L̂_5   =   Ĉ_12  =  Ĉ_80  =  Ĉ_85  =   0   , 
    λ_6^A     =   λ_16^A   =  λ_12^S   =  λ_16^S   =    0   , 
    λ_6^AA   =    -  F^2/16   F_A^2  , 
    λ_1^SA   =   1/√(2)  F_A ( c_d   -  F^2/8   c_m)   , 
    λ_2^SA   =    -  c_d/2  √(2)  F_A  .

It is interesting to observe that the low-energy couplings of the  Lagrangians vanish. This strengthens the notion of resonance dominance of the chiral couplings.



 §.§ ⟨  S   V_μ  V_ν ⟩


We proceed analogously with the ⟨ S V_μ  V_ν⟩ Green function defined by:

    ( Π^ijk_SVV)_μν  =   i^2  ∫ d^4x   d^4y  
    e^i ( p_1 · x + p_2 · y ) ⟨ 0 | T { S^i(0) V_μ^j(x) V_ν^k(y) } |
    0 ⟩  ,

where

    V^i_μ(x)   =  ( q̅γ_μλ^i/2 q )(x)  ,

and the scalar current as defined in Eq. (<ref>).
In the SU(3) limit it satisfies the Ward identities:

    p_1^μ( Π^ijk_SVV)_μν    =     0   ,
    
    
     p_2^ν( Π^ijk_SVV)_μν    =     0  .

Its general structure is given by:

    ( Π^ijk_SVV)_μν    =      d^ijk B_0 [
       F_V(p_1^2,p_2^2,q^2)   P_μν  +
      G_V(p_1^2,p_2^2,q^2)   Q_μν]   ,

where P_μν and Q_μν have been defined in Eq. (<ref>).
The short-distance behaviour of the ⟨ SVV ⟩ function, at leading order in the momenta
expansion, reads [It is possible to vary the high energy behavior of the Green function as

    lim_λ_1, λ_2 →∞( Π^ijk_SVV)_μν( λ_1 p_1, λ_2 p_2 )
     .

Since λ_1,λ_2 arbitrarily go to infinity, the matching in the short distance region should be fulfilled for each momentum independently.]:

    lim_λ→∞( Π^ijk_SVV)_μν( λ p_1, λ p_2 )
        =     -    d^ijk  B_0 F^2 1/λ^2 1/p_1^2   p_2^2   q^2 [ 2   Q_μν  +  ( p_1^2 + p_2^2 + q^2 ) P_μν]  
           +   O( 1/λ^3)   ,  
    lim_λ→∞( Π^ijk_SVV)_μν( λ p_1, p_2 )     =   
    -2   d^ijk 1/λ Π_VT(p_2^2)/p_1^2  P_μν  +
      O( 1/λ^2)   ,
    lim_λ→∞( Π^ijk_SVV)_μν( p_1, λ p_2 )     =   
    -2  d^ijk 1/λ Π_VT(p_1^2)/p_2^2  P_μν  +
      O( 1/λ^2)   ,  
    lim_λ→∞( Π^ijk_SVV)_μν( λ p_1, q-λ p_1 )     =    O( 1/λ^2)   ,

and Π_VT(p^2) is defined by:

    ( p_ρ g_μσ  -   p_σ g_μρ)  δ^ij Π_VT(p^2)   =  ∫ d^4x   e^ip· x⟨ 0 |   T { V_μ^i (x) ( q̅ σ_ρσ λ^j/2  q ) (0) }  | 0 ⟩  .

Let us compute now the F_V and G_V functions (<ref>) in the RChT formalism. Analogously to the
previous Green function we denote our Lagrangian as:

    L_  =    L_(2)^  +   L_(4)^ +    L_(6)^  +   L_(2)^V   +
      L_(2)^S   +   L_V  ,

where L_(2)^ is defined in Eq. (<ref>), L_(4)^ and L_(6)^ are defined in Eq. (<ref>), L_(2)^V and L_(2)^S are specified in Eq. (<ref>) and L_V includes interaction terms between scalar, vector resonances, and external currents. There is a key difference between the operators needed to match the Green function in the ⟨ S A_μ A_ν⟩ case and the present ones. The Lagrangian L_A only includes  those operators that, upon integration of the resonance, contributes to the ChPT O(p^6) Lagrangian. Contrary to the ⟨ S A_μ A_ν⟩ case, these operators are not enough to achieve the matching in the ⟨ S V_μ V_ν⟩ case. More precisely, if we only include the operators in Table <ref> we would get G_V (p_1^2,p_2^2,q^2) = 0 and, therefore, we would not be able to fulfill the matching. We thus need to include additional operators that are listed in Table <ref>. They have the chiral structure: ⟨ R χ(p^6) ⟩, ⟨ RR χ(p^4) ⟩ and ⟨ RRR χ(p^2) ⟩ and yield contributions to both F_V and G_V.




The complete set of diagrams contributing to ⟨ S V_μ V_ν⟩ is given in Figure <ref>.



The resulting expressions for the F_V and G_V functions are:

    F_V(p_1^2,p_2^2,q^2)     =     -   32  Ĉ_61  -   32  λ_15^S   P_S   +   16  √(2) ( λ_6^V   + λ_3^SV  P_S  ) ( P_V(p_1^2)   +   P_V(p_2^2)   )  
           -    16  ( λ_6^VV +  λ^SVV  P_S  )    P_V(p_1^2)   P_V(p_2^2)  
           -    4  ( ( 2  κ_2^SVV  +   2  κ_3^SVV  + κ_6^SVV) (  p_1^2   +   p_2^2  )  
                -   ( 2  κ_3^SVV  -   4  κ_4^SVV  +   2  κ_5^SVV  +  κ_6^SVV )   q^2
     )   P_S   P_V(p_1^2)   P_V(p_2^2)  
           +   4 √(2) ( 2κ_1^SV-2κ_3^SV+κ_4^SV+κ_5^SV)   P_S  ( p_1^2   P_V(p_2^2)   +   p_2^2   P_V(p_1^2)  )  
           +   4 √(2) ( 2κ_3^SV-κ_4^SV+κ_5^SV)q^2   P_S  (P_V(p_1^2)   +   P_V(p_2^2)  )  
           +   4 √(2) ( 2κ_3^SV+κ_4^SV-κ_5^SV)  P_S  (p_1^2 P_V(p_1^2)   +  p_2^2  P_V(p_2^2)  )  
           +   8   (κ_1^S+2κ_3^S ) q^2   P_S  -   8  ( κ_1^S   + 2  κ_2^S+2 κ_3^S) (  p_1^2   +   p_2^2   )   P_S  
           -    4  ( ( κ_1^VV+2κ_2^VV+2κ_3^VV) (  p_1^2   +   p_2^2  )  
    -   (  κ_1^VV+2κ_3^VV) q^2   )   P_V(p_1^2)   P_V(p_2^2)  
           +    4 √(2) ( 2κ_1^V+2κ_3^V+κ_4^V+κ_5^V) ( p_1^2   P_V(p_1^2)   +   p_2^2   P_V(p_2^2)  )  
           +   4 √(2) ( 2κ_3^V-κ_4^V+κ_5^V)q^2 (P_V(p_1^2)   +   P_V(p_2^2)  )  
           -   4 √(2) ( 2κ_3^V-κ_4^V+κ_5^V) (p_1^2 P_V(p_2^2)   +  p_2^2  P_V(p_1^2)  )    ,

and

    G_V(p_1^2,p_2^2,q^2)     =      8  ( κ_1^SVV  -   2  κ_2^SVV)   P_S   P_V(p_1^2)   P_V(p_2^2)   -  32  κ_2^S   P_S  
           +   8  √(2) ( κ_1^SV  -  κ_2^SV )   P_S  (   P_V(p_1^2)   +   P_V(p_2^2)  ) 
           +   8  √(2) ( κ_1^V  -  κ_2^V )  (   P_V(p_1^2)   +   P_V(p_2^2)  ) 
     -  16  κ_2^VV  P_V(p_1^2)   P_V(p_2^2)   ,

where P_S has been defined in Eq. (<ref>) and

    P_V(p^2)   =  F_V   -   2  √(2) λ_22^V   p^2/M_V^2   -   p^2  ,

with M_V the mass of the nonet of vector resonances in the U(3) and chiral limit.
By imposing the constraints on Eqs. (<ref>,<ref>,<ref>) and (<ref>), we obtain:

    κ_2^S=κ_2^VV    =     0   , 
    κ_1^S  +   2  κ_3^S    =     0   , 
    κ_1^VV  +   2  κ_3^VV    =     0   , 
    κ_1^SV  -  κ_2^SV    =     0   , 
    
    2 κ_3^SV  +  κ_4^SV - κ_5^SV    =     -2√(2) λ_15^S/F_V  , 
    
    2 κ_1^SV  -  2 κ_3^SV  +  κ_4^SV  + κ_5^SV    =     0   , 
    
     2 κ_3^SV  -  κ_4^SV  + κ_5^SV    =    4 λ_6^V/c_m+M_V^2/c_m(2κ_1^V+2κ_3^V+κ_4^V+κ_5^V)   ,
    
    κ_1^V  -  κ_2^V    =     0   , 
    
    2 κ_3^V  -  κ_4^V  + κ_5^V    =    0   , 
    κ_1^V  +  2  κ_3^V  +  κ_5^V   =     -√(2) Ĉ_61/F_V  , 
    κ_1^SVV  -   2  κ_2^SVV    =    F^2/4   c_m   F_V^2 
    
    2 κ_3^SVV  - 4  κ_4^SVV  +  2 κ_5^SVV  +  κ_6^SVV    =     - 4 λ_6^VV/c_m + F^2/4 c_m   F_V^2  ,
    
    
    2  κ_2^SVV  +   2 κ_3^SVV  +  κ_6^SVV    =      -  F^2/4   c_m   F_V^2  -   4√(2)λ_3^SV/F_V - √(2) M_S^2( 2 κ_3^SV  -  κ_4^SV  +  κ_5^SV)/F_V
           - √(2) M_V^2( 2 κ_3^SV  +  κ_4^SV  -   κ_5^SV)/F_V .

Notice that, in this case, the local contribution from L_(6)^, namely Ĉ_61, is not forced to vanish
by the short-distance constraints. Our Lagrangian, defined in Eq. (<ref>),  generates both F_V and G_V functions, and is able to satisfy the short-distance relations.
Incidentally, the matching procedure in Eqs. (<ref>,<ref>) provides an expression for the vector-tensor correlator defined in Eq. (<ref>), namely:

    Π_VT(p^2)   =  B_0   F^2/p^2 - M_V^2  ,

that agrees with the result in <cit.>.



 §.§ RChT coupling constants
 
The relations between the RChT couplings obtained in Eqs. (<ref>,<ref>) rely on the assumptions of short-distance QCD asymptotic behavior and single resonance approximation. We may wonder how reliable are those assumptions. If our implementation of large-N_C was exact (i.e. if we had included an infinite number of resonances) we could argue that our computation should receive ∼33  % one-loop corrections. In practice this is a rough estimate because we cannot evaluate the error introduced by imposing the asymptotic behavior. Because of these uncertainties, one should expect slight modifications to the relations obtained in Eqs. (<ref>,<ref>). In our opinion the largest source of uncertainty arises from the lack of a more thorough implementation of the large-N_C description.
It is well known that the phenomenology of hadron processes indicates that large-N_C is a reasonable assumption for spin-1 related processes, but fails for scalar (vacuum) quantum numbers. [As a general setting, meson-vector form factors are well described in a N_C  →∞ framework in RChT. On the contrary, a resummation of many loops is usually required to provide a reasonable account of scalar form factors.] In this case, higher-order 1/N_C corrections seem to be particularly relevant. Let us consider, for instance, the case of the c_d and c_m couplings in Eq. (<ref>) with the constraints in Eq. (<ref>). One would conclude that in the single resonance approximation we have:

    c_d   =   c_m   =  F/2  .

Taking F = 92.4 we get c_d=c_m = 46.2. However, the phenomenology of different processes (I=1/2 and I=3/2 Kπ s-wave scattering, a_0(980) decay) gives 13     c_d    40 and 30     c_m     100 (see <cit.> and references therein). While the condition 4   c_d   c_m   =   F^2 is rather well satisfied, there seems to be some tension between the phenomenological values of c_d and c_m and the relation c_d=c_m. Given the large uncertainties, we cannot reliably estimate the error of our large-N_C result (<ref>) (in single resonance approximation), but it could be off even by a factor of 3 (for c_d) or 2 (for c_m) in the worst case.
We conclude that our relations in Eq. (<ref>,<ref>) may be affected by errors of similar size to the case above. The order of magnitude is expected to be correct but notable deviations may arise. Unfortunately, we cannot constrain most of the couplings with the present phenomenological status. However we can get reliable estimates in certain couplings, such as c_d and c_m, which appear in the decays of a scalar to two pseudoscalars. We will pursue this in Section <ref>.
In summary, our present knowledge of the hadron scalar spectrum, and its decays, is rather poor <cit.> and the couplings involved are essentially unknown. On one hand, we need to identify which is the spectrum described by the RChT (or any other) framework. On the other hand, we lack the required experimental data to have a general vision of the accuracy of our results. In the next section we will try to clarify part of the phenomenological status of scalar resonances.




§ SCALAR COUPLINGS
 
Which are the, experimentally identified, scalar states present in our Lagrangian? As commented at the end of Section <ref>, there is almost no discussion on the identification of the vector and axial-vector resonances of the RChT Lagrangian. They are, in fact, the lightest hadron resonances in the spectrum with those quantum numbers. Scalars (and glueballs) are different. They carry the vacuum quantum numbers and their identification (for M   2) generates controversy. Here, we will comment first several, more or less agreed,
features and we will propose a scheme.
As discussed in Section <ref> the lightest scalar resonance, namely the isosinglet f_0(500), corresponds to a wide ππ s-wave that does not survive the N_C →∞ limit. Increasing in mass we have K_0^*(700), the isotriplet a_0(980) and the isosinglet f_0(980). The next scalar appears at around 1.4. Hence, naively, one could consider that the first U(3) nonet of scalar resonances is the one with those states: S_L   =  {f_0(500), K_0^*(700), a_0(980), f_0(980) }. Following this scheme, determined by the mass, the next nonet would be: S_H   =  {f_0(1370),K_0^*(1430),a_0(1450),f_0(1500) }.
Until ∼ 2 there is another isosinglet scalar: f_0(1710). Other scalars appear around 2. Needless to say that the physical states do not need to correspond exactly with the basis in the Lagrangian and mixing between those with the same quantum numbers surely arise. If our assumption, relying on the mass, was correct, we could conclude that S_L would correspond to the nonet that vanishes at N_C →∞, as it includes the f_0(500). A thorough analysis in this limit was carried out in Ref. <cit.>. Their conclusion was that the most favored candidates for the leading nonet in the infinite number of colors limit was: S_∞  =  { f_0(980),K_0^*(1430),a_0(1450),f_0(1500)}.
Another aspect of the spectrum of scalars is related with their quark content. This is of no relevance for the RChT Lagrangian: it can allocate any quark content. However it is suitable to collect this information here. We will reduce our comment to [q̅  q] and [q̅  q][q̅  q] states (see <cit.> and references therein). One aspect that distinguishes the quark structure of the nonets is that, in the ideal mixing case, the tetraquark multiplet has an inverted spectrum: the isodoublet is heavier than the isotriplet. We see that this feature (the order in the spectrum) is clearly described by S_L above, while they are essentially degenerated (within errors <cit.>) in the case of S_H. This feature could be the result of a violation of the ideal mixing. There are also other reasons to conclude that the light nonet corresponds to the tetraquark structure while
the heavy one is the usual [q̅ q] <cit.>.
In this section we will identify the nonet of scalar resonances in our RChT Lagrangian with the S_H nonet above. We will also consider the singlet f_0(1710) and a general mixing between the isosinglet fields that generates the physical states, including a possible glueball.  As commented in Subsection <ref>, the phenomenology seems to indicate that the N_C →∞ limit is rather poor when scalars are involved. Hence, in our analysis, we will include subleading contributions into the Lagrangian in order to accommodate the experimental figures within their large errors. This will allow us to get more accurate determinations of the leading c_d and c_m couplings.
Similar studies have been carried out in the last years, see for instance <cit.> and references therein.



 §.§ S → PP: isodoublet and isotriplet decays
 
We will consider a RChT framework with violation of the N_C →∞ limit in the tree level Lagrangian. More precisely, we will consider
terms with more than one trace in flavour space. Previous studies <cit.> have pointed out a non-negligible mixing between the I=1,1/2 states of both nonets S_L and S_H. Hence we will include a mixing between them. The Lagrangian reads:

    L_I=1,1/2     =      c_d^L  ⟨  S_L   u_μ  u^μ ⟩  +  α_L  ⟨  S_L   u_μ ⟩ ⟨ u^μ ⟩  +   c_m^L  ⟨  S_L  χ_+ ⟩
            +   c_d^H  ⟨  S_H   u_μ  u^μ ⟩  +  α_H  ⟨  S_H   u_μ ⟩ ⟨ 
    u^μ ⟩  +   c_m^H  ⟨  S_H  χ_+ ⟩  ,

after diagonalization. This introduces two mixing angles:

    ( [ a_0,L; a_0,H ])     =    ( [   cosφ_a   sinφ_a; - sinφ_a   cosφ_a ])  ( [  a_0(980); a_0(1450) ])     ,  
          
    ( [ K^*_0,L; K^*_0,H ])      =    ( [   cosφ_k   sinφ_k; - sinφ_k   cosφ_k ])  ( [  K_0^*(700); K^*_0(1430) ])      .

The mixing angles φ_a and φ_k are not fixed. In Ref. <cit.> the values quoted are φ_a = π/4 and
φ_k ∼ 0.17  π. We will consider them as free parameters. The lack of data on the FSI phase shifts for the decays of these fields prevents the inclusion of these effects in our analysis. The amplitudes for such decays are collected in Subsection <ref> of Appendix <ref>.



 §.§ S → PP: isosinglet decays
 
As commented before, we are interested in the description of the decays of the f_0(1370), f_0(1510) and f_0(1710). Although we identify the first two as those of the S_H multiplet and the third as a possible glueball, the real situation can be much more cumbersome and the real physical states is surely a non-neglible mixing between the isosinglets of the S_H multiplet (namely S_8,S_0) and an extra singlet (S_1). A general rotation of them will provide the physical states:

    ( [ f_0(1370); f_0(1510); f_0(1710) ])   =   A  ( [ S_8; S_0; S_1 ]) ,

where

    A   =   ( [   cosγcosβcosα - sinγsinα   cosγcosβsinα + sinγcosα                - cosγsinβ;  -sinγcosβcosα - cosγsinα - sinγcosβsinα + cosγcosα                  sinγsinβ;                  sinβcosα                  sinβsinα                      cosβ ]).

Now we set up our RChT framework to describe these decays. Contrary to the first decays, we are not going to consider mixing between the light and heavy multiplets. This would give a complicated setting with many parameters and, as we will conclude, it is not necessary to provide a reasonable description of all the decays.
With these inputs the Lagrangian to study the f_0 → P P decays will be:

    L_I=0,S_1   =    c_d^H  ⟨  S_H   u_μ  u^μ ⟩  +   c_m^H  ⟨   S_H  χ_+ ⟩ 
    +  α_H  ⟨  S_H    u_μ ⟩ ⟨   u^μ ⟩  +  β_H  ⟨  S_H  ⟩ ⟨  u_μ  u^μ ⟩ 
            +  γ_H  ⟨  S_H  ⟩ ⟨   u_μ ⟩ ⟨  u^μ ⟩ 
    +   c_d'   S_1  ⟨  u_μ  u^μ ⟩  +   c_m'   S_1  ⟨ χ_+  ⟩  +  γ'  S_1  ⟨  u_μ ⟩ ⟨   u^μ ⟩  .

Furthermore, as the ππ and KK phase shifts are rather well known <cit.> we also incorporate the
parameterization of final state interactions as described in Appendix <ref>. The amplitudes for these decays are gathered in Subsection <ref> of Appendix <ref>.



 §.§ Results
 
The present experimental determination of the S → PP decay widths is rather poor. Many channels have not been observed or have large errors. 
As a result, we end up with more variables than experimental inputs. However, from our fit we can obtain a general idea of the current landscape.
We will fit our partial widths and ratios with the data collected in the rightmost column of Tables <ref> and <ref>.





We input the masses of the resonances from <cit.>, with the exception of the a_0(980) and f_0(1370). The first one is also fitted due to the sensibility of the results to its decay. For f_0(1370) we take the result put forward by <cit.> in the analysis of its dominant decay into four pions, M_f_0(1370) = 1.395. We take F = 92.4 for the decay constant of the pion.
Our results for the fit are presented in the central column of Tables <ref> and <ref>. As we can see, we obtain a reasonable description of most of the channels (being the clear exception the K_0^*(700) →π K decay). We get a null value
for Γ(f_0(1500) →ηη') since this decay is kinematically forbidden for the central value of the f_0(1500) mass.
The results for masses, couplings and parameters are collected in Table <ref>.



We are going to analyse, in turn, the outcome:


  a) We obtain the mixing angles between the I=0 states, α, β, γ with rather large errors. To illustrate the results let us change to the flavour basis, |S⟩, |N⟩, |G⟩, defined by:

    | S ⟩   ≡    | ss⟩  =   - √(2/3)  | S_8 ⟩ + 1/√(3)  | S_0  ⟩  , 
    
    | N ⟩   ≡   1/√(2)| uu + dd ⟩  =  1/√(3)   | S_8 ⟩ + √(2/3)  | S_0  ⟩ ,

being |G⟩ the singlet glueball. In this basis we have:

    ( [ f_0(1370); f_0(1500); f_0(1710) ])
      =  ( [ -0.82 ± 0.22  0.12 ± 0.49   0.57± 0.16;  0.07 ± 0.48 -0.95 ± 0.24  0.30 ± 0.25;  0.57 ± 0.14  0.29 ± 0.23  0.77 ± 0.09 ]) ( [ N; S; G;   ]) .

From this result we conclude that there is a dominant one-to-one identification between f_0(1370),f_0(1500) and f_0(1710) with N, S and G respectively. Notwithstanding there seems to be also a large mixing between f_0(1370) and f_0(1710) with the N and G states.
Our result agrees with solution II of Ref. <cit.>. Their solution I switches the roles of f_0(1500) and f_0(1710). Different models and different settings can be found in the literature. Our conclusion differs from the one in Ref. <cit.> because although they agree on identifying the f_0(1710) mostly with the glueball, they find that f_0(1370) is dominantly | S ⟩ and f_0(1500) is dominantly |N⟩. This later identification of f_0(1370) is also found in Ref. <cit.>, though with a noticeable four-quark component too. In Ref. <cit.> it was concluded that f_0(1500) was mostly glueball but f_0(1710) was also sharing a large component. Ref. <cit.> provides two scenarios: In one of them f_0(1710) is dominantly glueball; in the other this role corresponds to f_0(1500).
In relation with the mixing between the light and heavy nonets of scalar resonances, our results differ from those of Ref. <cit.>,  and we find a tiny mixing for the a_0 states and an almost inverted situation for the K_0^* states.

  2/ The couplings in Eq. (<ref>), c_d',c_m' and γ', involving the extra singlet S_1 (glueball), are consistent with zero. This indicates that the glueball component only arises through the mixing with the I=0 singlets of the nonet.

  3/ The rest of RChT couplings show an interesting trend. Although with large errors, the expected  1/N_C suppression  between the leading and next-to-leading terms does not seem to be realized. They are essentially of the same order. We verify that both multiplets satisfy the condition in Eq. (<ref>): c_d^L   c_m^L > 0 and c_d^H   c_m^H > 0, but we notice that the relation c_d=c_m is approximately satisfied only by the light multiplet c_d^L ∼ c_m^L. Meanwhile the heavy multiplet deviates from this relation. None of them satisfies, numerically, Eq. (<ref>), though the light multiplet comes close.




§ CONCLUSIONS
 
The phenomenology of the lightest hadron scalars is rather clumsy. The issues of identification of the U(3) nonets, its nature and their decays embrace a thorough research and a large number of publications. Many aspects remain to be understood. In this work we have tried to put some light on the features and problems that have to be taken into account for a Lagrangian description of the scalar sector; in our case  within the Resonance Chiral Theory.
The greater part of the decays of scalar resonances involve the ⟨  S   V_μ  V_ν ⟩ and ⟨  S   A_μ A_ν ⟩ Green functions of QCD currents. We have analysed these within RChT, including the necessary operators in order to fulfill the short-distance requirements determined by the matching in Eq. (<ref>). As a result we found a set of relations between the couplings in our Lagrangian. These should be valid in the N_C →∞ limit and single resonance approximation.
Although the procedure that we have followed has given in the past many  successful predictions, we know that hadron scalar-involved amplitudes are not well behaved in the large-N_C limit. In order to assess our results, we have carried out a fit to S → PP decays in Section <ref>. In the fit we have included subleading contributions in 1/N_C, to analyze the behavior of our RChT description of such decays. The results of our study are indeed pointing out that operators that should be suppressed following large-N_C premises are in fact as relevant as the leading ones. Hence, at least part of the relations between the couplings involving scalars, in the N_C →∞ limit, may be largely violated. We have to stress, though, that the poor, and sometimes confusing, experimental determinations in most of the scalar decays could mislead this conclusion. It will be important to improve the experimental measurements in order to validate this scenario.
As a consequence of our study we also conclude that, within errors, f_0(1370) is dominantly a
|uu + dd⟩ state, f_0(1710) is dominantly a glueball, but both of them also have a noticeable mixing.
The f_0(1500) is dominantly a |ss⟩ state.
The results by other authors vary, however the use of different frameworks make the comparison difficult.
The study of hadron scalar resonances remains an open field. Their spectrum, classification and nature originate a rich debate. The large-N_C framework, already questioned in the study of these decays, does not seem to be the proper setting because of the large size of subleading corrections. However a solid conclusion will only be possible if a better experimental knowledge of the spectrum and decays is achieved.




§ ACKNOWLEDGEMENTS

We wish to thank Gerhard Ecker and Roland Kaiser for their participation in an early stage of this project.
This work has been supported in part by Grants No. FPA2014-53631-C2-1-P, FPA2017-84445-P and SEV-2014-0398 (AEI/ERDF, EU) and by PROMETEO/2017/053 (GV).
Ling-Yun Dai thanks the support from National Natural Science Foundation of China (NSFC) with Grant No. 11805059 and the Fundamental Research Funds for the Central Universities with Grant No. 531107051122. The work of J.F. was supported in part by the Swiss National Science Foundation (SNF) under contract 200021-159720.







§ APPENDICES





§ CHIRAL NOTATION
 
We collect briefly the basic notation used in both ChPT and RChT <cit.>. The Goldstone fields ϕ parameterize the elements u(ϕ) of the
coset space SU(3)_L ⊗ SU(3)_R / SU(3)_V: 


    u(ϕ) = exp{i/√(2) F Φ(ϕ)  } ,

where F is the decay constant of the pion in the chiral limit and

    Φ (ϕ)   =  ∑_i=1^8  λ_i  ϕ_i/√(2)  =  (
    [   1/√(2) π^0 + 1/√(6) η_8                       π^+                       K^+;                       π^- - 1/√(2) π^0 +
    1/√(6) η_8                       K^0;                       K^-                      K̅^0              - 2/√(6) η_8 ])
     ,

with λ_i the Gell-Mann matrices.
The nonlinear realization of SU(3)_L ⊗ SU(3)_R on resonance fields depends on their transformation  properties
under the unbroken SU(3)_V, the flavour group. Here we will consider massive states transforming as octets (R_8) or singlets (R_0),
with R=V,A,S,P for vector, axial-vector, scalar and pseudoscalar fields, respectively. In the large-N_C limit both become degenerate in the chiral limit and we collect them in a nonet field:

    R =  ∑_i=1^8 λ_i  R_i/√(2)   +  R_0/√(3) 1  .

We will use the antisymmetric representation for the spin-1 fields <cit.>.
In order to calculate Green functions of vector, axial-vector and scalar currents, it is
convenient to include external hermitian sources ℓ_μ(x) (left), r_μ(x) (right), s(x) (scalar) and p(x) (pseudoscalar).
With the fundamental building blocks u(ϕ), V_μν, A_μν, S, ℓ_μ, r_μ, s and p, the hadronic Lagrangian
is given by the most general set of monomials invariant under Lorentz, chiral, P and C transformations. At leading order in 1/N_C, the
monomials should be constructed by taking a single trace of products of chiral operators (exceptions to this rule are not of interest for our research). The chiral tensors χ(p^n), i.e. those not including resonance fields, can be labeled according to the chiral power counting. The independent building blocks of lowest dimension are:

    u_μ   =    i { u^†(∂_μ - i r_μ)u -
    u(∂_μ - i ℓ_μ) u^†}              [ O(p)]   ,
    
    χ_±   =     u^†χ u^†± u χ^† u
                                  
      [ O(p^2)]   ,
    
    
    f_±^μν   =    u F_L^μν u^†± u^† F_R^μν u
                                
     [ O(p^2)]    ,
    
    
    h_μν   =   ∇_μ u_ν + ∇_ν u_μ                                 [ O(p^2)]    ,


with

    χ= 2 B_0 (s+ip)   ,                 B_0 = - ⟨ 0 | u u | 0 ⟩/F^2  ,

and non-Abelian field strengths
F_R^μν = ∂^μ r^ν - ∂^ν r^μ -
i[r^μ,r^ν], F_L^μν = ∂^μℓ^ν - ∂^νℓ^μ - i [ℓ^μ,ℓ^ν]. The covariant derivative is
defined by ∇_μ X = ∂_μ X + [Γ_μ,X],
in terms of the chiral connection
Γ_μ = { u^† (∂_μ - i r_μ)u +
u (∂_μ - i ℓ_μ) u^†}/2
for any operator X transforming as an octet of SU(3)_V.
Higher-order chiral tensors can be obtained by taking products of
lower-dimensional building blocks or by acting on them with the
covariant derivative.






§ S → PP DECAY AMPLITUDES
 
The widths of the S → P_1 P_2 decays are given by:

    Γ (S → P_1P_2)  ≡Γ_i   =  λ^1/2(m_S^2, m_P_1^2, m_P_2^2 )/16 N_P_1P_2π M_S^3 |ℳ_S→ P_1P_2|^2    ,

with λ(a,b,c) = (a+b-c)^2-4ab. Notice that N_P_1P_2 is 2 for two identical particles such as π^0π^0,ηη. Here
we have taken into consideration the effect of mass of the final mesons in the phase space.
In Eq. (<ref>) the amplitudes ℳ_SP_1P_2 are given in the following subsections.




 §.§  I = 1, 1/2 decays
 
The couplings and mixing for the decays of a_0(980), a_0(1450), K_0^*(700) and K_0^*(1430), have been defined in
Eqs. (<ref>,<ref>). The decay amplitudes, defined in Eq. (<ref>),
of the isovectors and isodoublets in the S_H multiplet are:

    ℳ_a_0^+→ηπ^+^H     =    -1/√(3) F^2{ (M_a_0^2-m_η^2-m_π^+^2 ) [√(2)cosθ(c_d^Lsinφ_a+c_d^Hcosφ_a) 
                                     -sinθ((3α_L+2c_d^L)sinφ_a
     +(3α_H+2c_d^H)cosφ_a) ] 
                    +   2   m_π^2(√(2)cosθ-2sinθ)(c_m^Lsinφ_a+c_m^Hcosφ_a) } ,
    ℳ_a_0^0→ηπ^0^H     =     -1/√(3) F^2{ (M_a_0^2-m_η^2-m_π^0^2 ) [√(2)cosθ(c_d^Lsinφ_a+c_d^Hcosφ_a) 
                                     -sinθ((3α_L+2c_d^L)sinφ_a
    +(3α_H+2c_d^H)cosφ_a) ]  
                   +  2   m_π^2(√(2)cosθ-2sinθ)(c_m^Lsinφ_a+c_m^Hcosφ_a) } , 
    ℳ_a_0^+→η'π^+^H     =    -1/√(3) F^2{ (M_a_0^2-m_η'^2-m_π^+^2) [√(2)sinθ(c_d^Lsinφ_a+c_d^Hcosφ_a) 
                                     +cosθ((3α_L+2c_d^L)sinφ_a
    +(3α_H+2c_d^H)cosφ_a) ] 
                   +   2   m_π^2(√(2)sinθ+2cosθ)(c_m^Lsinφ_a+c_m^Hcosφ_a) }  ,
    
    ℳ_a_0^0→η'π^0^H    =   -1/√(3) F^2{ (M_a_0^2-m_η'^2-m_π^0^2 ) [√(2)sinθ(c_d^Lsinφ_a+c_d^Hcosφ_a) 
                                     +cosθ((3α_L+2c_d^L)sinφ_a + (3α_H+2c_d^H)cosφ_a) ] 
                   +   2   m_π^2(√(2)sinθ+2cosθ)(c_m^Lsinφ_a+c_m^Hcosφ_a) }  ,  
    ℳ_a_0^+→  K^+K^0^H     =     -1/F^2 ( (M_a_0^2-m_K^+^2-m_K^0^2)(c_d^Lsinφ_a+c_d^Hcosφ_a)
    +2 m_K^2(c_m^Lsinφ_a+c_m^Hcosφ_a ))  , 
    ℳ_a_0^0→  K^+K^-^H     =     -1/√(2) F^2( (M_a_0^2-2m_K^+^2 )(c_d^Lsinφ_a+c_d^Hcosφ_a)
    +2 m_K^2(c_m^Lsinφ_a+c_m^Hcosφ_a ) )  , 
    ℳ_a_0^0→  K^0K^0^H     =     -1/√(2) F^2( (M_a_0^2-2m_K^0^2)(c_d^Lsinφ_a+c_d^Hcosφ_a) +2 m_K^2(c_m^Lsinφ_a+c_m^Hcosφ_a))   ,


    ℳ_K_0^*^+→  K^+π^0^H     =     -1/√(2)F^2{(c_d^Lsinφ_k+c_d^Hcosφ_k) (M_K_0^*^+^2-m_K^+^2-m_π^0^2)
                   +(c_m^Lsinφ_k+c_m^Hcosφ_k) (m_K^2+m_π^2) }  , 
    ℳ_K_0^*^+→  K^0π^+^H     =     -1/ F^2{(c_d^Lsinφ_k+c_d^Hcosφ_k) (M_K_0^*^+^2-m_K^0^2-m_π^+^2)
                   +(c_m^Lsinφ_k+c_m^Hcosφ_k) (m_K^2+m_π^2) }  , 
    ℳ_K_0^*^0→  K^0π^0^H     =     -1/√(2)F^2{(c_d^Lsinφ_k+c_d^Hcosφ_k) (M_K_0^*^+^2-m_K^+^2-m_π^0^2)
                   +(c_m^Lsinφ_k+c_m^Hcosφ_k) (m_K^2+m_π^2) }  , 
    ℳ_K_0^*^0→  K^+π^-^H     =    -1/ F^2{(c_d^Lsinφ_k+c_d^Hcosφ_k) (M_K_0^*^+^2-m_K^0^2-m_π^+^2)
                   +(c_m^Lsinφ_k+c_m^Hcosφ_k) (m_K^2+m_π^2) }  ,
    
    ℳ_K_0^*^+→  K^+η^H     =    -1/ 2√(3)F^2{ (M_K_0^*^+^2-m_K^+^2-m_η^2)[-√(2)cosθ(c_d^Lsinφ_k+c_d^Hcosφ_k) 
                        -2sinθ( (3α_L+2c_d^L)sinφ+(3α_H+2c_d^H)cosφ) ]
                        +(c_m^Lsinφ_k+c_m^Hcosφ_k) [3√(2)cosθ m_π^2-5√(2)cosθ m_K^2-8sinθ m_K^2] }  , 
    ℳ_K_0^*^0→  K^0η^H     =    -1/ 2√(3)F^2{ (M_K_0^*^0^2-m_K^0^2-m_η^2) [-√(2)cosθ(c_d^Lsinφ_k+c_d^Hcosφ_k) 
                        -2sinθ( (3α_L+2c_d^L)sinφ+(3α_H+2c_d^H)cosφ) ]
                        +(c_m^Lsinφ_k+c_m^Hcosφ_k) [3√(2)cosθ m_π^2-5√(2)cosθ m_K^2-8sinθ m_K^2] }  .

Those for the decays of the S_L multiplet are:

    ℳ_a_0^+→ηπ^+^L     =    -1/√(3) F^2{  (M_a_0^2-m_η^2-m_π^+^2) [√(2)cosθ(c_d^Lcosφ_a-c_d^Hsinφ_a) 
                                    -sinθ((3α_L+2c_d^L)cosφ_a
    -(3α_H+2c_d^H)sinφ_a) ] 
                   +   2   m_π^2(√(2)cosθ-2sinθ)(c_m^Lcosφ_a-c_m^Hsinφ_a) } ,
    ℳ_a_0^0→ηπ^0^L     =    -1/√(3) F^2{  (M_a_0^2-m_η^2-m_π^0^2) [√(2)cosθ(c_d^Lcosφ_a-c_d^Hsinφ_a) 
                                     -sinθ((3α_L+2c_d^L)cosφ_a
    -(3α_H+2c_d^H)sinφ_a) ] 
                   +   2   m_π^2(√(2)cosθ-2sinθ)(c_m^Lcosφ_a-c_m^Hsinφ_a) } , 
    ℳ_a_0^+→η'π^+^L     =    -1/√(3) F^2{  (M_a_0^2-m_η'^2-m_π^+^2) [√(2)sinθ(c_d^Lcosφ_a-c_d^Hsinφ_a) 
                                     +cosθ((3α_L+2c_d^L)cosφ_a
     -(3α_H+2c_d^H)sinφ_a) ] 
                   +   2   m_π^2(√(2)sinθ+2cosθ)(c_m^Lcosφ_a-c_m^Hsinφ_a) } , 
    ℳ_a_0^0→η'π^0^L     =    -1/√(3) F^2{  (M_a_0^2-m_η'^2-m_π^0^2)[√(2)sinθ(c_d^Lcosφ_a-c_d^Hsinφ_a) 
                                     +cosθ((3α_L+2c_d^L)cosφ_a
    -(3α_H+2c_d^H)sinφ_a) ] 
                   +   2   m_π^2(√(2)sinθ+2cosθ)(c_m^Lcosφ_a-c_m^Hsinφ_a) }  , 
    ℳ_a_0^+→  K^+K^0^L     =     -1/F^2 ( (M_a_0^2-m_K^+^2-m_K^0^2)(c_d^Lcosφ_a-c_d^Hsinφ_a) 
                 +2 m_K^2(c_m^Lcosφ_a-c_m^Hsinφ_a) )  , 
    ℳ_a_0^0→  K^+K^-^L     =     -1/√(2) F^2( (M_a_0^2-m_K^+^2-m_K^0^2)(c_d^Lcosφ_a-c_d^Hsinφ_a)
    
                      +2 m_K^2(c_m^Lcosφ_a-c_m^Hsinφ_a) )  , 
    ℳ_a_0^0→  K^0K^0^L     =     -1/√(2) F^2( (M_a_0^2-2 m_K^0^2)(c_d^Lcosφ_a-c_d^Hsinφ_a) 
                     +2 m_K^2(c_m^Lcosφ_a-c_m^Hsinφ_a) )    ,


    ℳ_K_0^*^+→  K^+π^0^L     =     -1/√(2)F^2{(c_d^Lcosφ_k-c_d^Hsinφ_k) (M_K_0^*^+^2-m_K^+^2-m_π^0^2)
                    +(c_m^Lcosφ_k-c_m^Hsinφ_k) (m_K^2+m_π^2) }  , 
    ℳ_K_0^*^+→  K^0π^+^L     =     -1/ F^2{(c_d^Lcosφ_k-c_d^Hsinφ_k) (M_K_0^*^+^2-m_K^0^2-m_π^+^2)
                +(c_m^Lcosφ_k-c_m^Hsinφ_k) (m_K^2+m_π^2)}  , 
    ℳ_K_0^*^0→  K^0π^0^L     =     -1/√(2)F^2{(c_d^Lcosφ_k-c_d^Hsinφ_k) (M_K_0^*^+^2-m_K^+^2-m_π^0^2)
                   +(c_m^Lcosφ_k-c_m^Hsinφ_k) (m_K^2+m_π^2) }  , 
    ℳ_K_0^*^0→  K^+π^-^L     =    -1/ F^2{(c_d^Lcosφ_k-c_d^Hsinφ_k) (M_K_0^*^+^2-m_K^0^2-m_π^+^2)
               +(c_m^Lcosφ_k-c_m^Hsinφ_k) (m_K^2+m_π^2) }  ,
    
    ℳ_K_0^*^+→  K^+η^L     =    -1/ 2√(3)F^2{ (M_K_0^*^+^2-m_K^+^2-m_η^2)[-√(2)cosθ(c_d^Lcosφ_k-c_d^Hsinφ_k) 
                        +2sinθ( (3α_H+2c_d^H)sinφ-(3α_L+2c_d^L)cosφ) ]
                        +(c_m^Lcosφ_k-c_m^Hsinφ_k) [3√(2)cosθ m_π^2-5√(2)cosθ m_K^2-8sinθ m_K^2] }  , 
    ℳ_K_0^*^0→  K^0η^L     =    -1/ 2√(3)F^2{ (M_K_0^*^0^2-m_K^0^2-m_η^2)[-√(2)cosθ(c_d^Lcosφ_k-c_d^Hsinφ_k) 
                        +2sinθ( (3α_H+2c_d^H)sinφ-(3α_L+2c_d^L)cosφ) ]
                        +(c_m^Lcosφ_k-c_m^Hsinφ_k) [3√(2)cosθ m_π^2-5√(2)cosθ m_K^2-8sinθ m_K^2 ] }  .





 §.§ I = O decays
 
In the following amplitudes, i=1,2,3 and f_1 ≡ f_0(1370), f_2 ≡ f_0(1500) and f_3 ≡ f_0(1710):

    ℳ_f_i→π^+π^-, π^0 π^0    =    M_f_i^2-2 m_π^2/3   F^2 [√(6)   c_d^H  ( a_i1 + √(2)   a_i2)   +   6  √(3) β_H   a_i2  +    6   c_d'    a_i3]  
           +   4  m_π^2/F^2[ c_m^H/√(6) ( a_i1   +  √(2)  a_i2)   +   c_m'   a_i3] 
    
    ℳ_f_i→  K^+K^-, K^0 K^0    =    M_f_i^2-2 m_K^2/6   F^2 [√(6)   c_d^H  ( -    a_i1  +   2  √(2)   a_i2)   +   12  √(3) β_H   a_i2  +    12   c_d'    a_i3]  
           +  m_K^2/3   F^2[ √(6)   c_m^H ( -   a_i1   +   2  √(2)  a_i2)   +   12    c_m'   a_i3] ,


    ℳ_f_i→ηη    =    M_f_i^2-2 m_η^2/6   F^2 [ - √(6)  c_d^H   a_i1  +   12   c_d'   a_i3  +  
    2  √(3)  a_i2( 3 α_H + 6 β_H + 2 c_d^H+ 9 γ_H )   +   18  γ_H   a_i3. 
          .              -  cos 2 θ [ √(6)  c_d^H   a_i1  +   6  √(3)  a_i2  (  α_H   +    3  γ_H)   +   18    a_i3 γ' ] . 
          .              -   2  √(3) sin 2 θ ( 3  α_H   + 2   c_d^H )   a_i1] 
           +  1/9   F^2[ √(6)  c_m^H   a_i1 [ 3   ( 3 m_π^2 - 4 m_K^2 )   +   (m_π^2-4 m_K^2 ) ( cos 2 θ  +   2  √(2) sin 2 θ ) ] . 
                  . +   4  ( √(3)  c_m^H   a_i2 + 3 c_m'   a_i3) ( 3   m_K^2   +   (m_K^2 - m_π^2) ( cos 2 θ  +   2  √(2) sin 2 θ ) ) ] 
    ℳ_f_i→ηη'    =    M_f_i^2 - m_η^2 - m_η'^2/6   F^2 [ 2  √(3)  (3  α_H   +   2   c_d^H )   a_i1 cos 2 θ  -  . 
                          . -  ( √(6)  c_d^H   a_i1  +   6  √(3)  (α_H + 3 γ_H)   a_i2  +   18   a_i3 γ' ) sin 2 θ ] 
           +  1/9   F^2 ( 2  √(2)cos 2 θ  -  sin 2 θ) [ √(6)  c_m^H   a_i1  ( 4 m_K^2 - m_π^2)
    . 
                                      .  -   4 (m_K^2 - m_π^2) ( √(3)  c_m^H   a_i2  +   3   c_m'   a_i3) ] .

Here a_ij are the matrix elements of the A matrix in Eq. (<ref>).
In Eq. (<ref>), θ is the η-η' mixing angle defined by:

    ( [  η; η' ]) =  ( [   cosθ - sinθ;   sinθ   cosθ ])  ( [ η_8; η_0 ]) .







§ FINAL STATE INTERACTIONS IN F_I → PP DECAYS
 
We know that I=0, S=0 amplitudes have large FSI effects. Unfortunately we only have reliable information on the ππ and K K phase-shifts. Hence we can only consider the FSI effects in the decays with those final states.
We would expect that I=1 or I=1/2 final states should be less affected and, therefore, we will consider only f_i → PP decays (i=1,2,3 as in Appendix <ref>), with P=π,K.
Following Refs. <cit.>,  we can parameterize:

    ( [   ℳ_f_i →ππ; ℳ_f_i → K K ])^  =  √(S)( [   ℳ_f_i →ππ; ℳ_f_i → K K ])^  ,

where

    √(S)  =   O^T √(S_)  O  ,

with

    S_  =  ( [  e^2 i δ_ππ^I=0               0;               0 e^2 i δ_K K^I=0 ])          ,          O  =  ( [   cosω   sinω; - sinω   cosω ]) .

Here ω should be a new parameter to fit. For the phase-shifts we will only need δ_ππ^I=0(M_f_i^2) and
δ_K K^I=0(M_f_i^2), because in two-body decays always s=M^2 being M the mass of the decaying particle. The phase shifts are given by the extended K-matrix fit following <cit.>, up to 1.8. We will consider the results in Table <ref>.





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