# Constraints on the \(\mathbf {\omega \pi }\) form factor from analyticity and unitarity

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## Abstract

Motivated by the discrepancies noted recently between the theoretical calculations of the electromagnetic \(\omega \pi \) form factor and certain experimental data, we investigate this form factor using analyticity and unitarity in a framework known as the method of unitarity bounds. We use a QCD correlator computed on the spacelike axis by operator product expansion and perturbative QCD as input, and exploit unitarity and the positivity of its spectral function, including the two-pion contribution that can be reliably calculated using high-precision data on the pion form factor. From this information, we derive upper and lower bounds on the modulus of the \(\omega \pi \) form factor in the elastic region. The results provide a significant check on those obtained with standard dispersion relations, confirming the existence of a disagreement with experimental data in the region around \(0.6\, \text {GeV}\).

## Keywords

Form Factor Spectral Function Operator Product Expansion Elastic Region Real Analytic Function## 1 Introduction

Recent years have seen strongly increased interest in the transition form factors of light mesons [1]. One of the main reasons is the fact that pseudoscalar (\(\pi ^0, \eta , \eta '\)) pole terms constitute some of the most important contributions to hadronic light-by-light scattering, which is soon to become the biggest stumbling block in a more accurate theoretical determination of the standard model prediction for the muon’s anomalous magnetic moment; see [2] and references therein. The strength of these pole terms is determined by the singly- and doubly-virtual form factors for \(\pi ^0,\,\eta ,\,\eta ' \rightarrow \gamma ^*\gamma ^{(*)}\). However, for the virtuality of one of the photons fixed to the mass of one of the light isoscalar vector resonances \(\omega \) or \(\phi \), these form factors are intimately linked to vector-meson transition form factors that can be measured in decays such as \(\omega \rightarrow \pi ^0\ell ^+\ell ^-\) or \(\phi \rightarrow \eta \ell ^+\ell ^-\). In all likelihood, these vector-meson conversion decays present one of the few opportunities to measure the doubly-virtual \(\pi ^0\) or \(\eta \) form factors with very good precision. Furthermore, they present an essential ingredient to a more advanced theoretical understanding of hadronic light-by-light scattering [3, 4].

Recent dispersive treatments [5, 6] of the \(\omega \pi \) electromagnetic form factor \(f_{\omega \pi }(t)\) are in disagreement with experimental data in the region around \(0.6\, \text {GeV}\) [7, 8, 9], which show strong deviations from even approximate vector-meson-dominance behavior [10]. The main ingredient in the dispersion relation is unitarity, which allows one to express the discontinuity of the form factor in terms of the P partial wave of the process \(\pi \pi \rightarrow \omega \pi \) [6, 11] and the pion electromagnetic form factor, quantities determined with precision. Strictly speaking, this relation is valid only in the elastic region, \(4 m_\pi ^2\le t<16m_\pi ^2\). Due to the strong phase-space (and chiral) suppression of multiparticle intermediate states, the elastic regime approximately extends up to \(\omega \pi \) threshold in the P wave, \(4 m_\pi ^2\le t<t_+ = (m_\omega +m_\pi )^2\), above which the inelasticity in pion–pion scattering is assumed to be dominated by the \(\omega \pi \) intermediate state [11]. However, due to the lack of information on the discontinuity in the inelastic region, elastic unitarity is assumed to be valid also at higher energies in the evaluation of the dispersion integral. This assumption may affect the precision of the theoretical treatment. Having in view the disagreement with some experimental data on \(|f_{\omega \pi }(t)|\) below \(t_+\), it is of interest to investigate the form factor in a more model-independent framework, which avoids this assumption.

In the present paper we exploit alternative information on the form factor above \(t_+\). We use a method proposed originally by Okubo [12, 13] (before the advent of QCD), which leads to bounds on form factors by exploiting the positivity of the spectral function of a suitable current–current correlator. This technique, known as *method of unitarity bounds*, has been resuscitated in the QCD era and was applied to a variety of form factors of heavy and light mesons [14, 15, 16, 17, 18, 19, 20, 21] (for a review and more references see [22, 23]). In the present study, we use a dispersion relation for the polarization function of two isovector vector currents, calculated by operator product expansion (OPE) in the Euclidean region, and exploit unitarity for the spectral function. Including \(\pi \pi \) and \(\omega \pi \) states in the unitarity sum, we derive an upper bound for an integral on the modulus squared of the \(\omega \pi \) form factor along the cut from \(t_+\) to infinity. From this condition and the known discontinuity in the unphysical region \(4 m_\pi ^2\le t< t_+\), we derive bounds on the form factor in the latter region and compare them with the experimental data and the results of the standard dispersion relations.

In this work, we have adapted the original method of unitarity bounds, making it suitable for the information available on the \(\omega \pi \) form factor, namely its discontinuity across the cut below the inelastic threshold. To solve the corresponding optimization problem we use analytic techniques already applied to scattering amplitudes [24, 25]. We also had to take into account the fact that, unlike most form factors studied up to now, the \(\omega \pi \) form factor is not a real analytic function.^{1} We have therefore made the proper generalization of the formalism to the case of analytic functions that are not of real type. To our knowledge such a formulation appears here for the first time and represents an important generalization of methods existing in the literature. The above modifications may have applications in the study of other form factors as well.

In Sect. 2 we briefly summarize the standard dispersive treatment of the \(\omega \pi \) form factor. In Sect. 3 we review the formalism of unitarity bounds, which leads to the integral constraint on the modulus squared of the form factor along the cut. In Sect. 4 we solve the optimization problem and derive upper and lower bounds on \(|f_{\omega \pi }(t)|\) for \(t\) below \(t_+\). Section 5 contains the numerical results of our work and Sect. 6 our conclusions.

## 2 Standard dispersive treatment of \(f_{\omega \pi }(t)\)

In [5] the pion vector form factor has been reconstructed from an Omnès representation [27] using pion–pion phase shifts [28, 29] as input. The partial wave \(f_1\) was calculated in [11] using Khuri–Treiman techniques [30].^{2} Besides not fulfilling a straightforward variant of Watson’s final-state phase relation [31], it also has the peculiarity of showing a singular behavior at the pseudothreshold \(t_-=(m_\omega -m_\pi )^2\), which can be understood perturbatively from the analytic structure of certain non-trivial two-loop Feynman diagrams (see e.g. [32]). Note that this does not imply any singular behavior of \(f_{\omega \pi }(t)\) near \(t_-\): the form factor remains regular on the upper rim of the cut.

## 3 Consequences of perturbative QCD, analyticity, and unitarity

## 4 Bounds on \(|f_{\omega \pi }(t)|\) below \(t_+\)

In this section, we sequentially formulate the extremal problem, present its solution, and also discuss the special case of the optimal solution for the case of a real analytic form factor.

### 4.1 Formulation of an extremal problem

*i.e.*a function analytic and without zeros in \(|z|<1\), its modulus on \(|z|=1\) being equal to \(\sqrt{w_{\omega \pi }(\tilde{t}(z), -Q^2) |\mathrm{d}\tilde{t}(z)/\mathrm{d}z|}\), where \(\tilde{t}(z)\) is the inverse of (16). The construction of the outer functions is explained in [46] (see also the review [22]). Using the expression (11) of \(w_{\omega \pi }(t, -Q^2)\), we obtain for \(C(z)\) the exact analytic expression

^{3}

### 4.2 Solution of the extremal problem

### 4.3 Optimal solution for real analytic functions

*i.e.*it satisfies \(f_{\omega \pi }(t^*) =(f_{\omega \pi }(t))^*\). A similar, more general, optimization problem for real analytic functions was solved previously in [24, 25] with a slightly different method. Below we shall use the method of Lagrange multipliers applied above, adapting it to real analytic functions. In this case the discontinuity of the form factor across the cut is equal to \( 2 i \,\mathrm{Im}\,f_{\omega \pi } (x+i\epsilon )\), and from (20) we obtain the discontinuity \(\varDelta (x)\) of \(h(x)\) for \(x\in (x_\pi , 1)\) as

We end this section with a remark that might be useful for improving the bounds. From general arguments [22] and the expressions given above, it follows that the bounds depend monotonically on the value of \(I\) in the \(L^2\)-norm constraint (12): smaller values of \(I\) lead to narrower allowed intervals for \(|f_{\omega \pi }(t)|\!\) at \(t\!\!<\!t_+\). Therefore, the bounds can be made tighter in principle by taking into account more intermediate states, besides the \(\pi \pi \) pairs, in the unitarity relation (9) for the QCD correlator. Many of these are (re)measured exclusively with excellent precision in the ongoing quest to improve on the hadronic vacuum polarization contribution to the muon’s anomalous magnetic moment (see e.g. [47]). The positive contribution of these states can be subtracted from the QCD value of the correlator as in (13), reducing the value of \(I\). To establish the practical effect on the bounds requires further investigations.

## 5 Results

In the calculation of the bounds we have employed the discontinuity of the \(\omega \pi \) form factor in the range \((4 m_\pi ^2, t_+)\) from two different analyses: the recent dispersive treatment reported in [5] and the older work [26]. For the quantity \(I\) we used the estimate given in (15). It turns out that the value of the form factor at \(t=0\) plays a significant role in producing stringent constraints. Although in principle \(f_{\omega \pi }(0)\) can be complex, we assumed that it only has a small phase which can be neglected [5]. The upper and lower bounds given below were obtained using as input the central value \(f_{\omega \pi }(0)=2.30\, \text {GeV}^{-1}\).

We checked that the bounds are quite stable with respect to the variation of the input: by varying \(f_{\omega \pi }(0)\) inside the error quoted in (6), the upper bounds in the region of interest are changed by at most 2.5 %. Also, the uncertainty of the quantity \(I\) quoted in (15) affects the bounds by at most 2 %. As the experimental errors are currently rather in the 10–20 % range, we refrain from displaying these small variations in the bounds graphically and only discuss the central results.

Figure 1 shows that, although the allowed ranges for the ratio \(|f_{\omega \pi }(t)/f_{\omega \pi }(0)|^2\) calculated with (41) and (47) are rather large, the upper bounds exclude some of the data points from [7, 8, 9] in the region above \(0.6\, \text {GeV}\). Specifically, the exclusion is at the \(4\sigma \) level for the last point (at \(0.63\, \text {GeV}\)) from [9], and at about the \(2\sigma \) level for the other points lying above the upper bounds. On the other hand, the dispersive calculation performed in [5] is situated in the allowed range for the modulus derived here. Note also that in [48, 49], \(f_{\omega \pi }(t)\) has been calculated based on a low-energy effective theory including explicit vector-meson degrees of freedom. We have checked that, although the representation [48, 49] rises more quickly than the dispersive ones [5, 26] and follows the data more closely than most others, it still lies comfortably inside the bounds.

## 6 Discussion and conclusions

The present study was motivated by the discrepancies noticed recently between the theoretical calculation of the \(\omega \pi \) form factor by a dispersion relation and some of the data reported in [7, 8, 9]. Our aim was to avoid the assumptions made on the discontinuity of the form factor above the threshold \(t_+\), where the elastic unitarity (2) is no longer valid. To this end we have resorted to the formalism of unitarity bounds. The central point of the formalism is the derivation of an integral condition on the modulus squared of the form factor from \(t_+\) to infinity, which can be calculated using OPE and perturbative QCD in the Euclidean region for a suitable correlator, together with unitarity and positivity of the spectral function. In the present case it was convenient to consider the tensor (7) of two isovector currents.^{4} From the integral condition, by using techniques of analytic interpolation theory [46], one can derive bounds on the form factor or its derivatives at points inside the holomorphy domain.

In this work we have considered a modified version of the standard formalism, suitable for including the information available on the \(\omega \pi \) form factor, namely the discontinuity (2) known in the elastic region. For real analytic functions, the bounds (47) are consistent with the more general results given in [24, 25]. We have also derived bounds on the modulus of the form factor in the case when, due to rescattering effects, it is not an analytic function of real type. This is a generalization of the formalism of unitarity bounds, considered for the first time in this paper. The framework is not specific to the \(\omega \pi \) form factor and could easily be adapted to the analysis of other form factors. The derivation uses the maximization of the modulus of a difference of complex numbers in the last step, which implies that strictly speaking the bounds (41) are not optimal. In practice, for the real analytic form factor with the discontinuity from [26], the predictions of (41) are almost indistinguishable from the optimal bounds (47) in the energy region of interest.

The numerical results show that several experimental data around \(0.6\, \text {GeV}\) are situated above the upper bounds derived in the present paper, using two different evaluations of the discontinuity (2) in the elastic region. Having in view the model-independent treatment of the region above the inelastic threshold \(t_+\) adopted in our analysis, the disagreement signals possible problems with the experimental data. Adding to the seeming inconsistency between data on \(f_{\omega \pi }(t\le t_-)\) from \(\omega \rightarrow \pi ^0\ell ^+\ell ^-\) and data on \(f_{\omega \pi }(t\ge t_+)\) from \(\mathrm{e}^+\mathrm{e}^-\rightarrow \omega \pi ^0\) [50, 51, 52] (see e.g. the compilation in [1] or the continuation of the calculation [5] above \(t_+\) shown in [53]), our results therefore strongly support the desirability of renewed experimental efforts to measure the \(\omega \) conversion decay in a more exclusive setting [54, 55].

## Footnotes

- 1.
A function \(F(t)\) analytic in the \(t\)-plane cut for \(t\ge 4 m_\pi ^2\) is of real type if it satisfies the condition \(F(t^*)= (F(t))^*\). In particular, this implies that the function is real on the real axis for \(t<4 m_\pi ^2 \), while its discontinuity across the cut can be written as \(\mathrm{disc}\,F(t)\equiv F(t+i\epsilon )- F(t-i\epsilon )=2 i \, \mathrm{Im}\,F(t+i\epsilon )\).

- 2.
- 3.
- 4.
It is easy to see that, if one considers the full electromagnetic current in (7), the QCD contribution (14) will be scaled by a factor greater than unity (equal to 4/3), and therefore the formalism will lead to weaker bounds. They can be improved if one includes the most important low-energy isoscalar states, approximated by narrow \(\omega \) and \(\phi \) resonances, in the unitarity sum, which will be subtracted along with the two-pion contribution as in (13). As the bounds obtained in the present work correspond to subtracting the

*full*isoscalar contribution, they are obviously slightly stronger than those based on the full current in such a procedure.

## Notes

### Acknowledgments

We would like to thank Stefan Leupold for useful comments and suggestions. Financial support by the Ministry of Education (Contract PN 09370102/2009), by CNCS-UEFISCDI (Program Idei, Contract 121/2011), by the DFG (SFB/TR 16, “Subnuclear Structure of Matter”), and by the project “Study of Strongly Interacting Matter” (HadronPhysics3, Grant Agreement No. 283286) under the 7th Framework Program of the EU is gratefully acknowledged.

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