Analysis of the $b_1$ meson decay in local tensor bilinear representation

We explore the validity of vector meson dominance in the radiative decay of the $b_1(1235)$ meson. In order to explain the violation of the vector meson dominance hypothesis in this decay process, we investigate a model where the $b_1$ meson strongly couples with the local current in tensor bilinear representation. The tensor representation is investigated in the framework of the operator product expansion and we found a low energy decay process that does not follow the usual vector meson dominance hypothesis. The $\omega$-like intermediate meson state of quantum numbers $I^G(J^{PC}) = 0^{-}(1^{--})$ is found to have a nontrivial role in the decay process of the $b_1$ meson. The spectral structure of the $\omega$-like state is found to be close to a $\pi$-$\rho$ hybrid state, which provides a mechanism that evades the usual vector meson dominance hypothesis. Precise measurements of various decay channels of the $b_1$ meson are, therefore, required to unravel the internal structure of axial vector mesons.

If the U(1) symmetry part of the vector meson field is conserved, the vector meson field plays the role of the external electromagnetic field source through the direct coupling in the form of L γω ∼ A µ V µ [4][5][6][7][8]. This argument is based on the assumption that the vector meson has the same representation as the electromagnetic current. 3 According to the VMD hypothesis, it is natural to assume that the radiative b 1 decay is dominated by the intermediate ω meson, namely, it leads to the decay chain of b 1 → π ± ω → π ± γ. However, as will be discussed in Sec. 2, such VMD model underestimates the decay width of b 1 → π ± γ [13,14] and it causes difficulty in understanding the observed decay properties of the b 1 meson. This implies that there might be a missing source or mechanism for the radiative decays of the b 1 meson, which dominates over the mechanism proposed by the VMD hypothesis. The present work is motivated by the idea that, if the intermediate spin-1 mesonic state is not in exactly the same representation with the electromagnetic current, then there could be an additional process for the b 1 radiative decay that cannot be ascribed to the VMD hypothesis alone. In other words, another representation for spin-1 mesonic states can be allowed if it overlaps with the U(1) gauge boson in low energy regime where flavor symmetry is broken down to SU(2) V [15,16]. In this case, the state with the quantum numbers of the ω meson, usually represented by chiral vector current, can also be represented in the helicity mixed space via a tensor current. We denote this state asω. In this helicity mixed representation, the b 1 → πω decay can be regarded as a production process of a soft-pion as the b 1 andω states are chiral partners to each other.
In this study, the properties of the tensor representation for spin-1 mesonic current is reviewed and their current correlation function is calculated in the framework of operator product expansion (OPE) [17]. We first discuss the spin-1 mesonic state interpolated by tensor currents and the b 1 decay process in the context of chiral rotation [18,19]. The distinct nature of the spin-1 mesonic state in the tensor current correlation is analyzed through QCD sum rules [20][21][22][23][24][25]. In the OPE of tensor current correlator, the four-quark condensates appear as the leading quark contribution. Although the vacuum saturation hypothesis for the four-quark condensate has been widely used in previous studies, it does not reflect all the possible vacuum structure of QCD. The factorization scheme is modified to take into account the non-factorizable part and quark correlation pattern in topologically nontrivial gauge configuration [26][27][28][29][30][31][32]. Within the usual limit of the factorization parameters, we will show that the tensor current can strongly couple to the b 1 meson as well as to the π-ρ continuum state that has the quantum numbers of the ω meson. Analyzing the b 1 → πω decay within the soft pion limit, we find that the ω-like mesonic stateω, which appears in the strong decay of the b 1 into the pion, would not be a simple one-particle state but an anomalous hybrid state. Our analyses will show that this process would be a source for b 1 radiative decays instead of the usual VMD mechanism. This paper is organized as follows. In section 2, we discuss the problems in the phenomenology of the b 1 meson decays. In section 3, a brief review on local bilinear representations of spin-1 mesonic state and their evolution in chiral symmetry breaking is presented. Detailed OPE for correlation functions and corresponding spectral analyses with four-quark condensates are presented in section 4. Section 5 contains discussion and conclusions. We leave the details on tensor currents and technical remarks on sum rule analysis in Appendixes.
2 Phenomenology of the b 1 meson In this Section, we review the phenomenology of the b 1 meson decays. The major strong decay channel, i.e., b 1 → πω, is discussed and then its radiative decays is discussed in connection with the VMD hypothesis.

Strong decay
In quark models, the b 1 (1235) state is described as a member of the 1 P 1 nonet (in the 2S+1 L J notation), where the qq pair is in the relative P -wave state, having quantum numbers J P C = 1 +− . Other members of this nonet may include the h 1 (1170) and h 1 (1380) which are iso-scalars, and K 1 (1270) or K 1 (1400) that has one unit of strangeness [14].
The strong decay of the b 1 meson is described by the amplitude of an axial-vector meson (J P = 1 + ) decay into a vector meson (J P = 1 − ) plus a pseudoscalar meson (J P = 0 − ), which reads where ε µ (1 ± , p) represents the polarization vector of J P = 1 ± meson with momentum p. Here, p A and p V are the momenta of the axial-vector meson and the vector meson, respectively, and the momentum of the pseudoscalar meson p P is determined by the energymomentum conservation, p A = p V + p P . The most general form of the amplitude Γ µν reads [33] where f i 's are form factors. If all of the particles in the decay process are on-mass shell, only the two terms, f 1 and f 2 , are non-vanishing because of ε(1 − , p V ) · p V = ε(1 + , p A ) · p A = 0. Therefore, the decay amplitude for b 1 → πω constrained by gauge invariance reads [33] where the momenta of the b 1 and ω mesons are represented by q and k, respectively, and Because of the transversality condition, ε(p) · p = 0, however, the terms with q µ or k ν vanish, and we can rewrite it effectively as where M b 1 and M ω are the masses of the b 1 meson and ω meson, respectively. Then the decay width is calculated as where E ω is the energy of the ω meson, E ω = M 2 ω + |k| 2 . Also from the definitions of the S-and D-wave amplitudes, which gives The magnitude of the momentum k is where λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx is the Källén function. This gives |k| = 346.5 MeV with M b 1 = 1.230 GeV, M ω = 0.783 MeV, and M π = 0.140 GeV. From the two experimental data, One can relate the above information with the couplings of the effective b 1 ωπ interaction Lagrangian, where ω µ and b µ 1 are the ω meson field and the b 1 meson field, respectively, and their field strength tensors are The above Lagrangian gives Comparing with Eq. (2.5) we have Then we can estimate the coupling constants as

Radiative decay and vector meson dominance
The radiative decay of b 1 → πγ was investigated in quark models based on the VMD hypothesis. The VMD hypothesis leads to where ω ⊥ represents the transverse component of the ω vector meson and g ω is the coupling of the ω meson to the photon in VMD that is estimated as g ω ≈ 17.14. The magnitudes of the momenta of the decays are |p γ | = 607.0 MeV and |p π | = 346.5 MeV. Here, n shows the characteristic momentum dependence of the decay process, of which values will be discussed later.
The decay amplitude of the strong decay of the b 1 into the transverse ω meson can be estimated following Ref. [13]. The transverse and longitudinal decay amplitudes of the b 1 → ωπ decay are related to the S and D wave amplitudes as which gives In Ref. [13], n = 3 is used in the expression of Eq. (2.20), which gives This seems to be successful to explain the observed value Γ(b 1 → γπ) expt. = 230 ± 60 KeV. However, it was pointed out in Ref. [14] that the VMD prediction is sensitive to the interaction form which determines the value of n. Using the covariant oscillator quark model, the authors of Ref. [14] predicted This is very similar to the value obtained with n = 1 in Eq. (2.20). However, the general form of the b 1 ωπ interaction contains two terms as shown in Eq. (2.7). In fact, the expression for the decay width (2.7) contains terms with n = 1, 3, and 5. Following Ref. [13], we estimate the radiative decay width of b 1 by replacing k by p γ and in the decay width formula of Eq. (2.7). Taking into account that Γ(b 1 → ω ⊥ π) ≈ 126 MeV, we obtain Γ(b 1 → γπ) ≈ 160 keV, (2.27) which is about 2/3 of the measured quantity.

Spin-1 mesons in local tensor bilinear representation
Throughout this study, the isospin flavor matrices are denoted as T 0 = I/2, T a = σ a /2, where I is the 2 × 2 unit matrix and σ a is the Pauli matrix. These matrices are normalized as Tr [T A T B ] = δ AB /2. Here, capital romans (A = 0 − 3) denote isospin (0 ⊕ 1) indices and lower-case romans (a = 1 − 3) denote isovector indices. We will use barred romans (ā = 1 − 8) to denote adjoint color indices in gauge interactions. Mesonic quantum numbers are represented by I G (J P C ) .

Brief review on local bilinear representation of spin-1 mesonic state
The simplest local representation of the b 1 meson isqT a γ 5 ← → D µ q. This current is in helicity mixed representation 1 2 , 1 2 ⊕ 1 2 , 1 2 , where the first and second numbers in the bracket show the SU(2) L and SU(2) R representations, respectively. The derivative leads to additional p 2 factors in the Wilson coefficient of OPE terms, which makes the OPE dominated by the continuum contribution [23]. In the same representation, tensor currentq T a σ µν q, which does not contain the covariant derivative, can also be considered. This current can couple to four different spin-1 mesonic systems. In the non-relativistic limit, where p µ → (m, 0), depending on the intrinsic quantum numbers, the spin-1 mesonic states of momentum p and polarization λ have the following relations: [16] ω : where the bar notation is used to distinguish the mesonic states of the helicity mixed tensor current representation from those of chiral vector representation. For example, theω is the state of tensor current representation having I G J P C = 0 − (1 −− ) as the usual ω meson.
Here,¯ (λ) µ is the polarization vector and the vector meson coupling is given by f T V in tensor representation. In a boosted frame, p µ → ( m 2 + p 2 , p), the covariant generalizations readω The tensor current couples only to the transverse polarization of spin-1 mesonic state [15,24,25] because of the relation, where¯ t(λ) µ denotes the transverse polarization vector. However, this current couples to both the parity-even and parity-odd modes, and each parity mode can be separately projected out from the current-current correlation as which defines the projection operators as with the relation [P (±) P (∓) ] µμ;νν = 0. Considering the assumed symmetries, the four mesonic modes reside in the same symmetry group U(2) L × U(2) R . The group structure allows the transformation among these modes depending on quantum numbers. For example, both theω state and theh 1 state have isospin 0 and they can be transformed to each other through the U A (1) transformation. The same is true for the isospin-1 states, theρ and theb 1 . Similarly, the chiral partner states can be transformed to each other through the axial SU(2) flavor rotation [16], namely, the pair ofω and theb 1 , and the pair of theρ and theh 1 . If all the symmetries are conserved, there should be no correlation between the two different representations of mesonic states. However, when chiral symmetry is broken at low energy regime, the mesonic states can have overlaps. Since the vector (or chiral) representations of the ω and ρ states read the overlap between the vector and tensor representations can be described as where P α;µμ = g αµ pμ − g αμ p µ projects only the transverse polarization. We also use the polarization sum in the SU(2) V diagonal phase, i.e., λ (λ) α¯ (λ) * µ = −g αµ . Algebraic details for tensor currents and their transformations are presented in Appendix A.

b 1 meson decay in the soft pion limit
mesonic states transform to each other under SU(2) L × SU(2) R symmetry. In the SU(2) V phase, if the b 1 (1235) meson state couples strongly to the tensor current (J a µμ ), the overlap for the decay of b 1 (1235) → πω(782) can be expressed as withm being the meson mass in the tensor representation. In the soft pion limit, the overlap can be rewritten as and R 1 (q) represents |q|-dependent contributions in the limit of lim q→0 R 1 (q) = 0. Comparing Eqs. (3.17) and (3.18), the |q|independent term can be obtained as If theω is identified as the ω(782) state, one can obtain the strength for the b 1 → πγ decay from the overlap for b 1 → πω with the VMD hypothesis. This process can be written as the following current correlation function in the soft pion limit: with m being the meson mass in chiral representation and R 2 (q) represents the |q|-dependent contributions with the condition that lim q→0 R 2 (q) = 0. The overlap f ω f T ω ω(k, λ) |ω(k, λ) can be calculated from the time-ordered correlator as If chiral symmetry is restored, the correlation (3.21) will vanish because the representation spaces of currents are totally disconnected. Diagrammatical descriptions of this process are depicted in Fig. 1. Therefore, Eq. (3.20) describes the decay mechanism ofb 1 → πγ that evades the usual VMD hypothesis. Then, the interpretation of theb 1 andω is required in connection with physical states, namely, to see whether and how the tensor representation can describe observed mesons.

Operator product expansion and spectral sum rules
Understanding the nature of theω andb 1 states requires their relations to the physical ω(782) and b 1 (1235) mesons. For this purpose, we analyze the spectral sum rules for the corresponding masses of these mesonic states. The correlation function can be calculated via OPE in the k 2 → −∞ limit as in Refs. [17,[20][21][22][23][24][25].

OPE of the correlators
To interpolate the mesonic state with given quantum numbers we employ the tensor current, where flavor matrix T 0 corresponds to the (ω,h 1 ) current and T a corresponds to (ρ a ,b a 1 ) current. The correlator can be obtained as [24,25] where nonzero OPE terms are found as with G ρσ = D ρ A σ − D σ A ρ being the gluon field strength tensor. 6 The correlator (3.21) can be calculated as where

Spectral sum rules
The invariants can be weighted to emphasize the ground state contribution as follows: where '∓' denote parity-odd and parity-even modes and the OPE continuum contribution (k 2 > s 0 ), where the excited states are assumed, is subtracted as Detailed arguments for the weighting scheme are given in Appendix B. The value of the gluon condensate is taken as (α s /π)G 2 = (0.33 GeV) 4 [20][21][22]. By using the factorization hypothesis, q αqβ q γqδ ρ,I q αqβ ρ,I q γqδ ρ,I − q αqδ ρ,I q γqβ ρ,I , (4.12) where the color indices are omitted, the four-quark condensates are estimated in factorized forms as where qT 0 q vac is obtained from the Gellmann-Oakes-Renner relation [19], with m π = 138 MeV and f π = 93 MeV. This leads to qT 0 q vac −(254 MeV) 3 for m q = 5 MeV. The parameters in Eqs. (4.13)-(4.15) have the same value, namely, α A = β A = γ = 1, in the usual factorization hypothesis. Hereafter, each parameter set with explicit isospin dependence, i.e., α 0 , β 0 , α a , and β a with a = 1, 2, 3, represents the factorization of the isoscalar and isovector four-quark condensates, respectively. As can be seen from Eqs. pair, the factorization parameters α, β, γ can take different values. This is so because, when there is an isospin operator in the quark antiquark pair, the quark-disconnected diagrams vanish identically. Furthermore, even for isosinglet quark antiquark pair, depending on the chiral and color structure, the quark-disconnected diagrams are expected to have different contributions. Therefore, in the present work, we choose the parameter set taking into account the following three conditions depending on the color and isospin structure of the quark antiquark pair: (a) possible correlation types that depend on chiral symmetry breaking, (b) nontrivial topological contribution from color gauge group, and (c) stability of the Borel curves. For isovector quark-antiquark pair, the usual factorization provides quarkconnected-like condensates. The condensates are given in the local limit of the pair of gauge equivalent nonlocal bilinear [26]. The colored pieces are part of the Dirac eigenmodes in the nonlocal gauge equivalent structure of the mesonic current. So the isoscalar condensates are considered to have nonzero quark-disconnected-like contribution if the colored spinless bilinear appears in pair to produce the totally color singlet configuration. Considering the possibility of this contribution, which does not appear in the usual factorization scheme, the quark-disconnected-like and quark-connected-like contributions are encoded in the isoscalar and isovector parameters, respectively. This typically leads to |α 0 | < |α a | and |β 0 | < |β a |. Also, if one considers topologically nontrivial gauge contributions, the spinless but partyodd combination of quark bilinear can have nonzero correlation in the same way as argued in Refs. [28,29]. For the condensate composed with pseudoscalar bilinear in Eq. (4.15), the nontrivial contribution may appear as an alternating series in winding number of the color gauge. Sum rule analysis with the parameters taken according to these criteria are found to produce stable Borel curves.
We first take (α 0 = 0.8, α a = 1.0) as the reference parameter set, which is similar to the usual factorization. 7 To ensure that the ground state contribution becomes dominant and the OPEs are convergent, the Borel windows for theω andb 1 sum rules are chosen to be in the range of 0.6 GeV 2 < M 2 < 1.2 GeV 2 and 0.6 GeV 2 < M 2 < 2.0 GeV 2 , respectively. Plotted in Fig. 2 are spectral structures of the invariants.
If the ground state is the one-particle mesonic state as explained in Sec. 3, the phenomenological structure can be written as However, it is also possible that the tensor current (4.1) can couple to some kinds of hybrid states, which are composed of two particles, if the mass scale of the hybrid is compatible with the one-particle mesonic state. The hybrid state containing the quantum number of the ω meson can be inferred from the anomalous interaction that is written as [10] where ω µμ is the ω field strength tensor. As the ω andω states share the same quantum , the phenomenological interpolating current for theω state can be written as where the trace is summed over flavor indices. The corresponding phenomenological structure of Πω (∓) (k 2 ) can be obtained from the correlator of Jω µμ as where m ρ is the ρ meson mass and the invariant function for each parity mode can be calculated as

22)
unstable Borel curves and controversial results for the parity-even mode sum rules. Therefore, we do not adopt these parameter sets in our study. More detailed sum rules analyses forω andb1 states with (a A < 0, b A < 0) are reported in Appendix B.  where R ∓ (k 2 ) is a regular polynomial and the m π → 0 limit has been taken. One can verify that the imaginary part appears in the range of m 2 ρ /k 2 < x < 1. The weighted Πω − (k 2 ) can be obtained as where the continuum threshold s 0 is taken to be the same as that assigned in W subt.
Similarly, the loop-like phenomenological current forb 1 state can be defined as The corresponding invariant can be obtained from the parity-even mode of Eq.
Then the sum rule for the ground states interpolated by the tensor current reads where ch y. ∓ represents the coupling between the tensor current (4.1) and the hybrid state. Equation (4.26) leads to the hybrid-subtracted mass sum rules as By assuming only the one-particle meson state for the ground state, the obtained masses of theω andb 1 modes are plotted in Fig. 3. Depending on the values of α 0 and β 0 , the mass of the isoscalar parity-odd mode is found to vary from 1.0 GeV to 1.2 GeV, which is much higher than the physical ω(782) mass. On the other hand, the mass of the isovector parity-even mode shows week dependence on the parameter set, and is in good agreement with the physical b 1 (1235) mass when (α a = 1, β a = 0.5) is employed. Because of the large mass difference between the ω(782) andω, it is unnatural to identify theω state as the physical ω(782).
Considering the mass range determined by the Borel curve in Fig. 3(a)  resonance states. However, such possibility is unlikely as the mass of the lowest excited state ω(1420) [1] is much larger than the mass range obtained in the sum rules analysis for theω. Therefore, we explore the possibility that the employed current couples to a hybrid state composed of two particles. If the tensor current (4.1) strongly couples to the one-particle mesonic state (hybrid state), the sum rules for the one-particle meson pole residue f T ∓ (the tensor current-hybrid coupling ch y. ∓ ) should provide a stable behavior in the Borel window. The sum rules for each coupling obtained from Eq. (4.26) read where m 2 ∓ , f T ∓ , and ch y. ∓ are input parameters. If the one-particle pole is dominant in the ground state, the Borel curve for f T ω will be stable. However, as can be seen in Fig. 4(a), the Borel curve is not stable, suggesting that something is missing or the assumption is not correct. On the other hand, ch y. ω provides relatively stable behavior in the proper Borel window even with the no-pole scenario (f T ω = 0) and provides a plateau value ch y. ω 7.0 with f T ω = 0.05 as shown in Fig. 4(b). In these optimized curves, f T ω reduces to 50% of its original scale when ch y. ω is varied from 0 to 7.0, while ch y. ω reduces only to 85% of its original scale when f T ω varies from 0 to 0.05. This leads to the conclusion that the ground state information in the tensor current correlator is dominated by the hybrid state.
Theb 1 state can be analyzed in a similar way. As shown in Fig. 5, the Borel curves for the couplings in theb 1 state shows the opposite tendency in comparison with theω analyses. Namely, the computed f T     > 0.11 is assigned. Therefore, theb 1 state is identified as the physical b 1 (1235) meson state considering the mass scale shown in Fig. 3(b).
By assuming only the π-ρ hybrid (i.e., theω) or b 1 (1235) state for the ground state, the corresponding coupling sum rules are plotted in Fig. 6(a) and Fig. 6(b), respectively. We also found that ch y. ω with f T ω = 0 shows stable Borel behavior and the plateau scale becomes moderate when s 0 ≥ 1.44 GeV 2 , which means that theω state can be interpreted as a hybrid state. Analogously, since f T b 1 with ch y. b 1 = 0 provides a plateau structure and the plateau scale becomes moderate when s 0 ≥ 3.24 GeV 2 , the one particle b 1 (1235) state would be good enough to approximate theb 1 state.

Discussion and Conclusion
The b 1 (1235) meson state mostly decays into the πω state by the strong interactions and the πγ state by the electromagnetic interactions. As the VMD scenario is successful in explaining the decays and form factors of ground state hadrons, low-energy phenomenology usually assumes that the photon always sees hadrons through vector meson states [6,7,13,14]. However, for the electromagnetic decays of the b 1 (1235) meson, the VMD hypothesis cannot fully explain the measured Γ(b 1 → πγ) [13,14]. In the present work, we explore the spin-1 mesonic states using the tensor representation for the purpose of searching for a VMD-evading process for the b 1 radiative decay. Because of the mixing of helicity states, the corresponding currents are not generally conserved in their parity or isospin eigenspace. If SU(2) L,R flavor symmetry is assumed, the four different spin-1 mesonic states reside in the grand U(2) L ⊗ U(2) R symmetry group [16].
As chiral symmetry is spontaneously broken, theb 1 state in the tensor representation decays into the chiral partnerω state by emitting a pion as the Goldstone boson. From the QCD sum rules analysis, we find that theb 1 state can be identified as the physical b 1 (1235) state, while it is problematic to identify theω state as the physical ω(782) meson. Theω state has the same spin and flavor quantum numbers as the ω(782) but it is found to have a much higher mass. Comparing the weighted OPE invariant with the anomalous loop-like spectral structure, theω state can be identified as the hybrid state of the π and ρ mesons. As there is no observed ω resonance state in the mass of around 1 GeV, theω state would be interpreted as an intermediate virtual state. It is well known that the vector current J 0 α =qT 0 γ α q couples strongly to the vector meson state and the the resulting spectral sum rules successfully reproduce the ω(782) mass [20][21][22][23]. Since the isoscalar component of J e.m. α is same as (e 0 /3)J 0 α , the nonzero invariant of Eq. (4.9) can be understood as the overlap between theω hybrid state and the physical ω(782) state, which then becomes the photon following the usual VMD scenario as depicted in Fig. 7 [6,7].
However, the loop structure of theω state suggests other mechanisms of radiative decays of the b 1 . The π-ρ hybrid state can have an anomalous electromagnetic interaction as If the π and ρ are on-shell, the vertex can be substituted with Eq. (4.18) by the VMD hypothesis [10]. However, as most part of the loop phase space is off-shell, the direct photon coupling channel (5.1) can be distinguished from the VMD channel which could be the additional mechanism for the b 1 radiative decay as depicted in Fig. 8.
If the b 1 strongly couples to the tensor current, its decay is mediated mostly by theω state that is interpreted as the ρ-π hybrid state. Then the major decay process of the b 1 meson would take the process of b 1 → πω → πω, which involves the anomalous ωρπ interaction. Because of the intermediateω state, the strong decay of the b 1 meson into two-pions, i.e., b 1 → ππρ, would be enhanced enough to be measured. However, because of the lack of experimental data, we cannot make a conclusion on the tensor representation of axial-vector mesons. For example, the decay of b 1 → ππππ was reported in 1960s [41], which concluded only that Γ(b 1 → π + π + π − π 0 )/Γ(b 1 → ωπ) < 0.5. Therefore, precise measurement of the details of the b 1 decay will shed light on our understanding on the structure and properties of axial-vector mesons and it can be performed at current experimental facilities.
A Tensor currents in SU(2) L × SU(2) R symmetry A.1 Transformation of tensor currents in the non-relativistic limit The quantum numbers of the mesonic state are defined at rest frame. Equations (3.1)- (3.4) show that each spin-1 state can be interpolated via tensor currents. Since the tensor current couples to both parity eigenstates, one can use dual tensor definition to describe a parity eigenstate in the opposite parity projection. For convenience, all the current is defined to interpolate spin-1 mesonic states in parity-odd projection. The infinitesimal U A (1) transformation on theb 1 current reads where the corresponding commutation relations are summarized as Similarly, one can rotate quark fields via infinitesimal SU(2) L × SU(2) R rotation as The transformation up to the leading order of θ a and δ a can be expressed in terms of explicit helicity states as where the rotation directions 'θ a = δ a ' and 'θ a = −δ a ' correspond to Q a v = d 3 x q † (x)T a q(x) and Q a 5 = d 3 x q † (x)T a γ 5 q(x), respectively. The corresponding commutation relations read Similar commutation relations can be obtained for other currents as The mixing properties are diagrammatically described in Fig. 9.

A.2 Transformation of tensor currents in relativistic generalization
The covariant currents are defined as J 0 µμ ≡qT 0 σ µμ q andJ a µμ ≡ − 1 2 µμαβq T a σ αβ q to describe theω andb 1 states, respectively, in the parity-odd projection at boosted frame.
qT A σ µν q The flavor axial charge commutation relations are Similarly, the other commutation relations can be obtained in covariant generalization as These lead to the same diagrammatical summarization presented in Fig. 9.

B.1 Four-quark condensates
In the OPE of the tensor current correlator, the four-quark condensate terms are the lowest mass-dimensional quark contribution. These condensates determine the asymptotic behavior of the Borel curves and distinguish parity eigenstates. The condensates in Eqs. and (4.14) give huge contribution to the sum rules because the Wilson coefficient of the invariant term of Eq. (4.6) is dominant over the others. The corresponding diagrammatical description can be found in Fig. 10(a). Unfortunately, the values of these four-quark condensates are unknown at present, and are thus estimated, for example, by adopting the vacuum saturation hypothesis. However, the usual factorization scheme does not reflect all the possible contributions in the condensates. In the condensates defined in Eqs. (4.13) and (4.14), isospin and color matrices appear in the condensates, and the usual factorization scheme only allows 'quark connected' correlations as in Fig. 11(a) when there are isospin matrices in the quark bilinears. Although isovector terms cannot have 'quark disconnected' contribution [ Fig. 11(b)], the isoscalar terms can have contributions even if they are colored. One can consider the non-local generalization of meson interpolating current, Then, in the very small but nonzero space-time separation = |y − x| = 0 1 which can be allowed in the relevant hadron size for the quark correlation average, gauge corrections are needed to fix the initial and final interpolated states on the equivalent gauge orbit [ Fig. 11(b)]. Therefore, if the pair of colored isoscalar bilinear appears in totally color blind configuration, there is no reason to discard disconnected contribution. Considering that the relative sign between two contributions is negative, the magnitude of the isoscalar parameter is assigned to be smaller than that of the isovector parameter: |α 0 | < |α a |, |β 0 | < |β a |.
We now consider the condensates containing pseudoscalar type bilinear as in Eq. (4.14). Since ψ 0 and γ 5 ψ 0 reside in the same eigenspace (λ = 0), the vacuum average of pseudoscalar bilinear does not vanish in the nontrivial gauge configuration [40]: where ν denotes the winding number, and the inner product between the modes in different eigenspace vanishes. If all the gauge configuration is summed up, the vacuum average can be written as where θ is the QCD vacuum angle. However, for the quark correlation in the four-quark vacuum average, the nontrivial topological contribution should appear in terms of ν 2 and would not vanish in the summation. In this study, the nontrivial contribution is assumed to appear as an alternating series of ν 2 in Eq. (4.14) while it does not in Eq. (4.13), which leads to |α A | > |β A |.
The condensate (4.15) contributes to the invariant piece (4.7) whose diagrammatical description is given in Fig. 10(b). As the vector bilinear is obtained from the field equation of the gluons, at least the sign of the usual factorization estimation has been justified in Euclidean space [20]. Therefore we take the usual vacuum saturation value of γ = 1. Furthermore, its contribution in Eq. (4.7) is much smaller than those from the other fourquark condensates so that a slightly different value for gamma will not change the main result of the present work.
B.2 Supplementary remarks for the sum rules with α A < 0, β A < 0 In the previous arguments, the proper sets (α 0 = 0.8, β 0 = 0.4) and (α a = 1.0, β a = 0.5) have been used for Borel sum rules for theω andb 1 states, respectively. If we use the reversed sign of the parameter set, the Borel curves for f T ∓ and ch y. ∓ are obtained as shown in Fig. 12 and Fig. 13, respectively. In this case, f T ω shows plateau-like behavior even in the one-particle pole limit (ch y. ω = 0). In the optimized condition, the stable Borel curves provide f T ω = 0.11 and ch y. ω = 2.0 as plateau values. It is found that the magnitude of f T ω reduces to 90% when we vary ch y. ω from 0 to 2.0, while ch y. ω becomes 20% of its original value when we change f T ω from 0 to 0.11. This means that theω state becomes dominated by the one-particle mesonic state with the set of (α 0 = −0.8, β 0 = −0.4).
However, this tendency becomes unclear in the case of theb 1 analysis. As one can find in Fig. 13   80% and 45% of their original magnitudes when we vary ch y. from 0 to 0.12, respectively. This observation then leads us to the conclusion that theb 1 state is dominated by the one-particle meson state within the set of (α a = −1.0, β a = −0.5). This result leads to the usual VMD scenario (b 1 → πω → πγ).
Considering that the Borel curve for f T b 1 is highly dependent of the parameter set, it is possible that the non-negligible part ofb 1 could be a π-ω hybrid state. However, this kind of hybrid state leads to a stable ω(782) state after pion breaking, where the VMD scenario for the radiative decay is still valid.

B.3 Borel sum rules
By assuming regularity of the correlation function, the invariants can be written in dispersion relation as where following relation has been used: (B.9) The residues located after the continuum threshold with finite s 0 can be subtracted as These results are used when E n is multiplied to all (M 2 ) n+1 terms in B[Π i (q 2 )]. This weighting scheme and subsequent spectral sum rules are known as the Borel transformation and Borel sum rules, respectively.