The study of ρ-ω mixing is a very interesting subject in hadron physics both theoretically and experimentally. The inclusion of ρ-ω mixing effect is crucial for a good description of the pion vector form factor in process, which quantifies the hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. On the experimental side, several experimental collaborations, such as KLOE^[1–2] and BESIII,^[3] have recently launched measurements of the with high statistics and high precision.

The ρ-ω mixing amplitude was assumed to be a constant or momentum-independent in the early stage of previous studies.^[4–5] The authors of Ref. [6] suspected the validity of the constant assumption and, based on a quark loop mechanism of ρ-ω mixing, they found that the mixing amplitude is significantly momentum-dependent. Since then, the use of various loop mechanisms for ρ-ω mixing is triggered in different models such as extended Nambu-Jona-Lasinio (NJL) model,^[7] the global color model,^[8] the hidden local symmetry model,^[9–11] and the chiral constituent quark model.^[12–13]

In this work, we aim at studying ρ-ω mixing in a model independent way by invoking Resonance Chiral Theory (RχT).^[14] It provides a reliable tool to study physics in the intermediate energy region.^[15–20] The tree-level calculation of ρ-ω mixing in the framework of RχT has been given in Refs. [21–22], however, the tree-level mixing amplitude turns out to be momentum-independent. In order to implement the momentum dependence, here we will calculate the one-loop contributions as shown in Fig. 1. The ρ-ω mixing can be induced either by strong isospin-violating or by electromagnetic effects. The former is proportional to the mass difference bewteen the quarks, i.e., and the latter is accompanied by the fine structure constant α. In the present study, only the mixing effects linear in or α are under our consideration. Apart from the overall factors or α, the large-N_C counting rule proposed in Ref. [23] is imposed to truncate our perturbative calculation. Specifically, our calculations are truncated at next-to-leading order in the expansion for the strong and electromagnetic contributions. The ultraviolet (UV) divergence from the loops is cancelled by introducing counterterms with sufficient derivatives and the involved couplings are assumed to be beyond the leading order in expansion as claimed in Ref. [24].

Fig. 1

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	Fig. 1 Feynman diagrams contributing to ρ-ω mixing.

We assess the impact of momentum-dependent ρ-ω mixing amplitude on the pion vector form factor by fitting to the experimental data extracted from the process and decay in the energy region of (650–850) MeV. Besides, the decay width of is implemented as a constraint in the fit. It is known that, provided isospin invariance holds, the isovector part of the pion form factor in the annihilation is related to the one in τ decays theoretically, via the conserved vector current assumption.^[25–26] Different effects of isospin breaking have been studied to describe the annihilation data and τ decays data simultaneously,^[26–34] such as the short distance and long distance corrections in the τ partial decay width to two pions, charged and neutral ρ mass and width difference, and ρ-ω mixing. In our study we will take into account all the above isospin breaking effects. Our fit result shows that the ρ-ω mixing amplitude is significantly momentum-dependent and its imaginary part is much smaller than real part. Based on the fitted values of the parameters, we also analyze the decay width of by including the effect of the ρ-ω mixing. It is found that the decay width is dominated by the ρ-ω mixing effect while the contribution from the direct coupling of is negligible.

This paper is organized as follows. In Sec. 2, we introduce the description of ρ-ω mixing. In Sec. 3, we present the theoretical framework and elaborate on the calculation of the tree-level and loop contribution of ρ-ω mixing. In Sec. 4, the fit result is shown and the related phenomenology is dicussed. A summary is given in Sec. 5.

2 Generic Description of -ω Mixing

In the isospin basis , we define and for convenience. The mixing between the isospin states of and can be implemented by considering the self-energy matrix

In above the vector-current conservation has been used to eliminate the longitudinal part proportional to p_μ. Furthermore, we have also neglected terms of , since they correspond to contributions at two-loop order and are beyond our accuracy. m_ρ and m_ω are bare masses of the ρ and ω mesons, respectively.

The ρ-ω mixing, i.e., mixing between the physical states of ρ⁰ and ω, is obtainable by introducing the following relation

Solving the above equation and neglecting higher-order terms of and , one obtains:

The two mixing parameters should be just different with each other slightly, see Ref. [35] for more details.

In this section we will calculate the mixing amplitude using RχT so as to study its momentum dependence. The information of ρ-ω mixing is encoded in the off-diagonal element of the self-energy matrix, which can be decomposed as

3.1 Resonance Chiral Theory and Tree-Level Amplitudes

In RχT, the vector resonances are described in terms of antisymmetry tensor fields with the normalization

with being the polarization vector. The kinetic Lagrangian of vector resonances takes the form^[14]

where M_V is the mass of the vector resonances in the chiral limit. Here the vector mesons are collected in a 2 × 2 matrix

Besides, the covariant derivative and chiral connection are defined by

The Goldstone bosons originating from the spontaneous breaking of the chiral symmetry are nonlinearly parametrized as

with F being the pion decay constant.

In the isospin limit, the standard Lagrangian describing the interactions between and Goldstone bosons or electromagnetic fields are given by

with the relevant building blocks defined by

Here are field strength tensors composed of the left and right external sources l_μ and r_μ, and F_V, G_V are real couplings.

The LO isospin-breaking effect is introduced by the Lagrangian

with and . Here v₈ is an unknown coupling constant. However, it can be determined by considering the mass relations of the vector mesons at in terms of the quark counting rule,^[21] which leads to: .

With the above preparations, one is now able to calculate the tree amplitudes. The tree-level strong contribution, corresponding to diagram (a) in Fig. 1, turns out to be

which is counted as , since . The tree-level electromagnetic contribution is from diagram (b) in Fig. 1 and the amplitude can be obtained by using the Lagrangian in Eq. (13):

with the N_C-counting order being due to . As mentioned in the beginning of this section, the leading-order strong and electromagnetic contributions indeed start at different orders in expansion.

3.2 Loop Contributions

The relevant loop diagrams contributing up to our accuracy are shown in the second and third lines of Fig. 1. Diagrams (c) and (d) contribute to the strong correction at , which are next-to-leading order compared to diagram (a). Likewise, with respect to diagram (b), diagrams (e)–(h) lead to electromagnetic corrections at next-to-leading order, i.e. . In our calculation below, the necessary isospin-breaking vertices are constructed based on the basic chiral building blocks taken from χPT^[36] and RχT.^[14]

(i) Diagram (c): Loop

The vertex of can be read from the Lagrangian in Eq. (13). For the isospin-violating vertex of , we construct the following Lagrangian

For convenience, we define the combination . The -loop contribution can be obtained by calculating the integral

where p and k denote the momenta of the external vector meson and either of the exchanged pions, respectively. After integrating, the structure function can be extracted, which reads

where and with and γ_E being the Euler constant.

(ii) Diagram (d): π-Tadpole Loop

According to the Lorentz, P and C invariances, the Lagrangian corresponding to the interaction of can be written down as follows:

Note that the term, which contributes to the contact interaction of ρ-ω mixing, also yields vertex. Though in Eq. (21) there are many terms with a large number of free couplings, the final result only depends on combinations of these couplings. For simplicity, the following two combinations are necessary, i.e.,

Furthermore, one can neglect the mass difference between the charged and neutral pions in the internal lines of loops, since the resultant difference is of higher orders beyond our consideration. As a result, the expanded form of Lagrangian (21) can be reduced simply to

With the above Lagrangian, the π-tadpole contribution to the ρ-ω mixing can be derived:

Eventually, the explicit expression of the strong structure function has the form of

(iii) Diagrams (e)–(h): Loops

In the loop diagrams (e)–(h), there are two types of vertices. The coupling of vector meson (V) as well as vector external source (J) to pseudoscalar (P) is labeled by VJP vertex for short. The interaction of two vector mesons and one pseudoscalar is called VVP vertex. The operators of VJP type are given in Ref. [37]:

and the ones of VVP type are

The involved couplings or their combinations can be estimated by matching the leading operator product expansion of Green function to the result calculted within RχT. Such a procedure leads to high energy constraints on the resonance couplings as follows:^[37]

The mass of vectors in the chiral limit, M_V, can be estimated by the mass of meson.^[38]

The loops diagrams (e)–(h) can be calculated simultanously if the effective vertices of and are used, where a “*” stands for an off-shell particle. The explicit expression for reads

where p and k denote the momentum of the vector meson and the photon, respectively. Analogically, for , one has

It should be stressed that there are two terms in each effective vertex. One corresponds to the case that the virtual photon is coupled to the VP system directly, while the other to the case that it is interacted through an intermediate vector meson. Note also that, throughout this work we only account for the corrections proportional either to or , which implies the calculation of electromagnetic contribution can be carried out in the isospin limit, i.e., .

With the help of the effective vertices, the loop contribution, i.e., the sum of the loops diagrams (e)–(h), can be expressed as:

The further calculation is straightforward but the result of the extracted electromagnetic structure function is too lengthy to be shown here. It is worthy noting that in our numerical computation we will use the high energy constraints in Eq. (28) together with the fitted parameters given in Ref. [18], therefore, all the parameters involved in are known.

(iv) Counterterms and Renormalized Amplitude

Up to now, the total contribution of ρ-ω mixing can be expressed as

which is still unrenormalized. The resonance chiral theory is unrenormalizable in the sense that the amplitude has to be renormalized order by order with increasing number of counterterms when the accuracy of the calculation is improved. In our case, the tree amplitudes, and , can only absorb the ultraviolet divergence proportional to p⁰. In order to cancel the , and stemming from the loop contribution and , additional counterterms are needed. For this purpose, we construct

We adopt the subtraction scheme and absorb the divergent pieces proportional to by the bare couplings in the counterterms. Consequently, the remanent finite pieces of counterterms can be written as:

with

In summary, the UV-renormalized mixing amplitude reads

where a bar indicates the divergences are subtracted. As discussed in Ref. [35], there is an important constraint on the mixing amplitude, namely, it should vanish as . Thus the final expression of the renormalized mixing amplitude should be

where an additional finite shift is imposed so as to guarantee that the constraint is satisfied.

In our numerical computation, the scale μ will be set to M_ρ and we use provided by particle data group (PDG).^[39] Furthermore, we can define

and in principle the unknown parameters in Eq. (35) are a, f₄, , , and .

4 The Effect of -ω Mixing on Pion Vector Form Factor

The mass and width of ρ meson are conventionally determined by fitting to the data of and ,^[39] where various mechanisms are introduced to describe the ρ-ω mixing effect. To avoid intervening by their ρ-ω mixing mechanisms, we do not employ their extracted values for the mass and width, rather, we set the mass M_ρ, the relevant couplings G_ρ and F_ρ to be free parameters in our fit. As for the width, an energy-dependent form will be imposed, which is supposed to be dominated by the two π decay channel:^[40]

For the narrow-width resonance ω, we take and from PDG.^[39] The physical coupling F_ω can be extracted from the decay width of . Using the Lagrangian , one can derive the decay width

The experimental data considered in this work are the pion form factor of the process^{[1–3,41–45]} and decay^[25,46] in the energy region of (650–850) MeV, and the decay width of .^[39]

The Feynman amplitude for the process , proceeding via virtual intermediate hadrons, i.e., ρ, ω and their mixing, is described by^[35]

Here the fourth term corresponds to higher-order contribution of isospin breaking, e.g., proportional to , which is beyond our accuracy and hence can be neglected. Including the contribution from the direct coupling of photon to the pion pair, the pion form-factor in annihilation reads

On the other hand, the expression of the pion form-factor in decay is

Our best-fitted parameters and the corresponding are compiled in Table 1. Our determination of the mass of ρ meson is in good agreement with the value reported in PDG.^[39] The fit results are plotted in the Fig. 2. One can see that the experimental data of pion form factor, especially the kink around the mass of ω in the process, is well described.

Fig. 2

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	Fig. 2 (Color online) Fit results for the pion form factor in the process (a) and process (b). The data of annihilation are taken from the OLYA and CMD,[41] CMD2,[42–43] DM1,[44] SND,[45] KLOE,[1–2] BESIII[3] collaborations. The τ decay data are taken from the ALEPH[46] and CLEO[25] collaborations. The solid lines are our theoretical predictions.

Table 1

Table 1

Table 1 The fit results of the parameters. . Fit results /MeV Gρ/MeV Fρ/MeV a/MeV−1 314.9/(242–8)=1.35

Table 1 The fit results of the parameters. .

In Fig. 3, contributions at different orders to the real and imaginary parts of the pion form factor are displayed. The leading-order contribution (mixing-effect irrelevant) includes the contact interaction and the ρ-mediated mechanism, namely the first two terms on the right side of Eq. (41). The next-to-leading-order contribution includes the ρ-ω mixing term and the direct coupling, namely the third term plus the forth term on the right side of Eq. (41). As expected, the isospin-breaking effects mainly affects the energy region around the masses of ρ and ω. It is found that the dominant contribution is from the imaginary part in that region. The isospin-breaking effects increase the absolute value of imaginary part around the ρ peak, and accounts for that the data are higher than the τ data in that region as shown in Fig. 2. Similar behavior has also been observed in Ref. [11] where the ρ-ω mixing was treated in hidden local symmetry model.

Fig. 3

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Fig. 3 (Color online) The real and imaginary parts of the fitted form factor . The black solid and red dashed lines represent our best results of the real and imaginary parts, respectively. The blue dotted and cyan dash-dot-dotted lines correspond to the leading order and the second order contributions of the real parts, respectively. The magenta dash-dotted and green short dash-dash-dotted lines denote the leading order and second order contributions of the imaginary parts.

In Fig. 4, we plot the real and imaginary parts of the mixing amplitude . It is found that the real part is dominant almost in all the region and its momentum-dependence is significant. Compared to the real part, the imaginary part is rather small. For the imaginary part, the contributions from loop and loop are of the same order, but with opposite sign. Note that the π-tadpole is real and s-independent as can be seen from Eq. (25). The smallness of the imaginary part is consistent with the observation in Refs. [5, 48], though therein the effect of direct was not taken into account and even in Ref. [5] the isospin breaking is considered to be purely electromagnetic origin. We also note that larger imaginary part is obtained in Refs. [8, 13] by using global color model and a chiral constituent quark model, respectively. However, our finding is more reliable in the sense that it is based on a model-independent description of the ρ-ω mixing and, moreover, constraint from experimental data is imposed by means of fitting.

Fig. 4

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	Fig. 4 (Color online) The real part (a) and imaginary part (b) of the mixing amplitude . The black solid lines represent our best fitted results. For the imaginary part, the red dashed and blue dotted lines correspond to the contribution of loop and loop, respectively.

The values of at physical masses of ρ or ω are interesting since they are related to the mixing parameters given in Eq. (6). To that end, we obtain: at , , and ; at , and . As expected, ϵ₁ and ϵ₂ come out to be almost the same. Note that, in the numerical calculation of ϵ_i, we have neglected the small imaginary part of the mixing amplitude as well as the widths of the ρ and ω resonances. This leads to a real number of ϵ_i and hence a probability interpretation can be assigned.

Using the central values of the fitted parameters in Table 1, we calculate the decay width of

From Eq. (44), we can find that the first term due to the direct is less than the second term due to the ρ-ω mixing by two orders. In other words, the direct coupling only affects the decay width less than one percent. Within 1σ uncertainties, our theoretical value of the branching fraction is , which agrees with the values given in PDG^[39] and by the recent dispersive analysis.^[49]

We have analyzed the ρ-ω mixing within the framework of resonance chiral theory. Based on the effective Lagrangians constructed under the guidance of various symmetries, we calculate the ρ-ω mixing amplitude up to next-to-leading order in large expansion. Importantly, the momentum-dependent effect is implemented due to the inclusion of loops in our calculation. The values of the resonance couplings are determined by fitting to the data of the pion vector form factor extracted from the process and decay. The decay width of is served an additional constraint in the fit as well. It is found that the imaginary part of the pion form factor is enhanced largely around the ρ peak. The ρ-ω mixing amplitude is dominated by its real part almost in all the region, which is significantly momentum-dependent. On the contrary, its imaginary part is relatively small. We also find that ρ-ω mixing plays a major role in the decay width of , and its contribution is two orders of magnitude larger than that from the direct coupling.