Consistent high-energy constraints in the anomalous QCD sector

The anomalousGreen function and related form-factors (pi^0 to gamma^* gamma^* and tau^- to X^- nu_tau vector form-factors, with X^-=(KKpi)^-, phi^- gamma, (phi V)^-) are analyzed in this letter in the large-N_C limit. Within the single (vector and pseudoscalar) resonance approximation and the context of Resonance Chiral Theory, we show that all these observables over-determine in a consistent way a unique set of compatible high-energy constraints for the resonance couplings. This result is in agreement with analogous relations found in the even intrinsic-parity sector of QCD like, e.g., F_V^2 = 3 F^2. The antisymmetric tensor formalism is considered for the spin-one resonance fields. Finally, we have also worked out and provide here the relation between the two bases of odd intrinsic-parity Lagrangian operators commonly employed in the literature.


Introduction
Chiral symmetry plays a crucial role in the structure of non-perturbative light-quark interactions in Quantum Chromodynamics (QCD). However, it becomes spontaneously broken, generating the corresponding chiral (pseudo) Goldstones ϕ a . Its low-energy interaction can be then described through an effective field theory Table 1: Monomials with one vector resonance field (O i V J P ) and two vector fields (O i V V P ) in the basis of Ref. [16].

The odd intrinsic-parity resonance Lagrangian
The operators in the RχT Lagrangian can be classified according to the number of resonance fields: ) with L G given by the operators with only pseudo-Goldstone fields and external vector and axial-vector sources. It contains the even-parity O(p 2 ) χPT Lagrangian [2,3,8] and the WZW term [6,7,16,17]. At leading order in the 1/N C expansion, the most general odd intrinsic-parity resonance chiral Lagrangian for processes involving one pseudo-Goldstone and two vector objects (two vector resonances, or one external source and one vector resonance) was derived in Ref. [16]: The monomials O i VJP and O i VVP are provided in Table 1. The antisymmetric tensor formulation is employed here to describe the spin-1 fields [8,9]. Analogous analyses with the spin-1 fields given in the four-vector (Proca) formalism can be found in Ref. [13,21], although the present work only studies the antisymmetric tensor representation.
We follow the notation and conventions of Ref. [15], where the chiral tensors entering L odd RχT are defined. M V is the vector resonance multiplet mass in the chiral and large-N C limits.
Within the same framework, and motivated by the analogous work on the even-intrinsic parity resonance Lagrangian accomplished in Ref. [15], the authors of Ref. [17] constructed the most general resonance chiral Lagrangian in the odd intrinsic-parity sector that can generate chiral low-energy constants up to O(p 6 ) [4,5]. They considered the contribution from the lightest resonance multiplets, in particular vector (V ) and pseudoscalar (P ) resonances. The latter was absent in the previous treatment in Ref. [16]. The part of the odd intrinsic-parity Lagrangian relevant for the V V P Green function and the related form-factors studied in this article is given by [17] where the corresponding operators can be read in Tables 2 and 3. It is possible to rewrite the resonance Lagrangian from Ref. [16] ( Table 1) in terms of the basis of monomials in Ref. [17] (Tables 2 and 3) by means of partial integration and the Bianchi and Schouten identities [5]. This exercise yields the following relations between theL odd RχT and the L odd RχT couplings: No high-energy constraint is considered for the derivation of these relations. The present study of the anomalous sector at high energies also requires the following pieces of the even-intrinsic parity Lagrangian [8]: 3 High-energy constraints

V V P Green function
We consider first the Green function V V P between two vector currents J µ,a V (x) and J ν,b V (y) and one pseudoscalar density J c P (z). In momentum space, the OPE prescribes a very precise short-distance behaviour for Π µν V V P (λp, λq, λr) when λ → ∞ [24]. Matching the V V P Green function prediction from the resonance chiral Lagrangian L odd RχT in eq. (2) and the previously referred OPE asymptotic behaviour yields [16] 4 c 3 + c 1 = 0 , If one incorporates pseudoscalar resonances and theÔ P V i operators from LagrangianL odd RχT in eq. (3), one now obtains [17] The five constraints derived in Ref. [17] for have been recast in the equivalent form in eqs. (11)- (15). The κ V l and κ V V m couplings have been rewritten in terms of the c i and d j couplings of the L odd RχT Lagrangian by means of the relations in eq. (4). This reproduces the first three constraints derived from L odd RχT in eqs. (6)- (8). Notice, however, that the inclusion of the lightest pseudoscalar resonance multiplet modifies the constraints (9) and (10).

τ − → (KKπ) − ν τ form-factors
A series of high-energy constraints were extracted from the analysis of the τ − → (KKπ) − ν τ decays [26] after demanding that the corresponding contribution to the spectral function of the vector-vector correlator vanished asymptotically 3 : In order to write the left-hand side of these equations we have employed the relations (4). The relations involving the V ϕϕϕ couplings (one vector field and three Goldstone fields) are omitted since they are irrelevant for our discussion [26].

Compatibility between constraints
In a first step, we find that the three τ − → (KKπ) − ν τ relations (16)- (18) are compatible with the V V P relations in eqs. (11)-(15) provided The first relation, eq. (25), was previously obtained in Ref. [17] (eq. (22)) after requiring the right shortdistance behaviour for both the V V P Green function and the π 0 → γγ * TFF.This condition obviously requires κ P V 3 = 0, i.e., the presence of a pseudoscalar resonance contribution.
The last constraint in (27), F 2 V = 3F 2 , is particularly interesting, as it was also previously derived in the even intrinsic-parity sector in an independent way. It was found in the high-energy analysis of the ππ vector form-factor at NLO in 1/N C [39] if the scalar resonance effects are disregarded. Likewise, the combination of the large-N C constraints for ππ vector form-factor (F V G V = F 2 ) [9] and ππ-scattering (3G 2 V = F 2 if the scalar resonance contributions are neglected) [40] also reproduces the condition F 2 V = 3F 2 . These two relations from ππ VFF and scattering also show up in the context of holographic models of QCD [33,41,42] when the derived sum-rules are restricted to the single vector resonance approximation. Indeed, recent studies in that field [33] are pointing out an interconnection between the even intrinsic-parity and anomalous sectors of QCD [33,42,43]. It is also worth to note that this value F V = √ 3F = 3G V was found to be a low-energy fixed point of the renormalized couplings F V (µ) and G V (µ) within the single vector resonance approximation, being NLO corrections in 1/N C of the order of Γ ρ /M ρ ∼ 20% [54].
The combined study of the ππ VFF and the π → γℓν axial-vector form-factor produces F V = √ 2 F = 2 G V if operators with two or more resonance fields are disregarded [9], although this is no longer so when they are taken into full consideration [22]. The large-N C study of the ππ partial-wave scattering amplitudes T I J (s) at high energies 4 yields the generalized version of the KSRF relation [44] including the effect of scalar resonances [40], Nonetheless, Ref. [45] observed that this relation left a residual logarithmically divergent behaviour in the T I J (s) at high energies. Moreover, eq. (28) is incompatible with the ππ VFF constraint F V G V = F 2 [9] and the odd-sector relation F 2 V = 3F 2 in eq. (27) if the scalar is taken into account and only the lightest multiplets are accounted for. On the other hand, phenomenologically, it is found to be reasonably well fulfilled and violations are mild: the resonance couplings extracted from pion and kaon scattering phase-shift fits to data were found to fulfill eq. (28) within 15 ↔ 20% violations [46], regardless of taking or not the c d → 0 limit. This slight tension can be relaxed if one discards the ππ partial-wave constraint in eq. (28) when only the lowest resonance multiplets are included. Alternatively, Ref. [45] advocated that the forward ππ scattering amplitude could be well described at large N C with just the lightest vector and scalar, obtaining the relation 2c 2 d + G 2 V = F 2 . This stems from assuming that the expected power behaviour from Regge theory at high energies (or a less divergent one). However, the latter constraint in combination with the ππ VFF (F V G V = F 2 ) and odd-sector one (F 2 V = 3F 2 ) leads to a scalar coupling c d = F/ √ 3 ≈ 50 MeV, which is far too large in comparison with previous phenomenological determinations where c d ∼ < 30 MeV (see Refs. [46,47] and references therein).
A good probe of the F 2 V = 3F 2 relation are the τ − → (πππ) − ν τ decays, even though its check is not free of ambiguities related to the treatment of the ρ(1450) resonance or the ππ rescattering effects producing the σ resonance. In Ref. [37] only the first contribution was included (phenomenologically) and fitting the differential decay widths as a function of the three-pion invariant masses yielded a deviation of ∼ 13% with respect to this prediction. The updated RχT TAUOLA currents [48] added also a modelization of the σ effect and benefited from the two-pion invariant mass distributions measured by BaBar [49] to reach good agreement with data [51]. The fitted value of F V is consistent with the previous prediction at one sigma, which is remarkable since the quoted error is slightly below 5%.
The consistent set of relations (27) can also be tested through the lightest meson (π 0 /η/η ′ ) TFF, which are an essential ingredient for predicting the related pseudoscalar-exchange hadronic light-by-light contribution to the muon anomalous magnetic moment. Refs. [17,52] obtain their best agreement with the π TFF data violating only the prediction for κ P V 3 in eqs. (27) by 4 ↔ 6%. The current understanding of the η − η ′ mixing in the double-angle mixing scheme allows the prediction of the related η − η ′ TFF [52], which are found in good agreement with data as well.
We find, therefore, that the minimal hadronical ansatz [38] -consisting on including as many resonance multiplets as needed to achieve a consistent set of short-distance constraints-reduces to the single (vector and pseudoscalar) resonance approximation for the V V P Green function and related anomalous form-factors. One must be aware that the OPE constraints from the V V P Green function in Sec. 3.1, where the three four-momenta p, q, r are taken to infinity at the same rate, depend crucially on the inclusion of both the lightest pseudoscalar and vector resonances. For instance, the total absence of P resonances leads to roughly a factor 2 difference with respect to alternative determinations of d 3 [50], far more important than the impact of considering F V = √ 2F or F V = √ 3F . Obviously, the unique consistent set of short-distance constraints (27) would be modified if the spectrum of the theory is enlarged.
Despite we deal with the odd-intrinsic parity sector, the constraint F V = √ 3F belongs also to the normal parity sector, where one can test the impact of heavier states. Therein, the study of two-meson vector form factors in hadronic tau decays has been sensitive to the effect of excited resonances thanks to the very good quality data taken at B-factories. In the discussion at the beginning of Sec. 4 in Ref. [35] (see also references therein) it is seen that the modifications induced by the excited resonances in the short-distance relations obtained within the single resonance approximation are -at most-of 5%.
This observation cannot be a priori generalized to the asymptotic relations involving couplings which describe interactions between more than one resonance field, where current data are not precise enough to settle this issue, although large deviations are not hinted by the time being.
In the anomaly sector, it is more difficult to quantify possible deviations and the impact from heavier states, due to both the higher complexity of the asymptotic constraints and the larger uncertainty of the measurements. The addition of the first excited vector meson multiplet does not change the analysis of asymptotic constraints derived from the V ϕ form factors [52]. However, it does modify the high-energy restrictions from the radiative pion form factor (see eqs. (A.9) and (A.10) in Ref. [53]). According to [53], the first two relations in our eq. (27) do not get modified by the inclusion of the first excited multiplet.
Nonetheless, the next three relations in (27) do indeed change. For instance, the (c 5 − c 6 ) constraint becomes [53] (c 5 − c 6 where the primed parameters refer to the excited vector V ′ and are defined in analogy to the respective V couplings. The second term on the left-hand side of eq. (29) modifies our original equation. Apart from dynamical reasons which might suppress c ′ 5 − c ′ 6 with respect to c 5 − c 6 , the factor reduces the effect of the former coupling combination by ∼ 0.3 [30]. More complicated equations are found in [53] involving d 3 , d 1 + 8d 2 and analogous and new excited resonances couplings. Likewise, there is no significant tension between the phenomenological analysis of the τ → η ( ′ ) π − π 0 ν τ data [31] and the short-distance constraints (27), which allows us to extract a conservative estimate of the impact of excited multiplets as less than 20% (see Ref. [31] for details).
New, more precise measurements of hadronic (radiative) tau decays, e + e − hadroproduction, vector meson decays, meson TFF and meson-meson scattering phaseshifts would be extremely helpful tools in increasing our knowledge of the hadronization of QCD currents in its non-perturbative regime and, in particular, in ascertaining the role of excited resonance multiplets in the corresponding dynamics and its effect in the short-distance relations obtained within the single resonance approximation. These are expected at Belle-II and forthcoming facilities.
Finally, we want to call the attention of the reader to the relations in eq. (4), which provide a dictionary between the two RχT bases L odd RχT [16] andL odd RχT [17] that can be useful in future comparisons.