Higher-order chiral Lagrangians with vector meson nonet in different representations

Quantum chromodynamics (QCD) is the fundamental theory describing strong interactions. However, due to non-perturbative effects, QCD often cannot provide precise predictions in the low-energy region. To study low-energy QCD involving pseudoscalar mesons, one can employ an effective theory at the hadronic level, leveraging chiral symmetry and its spontaneous breaking. This approach is known as chiral perturbation theory (ChPT) [1−5]. ChPT serves as an effective field theory of QCD at low energies. In ChPT, the chiral Lagrangian encompasses all effective interactions that respect Lorentz invariance, chiral symmetry, charge conjugation, and parity. Rather than an expansion in terms of the large strong coupling constant, ChPT is formulated as an expansion in the typical momentum scale (p). According to Weinberg's power-counting rule, tree-level and loop diagrams contribute terms proportional to powers of $ p/\Lambda $, where Λ represents the chiral symmetry-breaking scale. Thus, calculating a physical quantity with a specific accuracy requires only a finite number of interactions. Currently, pseudoscalar mesonic chiral Lagrangians have been constructed up to $ {\cal{O}}(p^8) $ in both $ S U(2) $ and $ S U(3) $ symmetries [4−10].

ChPT can be extended to higher-energy ranges to incorporate additional particles, such as vector mesons, which exhibit unique properties. With masses that fall between those of pseudoscalar mesons and baryons, vector mesons present challenges for consistent power counting [11, 12]. In the literature, vector mesons are sometimes treated as light degrees of freedom [13−28], while other studies treat them as heavy degrees of freedom [29−34].

Several representations are used to describe vector mesons: the traditional vector-field (or matter-field/Proca field) representation [35, 36], the antisymmetric tensor-field representation [4, 36, 37], the hidden local symmetry (HLS) representation [38−46], and the massive Yang-Mills representation [47−51]. This work focuses on the first three, as the massive Yang-Mills representation is seldom used today.

While distinct, these representations are interrelated. The vector-field representation, the most traditional, has its connections to other representations thoroughly documented in Refs. [36, 45, 52−54]. In the tensor-field representation, vector mesons are represented by antisymmetric tensors, which transform homogeneously under chiral symmetry [36, 37, 52, 54]. The vector-field and tensor-field representations share many similarities, with some studies treating them together [36, 52, 54]. The tensor-field representation offers certain advantages: it provides a straightforward generalization of the Dirac equation for spin-1/2 particles to spin-1 particles and corresponds to the $ (1,0) \oplus (0,1) $ representation of the Lorentz group, naturally encompassing two spin-1 degrees of freedom among its six degrees of freedom. In contrast, the vector-field representation uses the (1/2, 1/2) Lorentz representation, requiring an auxiliary condition to eliminate the unwanted spin-0 component. The tensor-field Lagrangian is also simpler in form than that of the vector-field representation. Further details on the relations between these two representations can be found in Refs. [36, 43, 52−54].

In the HLS representation, an artificial local symmetry is introduced into the nonlinear sigma model through the choice of specific field variables, which can be removed by selecting a gauge, such as the unitary gauge. This gauge symmetry is not a true symmetry but a redundancy in the description. The HLS representation's connections with other representations are discussed in Refs. [36, 43, 45, 55]. The massive Yang-Mills representation is a particular gauge choice within the generalized HLS model, determined by specific parameter values, with further relations to other representations elaborated in Refs. [36, 56−58].

Two types of equivalence relations exist among these representations [54, 55]. The first type indicates that different representations share the same structure without integrating out vector mesons [43, 45], while the second type arises when different representations yield equivalent results after vector fields are integrated out [36, 52, 54]. The first type of equivalence applies at the level of resonance Lagrangians, whereas the second applies at the level of pseudoscalar mesonic chiral Lagrangians. Importantly, the first-type equivalences imply the second-type, but the reverse is not necessarily true.

After integrating out the vector fields and comparing with pseudoscalar mesonic chiral Lagrangians [4, 5, 7, 59], the equivalence of these representations at the level of pseudoscalar mesonic chiral Lagrangians becomes evident [36, 52, 54]. A recent approach, the first-order formalism, enables the calculation of resonance contributions to low-energy constants (LECs) in ChPT at next-to-next-to-leading order (NNLO) by integrating out vector fields [54]. This method underscores the equivalence between the vector-field and tensor-field representations at the NNLO level.

Currently, although there is extensive research on ChPT with vector mesons, most studies focus on treating vector mesons as resonance states to investigate the properties of pseudoscalar mesons [60−66]. For such purposes, the lower-order chiral Lagrangian is usually sufficient. However, some studies concentrate on vector mesons themselves, analyzing aspects such as their masses [30, 67−72], electromagnetic properties [23, 73], decay constants [28, 74], radiative decays [16, 75], and scattering with other particles at high energies [22, 46, 76−78]. Research on vector mesons at the next-to-leading order (NLO) has also been conducted, and studies at the NNLO are emerging [63−66, 71].

To date, vector mesonic chiral Lagrangians have been developed up to NLO in the tensor-field and HLS representations [36, 38, 42, 63]. However, for studies focused on vector mesons themselves, lower-order Lagrangians may not offer the required accuracy. In this paper, we systematically construct chiral Lagrangians for the vector-field, tensor-field, and HLS representations, extending to NNLO and covering both the normal and anomalous cases, and the strong equivalence between the tensor-field and HLS representations is also analyzed. Additionally, to expand the application scope of the chiral Lagrangian, the singlet vector meson is also considered, i.e., the chiral Lagrangian is expanded to the vector meson nonet. This Lagrangian can be applied not only to low-energy vector mesons but also to heavy-vector-meson fields. For example, when $ J/\Psi $ is treated as a vector meson singlet, it can be used to study the interaction between $ J/\Psi $ and pseudoscalar mesons [79−81].

While multi-vector-meson interactions have been explored in earlier research, the complexity of these interactions in the vector-field and tensor-field representations has often restricted studies to single-vector-meson vertices. Consequently, our analysis concentrates on single-vector-meson vertices within these representations. Although higher-order chiral Lagrangians contain numerous terms, only a few play a significant role in specific physical phenomena. Advances in technology and the development of specialized software and computer algebra systems now enable efficient handling of the computational challenges posed by these complex terms.

This paper focuses on a tree-level analysis. Extending the discussion to include loop diagrams poses significant challenges, as the presence of vector mesons disrupts the conventional power-counting rule [68]. Specifically, loop diagrams introduce large imaginary components that violate power counting, and these terms cannot be absorbed using standard counterterms or renormalized parameters within the traditional renormalization scheme. Currently, the only viable solution to this problem is the complex-mass scheme [69, 82−91]. In this scheme, the masses and widths of vector mesons are assigned power-counting orders of $ {\cal{O}}(p^0) $ and $ {\cal{O}}(p^1) $, respectively. The power-counting rule for a vector meson propagator depends on the external momenta: it is counted as $ {\cal{O}}(p^0) $ if it carries large external momentum and $ {\cal{O}}(p^1) $ otherwise. Additionally, derivatives acting on vector mesons are assigned an order of $ {\cal{O}}(p^0) $. Meanwhile, the power counting for pseudoscalar mesons remains consistent with standard pseudoscalar mesonic ChPT. In the complex-mass scheme, power-counting violations can be controlled by introducing additional counterterms, albeit with more complex parameter structures. This approach preserves the power-counting rule, allowing for the application of the chiral expansion even at the loop level.

This paper is structured as follows. Sec. II provides a brief overview of pseudoscalar and vector mesons in ChPT. Secs. III to V detail the methods for constructing chiral Lagrangians with vector mesons in the vector-field, tensor-field, and HLS representations, respectively. Sec. VI outlines the resulting chiral Lagrangians, while Sec. VII examines the equivalence between the HLS and tensor-field representations. Finally, Sec. VIII provides a summary and some comments.

In QCD, the Lagrangian with external sources is

where $ \mathscr{L}_{{\rm{QCD}}}^{0} $ is the original QCD Lagrangian, and q denotes the light quark fields. Here, $ v^{\mu} $, $ a^{\mu} $, s, and p represent the vector, axial-vector, scalar, and pseudoscalar sources, respectively. For simplicity, the tensor source and the θ term are omitted. To facilitate comparison with the pseudoscalar mesonic chiral Lagrangian, $ a^{\mu} $ is considered traceless, and $ v^{\mu} $ is allowed to be non-traceless only in the two-flavor anomalous part; in other contexts, it is assumed to be traceless.

If the masses of the light quarks are neglected, the QCD Lagrangian exhibits a global $ S U(N_f)_L \times S U(N_f)_R $ chiral symmetry, where $ N_f = 2 $ or $ 3 $. This symmetry spontaneously breaks to an $ S U(N_f)_V $ symmetry, resulting in the emergence of pseudoscalar Goldstone bosons associated with the symmetry breaking. These bosons correspond to the lowest pseudoscalar mesons, represented by U. Under a chiral transformation, U transforms as $ U \rightarrow g_L U g_R^{\dagger} $, where $ g_R $ and $ g_L $ are elements of the $ S U(N_f)_L $ and $ S U(N_f)_R $ chiral groups, respectively.

For convenience, an auxiliary field u is introduced to describe the pseudoscalar meson fields, with $ u^2 = U $ [92, 93]

where $ F_0 $ is a low-energy constant, and $ \pi(x) $ represents pseudoscalar meson fields,

where $ \lambda_i $ are the Gell-Mann matrices. The chiral transformation of the pseudoscalar meson fields $ \pi(x) $ is nonlinear, often referred to as a nonlinear realization. Under a chiral transformation, u transforms as $ u \rightarrow g_L u h^{\dagger} = h u g_R^{\dagger} $, where h is a compensator field dependent on the meson fields. This transformation property of u facilitates the construction of chiral Lagrangians, which we will employ in this paper. For further details on ChPT, readers may consult Refs. [92−94] and the references therein.

The vector meson nonet is represented by a $ 3 \times 3 $ matrix in the vector-field representation,

The vector field $ V^\mu $ transforms as $ V^\mu\rightarrow h V^{\mu} h^{\dagger} $ under chiral rotations. In the tensor-field representation, $ V^\mu $ is replaced by an antisymmetric tensor $ W^{\mu\nu} $, where each element of $ V^{\mu} $ corresponds to an antisymmetric tensor field. The HLS representation is more complex and will be discussed in detail in Sec. V.

To account for finer effects such as $ \rho^0-\omega $ mixing [95] or $ \omega-\phi $ mixing [96], $ \rho^0 $, $ \omega_8 $, and $ \omega_0 $ should be included in the mixing. For ideal mixing,

This paper does not assume isospin symmetry. If one considers isospin breaking, such as the mass difference between $ \rho^0 $ and $ \rho^\pm $, then $ \rho^0 $, ω, and ϕ mixing must be included, and s should be replaced by $ {\rm{diag}}(m_u, m_d, m_s) $ in the calculation [30, 67].

The following three sections discuss the construction of chiral Lagrangians with vector mesons in different representations. In this paper, we restrict our focus to single-vector-meson vertices within the vector-field and tensor-field representations.

In addition to the vector meson fields, the building blocks in the vector-field representation are identical to those in the pseudoscalar mesonic chiral Lagrangians [7, 59].

where $ r^{\mu}=v^{\mu}+a^{\mu} $, $ \ell^{\mu}=v^{\mu}-a^{\mu} $, $ \chi=2 B_0(s+{\rm i} p) $, $ F_{R}^{\mu \nu}= \partial^{\mu} r^{\nu}-\partial^{\nu} r^{\mu}- {\rm i}\left[r^{\mu}, r^{\nu}\right] $, $ F_{L}^{\mu \nu}=\partial^{\mu} \ell^{\nu}-\partial^{\nu} \ell^{\mu}- {\rm i}\left[\ell^{\mu}, \ell^{\nu}\right] $, and $ B_0 $ is a constant related to the quark condensate. These building blocks exhibit the same chiral rotation properties as $ V^{\mu} $, i.e., $ X \rightarrow h X h^\dagger $, where X represents any building block. The covariant derivative is defined by

where $ \Gamma_{\mu} $ represents the chiral connection,

To construct the vector mesonic chiral Lagrangians, the vector fields $ V^\mu $, X, and their derivatives are considered. Some studies also introduce a building block $ \hat{V}^{\mu\nu}=\nabla^{\mu}V^{\nu}-\nabla^{\nu}V^{\mu} $ [36, 54]. However, since only single-vector-meson vertices are considered, the covariant derivatives acting on $ V^{\mu} $ can be transferred to other building blocks through partial integration, as will be discussed in Eq. (10). Therefore, aside from the kinetic term, this building block is not included.

In addition to chiral symmetry, chiral Lagrangians must also be invariant under parity (P), charge conjugation (C), and Hermitian conjugation (h.c.) transformations. The transformation properties of these building blocks have been extensively discussed in the literature [7, 36, 37, 59]. ChPT is typically expanded in powers of the momentum (p) of the external source, denoted as $ {\cal{O}}(p^n) $, where n is referred to as the chiral dimension (abbreviated as Dim). Table 1 summarizes these properties for reference.

The vector meson masses are approximately [97]

Their masses fall between those of the pseudoscalar mesons and baryons. These mesons can be considered either light or heavy degrees of freedom, depending on the context. In this paper, we assume $ V^\mu={\cal{O}}(p^0) $. The chiral dimensions of the other building blocks are consistent with those in the pseudoscalar mesonic chiral Lagrangians [11, 45]. Table 1 provides a summary of these power-counting rules.

Furthermore, each covariant derivative applied to the first six building blocks contributes an additional $ {\cal{O}}(p^1) $ order. For the anomalous part, the Levi-Civita tensor $ \varepsilon^{\mu\nu\lambda\rho} $ is also necessary. The aforementioned power-counting rule is effective at the tree level. For loop contributions, the complex-mass scheme should be employed, as discussed in the introduction.

The chiral Lagrangian must satisfy all the symmetries discussed above. Constructing a complete set that adheres to these symmetries is straightforward; however, the terms in this set are typically not linearly independent. To achieve a linearly independent chiral Lagrangian, the following linear constraint relations must be considered [6−9, 36, 37].

(i) Partial Integration. Given that derivatives applied to the entire Lagrangian can be disregarded, we obtain the following identity:

where A, B, and C are generic building blocks, ''$ \langle\cdots \rangle $'' denotes the flavor trace, and ''$\cdots $'' represents additional blocks as necessary. This relation indicates that one of the terms in Eq. (10) is redundant and can be omitted. Since our focus is on single-vector-meson vertices, the covariant derivatives initially applied to $ V^\mu $ can be systematically transferred to the other building blocks. This transfer eliminates the necessity for terms involving $ \hat{V}^{\mu\nu} $ or $ \nabla^\mu V^\nu $.

(ii) Equations of Motion (EOMs). The leading-order(LO) EOMs are crucial for the construction of chiral Lagrangians [7]. For pseudoscalar mesons, the LO EOM is

With this relation, $ \nabla^{\mu} u_{\mu} $ can be substituted with alternative expressions, thereby excluding it from the chiral Lagrangian.

For vector mesons, the LO Lagrangian is provided by [36]

This Lagrangian results in the Klein-Gordon equation and an auxiliary condition:

where ''$ \doteq $'' denotes equivalence up to higher-order terms or terms of the same order with fewer covariant derivatives. This notation is meaningful because terms of the same order but with fewer derivatives are already included according to our procedure for listing allowable terms. These equations imply that $ \nabla_{\nu} \nabla^{\nu} V^{\mu} $ and $ \nabla_\mu V^\mu $ can be transformed into alternative forms. By applying partial integration, Eq. (13) can be reformulated as

where O and $ O_{\mu} $ represent any products of building blocks with the appropriate indices. The omitted terms do not compromise the structural integrity of the chiral Lagrangian.

(iii) Bianchi Identity. Utilizing the definitions provided in Sec. III.A, the Bianchi identity is derived as follows:

where, for any building block X,

Two specific cases are

(iv) Schouten Identity. In the anomalous sections, the Levi-Civita tensor $ \epsilon^{\mu \nu \alpha \beta} $ is utilized. As there is no totally antisymmetric fifth-order tensor in four-dimensional spacetime, the Schouten identity implies

where $ S^{\mu} $ denotes any operator.

(v) Cayley-Hamilton Relations. All building blocks are $ N_f \times N_f $ matrices in flavor space. The Cayley-Hamilton theorem establishes relations among these matrices.

For SU(2),

where A and B are any $ 2 \times 2 $ matrices.

For SU(3),

where A, B, and C are arbitrary $ 3 \times 3 $ matrices.

The tensor-field representation is similar to the vector-field representation. The building blocks are nearly identical to those in Table 1, except that $ V^\mu $ is substituted with $ W^{\mu\nu} $. The transformation properties and power-counting rules for $ W^{\mu\nu} $ are also provided in Table 1.

In the tensor-field representation, the linear relations are similar to those in the vector-field representation, with the main distinction being the EOMs for vector mesons. The tensor field $ W^{\mu\nu} $ is antisymmetric, satisfying $ W^{\mu\nu} = -W^{\nu\mu} $. Utilizing the methods described in Refs. [37, 68, 98], Eq. (13) is replaced by

Additionally, an auxiliary condition is provided by [37, 52, 68, 98, 99]

For historical reasons, we adhere to the convention in Eq. (23), where the degrees of freedom $ W^{ij} $ are fixed, whereas $ W^{0i} $ are dynamic. Through partial integration, Eq. (23) results in the following two linear relations:

where $ R^{\mu\nu\rho} $ denotes an operator with three indices, and $ C^{\mu\nu} $ represents an operator with two indices.

The HLS representation differs significantly from the vector-field and tensor-field representations. This section offers a brief overview; further details are available in Refs. [38−40, 42, 43, 45, 100].

In the HLS representation, vector mesons are conceptualized as gauge fields, with their masses originating from the Brout-Englert-Higgs mechanism. The HLS framework is constructed on the $ G_{{\rm{global}}} \times H_{{\rm{local}}} $ symmetry, where $ G = S U(N_f)_L \times S U(N_f)_R $ signifies the global symmetry, and H represents the HLS. This $ G_{{\rm{global}}} \times H_{{\rm{local}}} $ symmetry can be broken down into an H symmetry, which corresponds to flavor symmetry. The U field from ChPT can be decomposed into two fields, $ \xi_L $ and $ \xi_R $, as

The two fields, $ \xi_L $ and $ \xi_R $, satisfy the following transformation properties under the $ G_{{\rm{global}}}\times H_{{\rm{local}}} $ symmetry:

where $ h(x) \in H_{{\rm{local}}} $ and $ g_{L,R} \in S U(N_f)_L \times S U(N_f)_R $. Two Maurer-Cartan 1-forms are

Their chiral transformation properties under the full symmetry are

The relations between $ \alpha_\perp^{\mu} $, $ \alpha_\parallel^{\mu} $, and the building blocks in the vector-field and tensor-field representations are $ \alpha_\perp^{\mu} = \dfrac{1}{2} u^{\mu} $ and $ \alpha_\parallel^\mu = {\rm i} \Gamma^{\mu} $. The covariant derivatives of $ \xi_{L,R} $ are expressed as

where $ K^{\mu} $ are the gauge fields associated with $ H_{{\rm{local}}} $. Under a chiral transformation, $ K^{\mu} $ transforms as

Using $ K^{\mu} $ fields, the covariant 1-forms are defined as

The quantity $ \hat{\alpha}_{\parallel}^{\mu} $ is associated with the vector field $ V^{\mu} $ in the standard vector-field representation as follows:

where ξ is a parameter associated with the redefinition of $ V^{\mu} $ in the HLS representation. The chiral transformation properties of $ \hat{\alpha}_{\perp}^{\mu} $ and $ \hat{\alpha}_{\parallel}^{\mu} $ are as follows:

The field strength of the HLS gauge boson is defined as

The LO chiral Lagrangian in this representation is

where $ F_\pi $ and $ F_\sigma $ are decay constants, g is the HLS gauge coupling constant, and $ \hat{\chi}_+ $ is defined in terms of the scalar and pseudoscalar external sources as follows:

The LO chiral Lagrangian comprises four essential building blocks: $ \hat{\alpha}_\perp^{\mu} $, $ \hat{\alpha}_\parallel^{\mu} $, $ V^{\mu\nu} $, and $ \hat{\chi}_+ $. However, these building blocks alone are insufficient for constructing higher-order chiral Lagrangians. To address this limitation, additional terms, including $ \hat{\chi}_- $, as well as the following building blocks, are required:

These equations yield the symmetric and antisymmetric indices for $ D^\mu \alpha_\perp^\nu $ and $ D^\mu \alpha_\parallel^\nu $. At NLO, only the antisymmetric building blocks, $ {\cal{A}}^{\mu\nu} $ and $ {\cal{V}}^{\mu\nu} $, are present, as noted in Ref. [45]. At higher orders, the symmetric forms of $ D^\mu \alpha_\perp^\nu $ and $ D^\mu \alpha_\parallel^\nu $ are also required. This paper denotes these symmetric tensors as $ {\cal{H}}^{\mu\nu} $ and $ H^{\mu\nu} $, respectively. Additionally, an alternative set of building blocks has been proposed in the literature [46], and we adopt the former convention for consistency. Under a chiral rotation, any building block Y transforms as $ Y \rightarrow h Y h^\dagger $. The covariant derivative of these building blocks is given by

The structure of these building blocks in the HLS representation differs significantly from that in other representations. Generally, the relations among these building blocks are quite complex. However, the situation can be simplified in the unitary gauge, as discussed in Refs. [42, 43, 45]. Table 2 outlines the relations between Eqs. (35), (36), (41)−(45), and Eq. (6) in the unitary gauge, along with the transformation properties of the building blocks. With these building blocks and their covariant derivatives, chiral Lagrangians of any order can be constructed.

Table 2 presents the power counting of the HLS building blocks, as detailed in Refs. [42, 45]. Additionally, the parameter g in the LO chiral Lagrangian is assigned a chiral order of $ {\cal{O}}(p^1) $ [45].

For convenience, we discuss all constraint relations in the unitary gauge. Consequently, most conclusions from Sec. III B remain applicable, provided that the relations in Table 2 are utilized. A notable difference arises in the EOMs, as detailed in Ref. [45]. The EOMs in the HLS representation are given by

Here, the parameter a is defined as $ a = F_\sigma^2 / F_\pi^2 $.

For $ V^{\mu\nu} $, as defined in Eq. (39), a relation analogous to the Bianchi identity can be derived:

Thus, we have established all linear relations within the HLS representation.

The LO chiral Lagrangians incorporating the vector meson octet were derived over 30 years ago [36−38]. In this work, we extend the discussion to include the nonet and higher-order Lagrangians. However, these higher-order Lagrangians involve a vast number of terms, and manually performing the calculations can lead to the inadvertent omission of some linear relations. Consequently, all calculations are performed with the aid of a computer.

One major challenge in these computations is symbolic manipulation. To our knowledge, no effective and widely accessible program currently exists to automatically generate all possible terms and associated linear relations. To address this, we employ a method that converts symbolic computations into numerical ones. This paper outlines the basic approach, with detailed explanations provided in Refs. [101−103].

To avoid symbolic complications, we assign unique numbers to each building block and index. For instance, we may assign $ V \to 11 $, $ u \to 12 $, $ \mu \to 1 $, $ \nu \to 2 $, and so on. Using this scheme, a term like $ V^{\mu} u_{\mu} u^{\nu} u_{\nu} $ can be represented by two vectors: one for the building blocks, $ (11, 12, 12, 12) $, and another for the indices, $ (1, 1, 2, 2) $. Compared to handling symbolic expressions, processing these vectors computationally is significantly more manageable. As a result, most of the tedious calculations are efficiently executed using this numerical approach.

The LO chiral Lagrangians are straightforward. We provide a brief summary and incorporate the singlet state. For the vector-field representation, the independent terms are listed in Table 3. The terms involving $ \langle V_{\mu} \rangle $, such as the fourth term in the $ S U(2) $ case, include only the singlet state.

For the tensor-field representation, the independent terms are presented in Table 4. In the $ S U(2) $ case, the third term includes only the singlet state.

For the HLS representation, the independent terms are presented in Table 5. Notably, the third term in both cases includes only the singlet state.

Because the expressions are very lengthy, the NLO result for the vector-field representation is provided in Table A1 in the supplementary material. The tensor-field and HLS representations are given in Tables 6 and 7, respectively.

The numbers of these chiral Lagrangians are presented in Table 8. Concerning the applications of these higher-order Lagrangian terms, it is theoretically possible to integrate out the vector field to connect the low-energy constants in the vector meson Lagrangian with those in the pseudoscalar meson Lagrangian. A related discussion on the vector meson octet is available in Ref. [104].

To verify our findings, we compared them with existing octet results and identified some redundant terms. The results for the octet tensor-field representation in the large-$ N_C $ limit are detailed in Ref. [63]. In the three-flavor case, by employing the linear relations from Sec. III B, we discovered an additional relation:

where $ {\cal{O}}^V_i $ denotes the i-th term listed in Table 1 of Ref. [63]. In conjunction with the relations provided in Eq. (25), three additional relations arise:

The terms on the left-hand side of Eqs. (49) and (50) are not independent and can be eliminated. As the large-$ N_c $ limit is not applicable in this context, it is necessary to incorporate the following terms at the $ {\cal{O}}(1/N_C) $ level into Ref. [63]:

Now, only 24 independent terms remain, denoted as $ {\cal{O}}^V_{i} $ for $ i=1,2,\ldots,28 $, excluding $ i=17, 19, 21, 22 $.

In the two-flavor case, only 16 terms remain, necessitating the further exclusion of $ {\cal{O}}^V_{3} $, $ {\cal{O}}^V_{4} $, $ {\cal{O}}^V_{9} $, $ {\cal{O}}^V_{23} $, $ {\cal{O}}^V_{24} $, $ {\cal{O}}^V_{25} $, $ {\cal{O}}^V_{26} $, and $ {\cal{O}}^V_{27} $.

For the anomalous octet component, the complete results are provided in Refs. [66, 105]. By employing the linear relations from Sec. III.B and Eq. (25), three additional relations are identified:

where $ {\cal{O}}^{V,a}_i $ denotes the i-th term in Eq. (25) of Ref. [66].

In the HLS representation for the octet, we replicate the findings from Refs. [42, 45], with a necessary correction. Specifically, for the three-flavor normal terms, the $ y_{17} $ term from Ref. [45] is related to other terms through the Cayley-Hamilton relation (22):

where $ Y_i $ corresponds to the term associated with $ y_i $. This relation arises because Eq. (22) is more general. The form used in Refs. [42, 45] is a specific case of Eq. (22). Further details are provided in Sec. 4.3 of Ref. [45].

For the three-flavor anomalous terms, some studies have presented similar results using different building blocks [41, 45, 100, 106, 107]. With these different building blocks, the results are

With the building blocks presented in this paper, the results are

These two sets of building blocks exhibit the following relations:

The relations of the LECs are

The NNLO results for the chiral Lagrangians are provided in Tables A2 and A3 in the supplementary material due to their extensive length. Note that in the vector-field representation, the number of independent terms at NNLO is significantly greater than in other representations. The quantity of terms is too extensive to be included here. This underscores an advantage of the tensor-field representation, which necessitates fewer terms compared to the vector-field representation. The numbers of these chiral Lagrangians are detailed in Table 9.