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Is there an isovector companion of the X(2175)?

In this letter we study the reaction e^{+}e^{-}\to\phi \pi^0 \eta with the \pi^{0}\eta system in an isoscalar s-wave configuration. We use a formalism recently developed for the study of e^{+}e^{-}\to\phi \pi\pi and e^{+}e^{-}\to\phi K^{+}K^{-}. The obtained cross section is within the reach of present e^{+}e^{-} machines. Measuring this channel would test the absence of an isovector companion of the X(2175) as predicted by the three body approach to this resonance.


I. INTRODUCTION
Recently, a detailed study of the reactions e + e − → φππ and e + e − → φK + K − was performed for √ s ≥ 2 GeV [1,2]. These processes involve the γφππ and γφK + K − vertex functions respectively which are calculated at low photon virtuality and low di-meson invariant mass using Resonance Chiral Perturbation Theory (RχP T ). The dynamics is the scalar poles. The related process e + e − → φf 0 has been also studied in the context of Nambu-Jona-Lasinio models [3].
In the e + e − → φππ case, the calculation reproduces the background in the total cross section but not the peak around 2175 MeV, discovered by BaBar [4] and confirmed by Bes in J/Ψ → ηφf 0 (980) [5] and Belle also in e + e − → φπ + π − [6].
Another proposal for the nature of X(2175), which turns out to be consistent with the observed mass and width, is a three-body φKK system [13]. The result of this framework is a neat resonance peak around a total mass of 2150 MeV and an invariant mass for the KK system around 970 MeV, quite close to the f 0 (980) mass. The state appears in the isospin I = 0 sector, and qualifies as a φf 0 (980) resonance. Interestingly, the theory also shows that there is no resonance in the isovector channel. This is the main motivation to study the reaction e + e − → φπ 0 η, whose final state is in a pure isovector state.
In this work we apply the same formalism as in [1,2] to the production of φπ 0 η. We start form the RχP T Lagrangian and follow the conventions in [14]. The relevant interactions in their notation are where We introduce the photon field through v µ = eQA µ and F µν L = F µν R = eQF µν (e > 0) where F µν denotes the electromagnetic strength tensor. At the energy of the reaction, √ s ≥ 1.7GeV, it is quite probable to excite higher states.
Here, we consider the excitation of virtual K * K states and their contribution to e + e − → φπ 0 η through the chain e + e − → K * K → φKK with the rescattering of the kaon pair to π 0 η as shown in Fig. (2).
The V V ′ P interaction in this diagram is dictated by the anomalous chiral Lagrangian [15] which we rewrite in terms of the tensor field as This Lagrangian was obtained in [15]. In this formalism no direct V P γ coupling emerges and this vertex is generated by the Lagrangian in Eq. (7) in combination with the V γ interaction in Eq. (3) (see the discussion above Eq. (56) of [15]).
The calculation of the invariant amplitude is similar to the previously studied φππ final state and we refer the reader to Ref. [1] for the details. Both RχP T contributions in Fig. (1) and the contributions of Fig. (2) turn out to be divergent. However, it is shown in it is shown there that the only divergent piece corresponds to the same scalar integral appearing in meson-meson scattering. The substraction constant required for this integral was discussed in [16,17] and we briefly review it below.
The total amplitude for e + (p where k 2 = (p + + p − ) 2 , V KKπη denotes the leading order on-shell amplitude for KK − π 0 η scattering, L µ ≡ v(p + )γ µ u(p − ) and the Lorentz covariant structures are given by The I, J functions entering Eq. (8) are given by with the divergent loop integral where µ is the dimensional regularization scale, i (l) = l 2 − m 2 i and Here σ(p 2 ) = 1 − 4m 2 K p 2 and the integrals in Eqs. (10,11) are given by The divergences in the loop integrals are contained in a(µ) in Eqs. (10,12) and we discuss its physical value in detail below.
From Eqs. (8,11) we can see that the dynamics is dominated by two main effects in the γ * φπ 0 η vertex function which occur at two different energy scales: the leading order terms for the KK − π 0 η on-shell scattering amplitude at the di-meson invariant mass scale, m πη , and the electromagnetic meson form factors at the energy of the reaction, √ s. The calculation of the γ * φπ 0 η vertex function involved in this amplitude is strictly accurate for low di-meson invariant mass and low virtuality of the photon. We improve this results in two respects: i) we replace the leading order result for the KK − π 0 η on-shell scattering amplitude by the unitarized amplitude containing the scalar poles and ii) we replace the leading order terms in the kaon form factor by the unitarized one and, following [2], the leading order terms of the K * K transition form factor are replaced by the complete form factor at the energy of the reaction as extracted from data.
The isovector s-wave KK − π 0 η unitarized scattering amplitude denoted t 0 KKπη gives a successful description of the corresponding cross section up to m πη ≈ 1GeV and is based on the imposition of unitary constraints in coupled channels of χPT. Following [16,17], unitarization reduces to the solution of the Bethe-Salpeter equation where T is the matrix of unitarized amplitudes of the desired isospin channel The elements of the matrix T (2) are the corresponding on-shell amplitudes calculated in χPT at O(p 2 ). A straightforward calculation yields The diagonal matrix G contains the loop integrals with G KK (m 2 πη ) given in Eq. (12) and with and The substraction constant has been fixed in analogy with Ref. [17] to a(µ 0 ) = 0.87 for µ 0 = 1.2 GeV matching Eq. (19) to the cutoff regularized integral with a cutoff parameter Λ = 1 GeV. It is related at different scales as a(µ) = a(µ 0 ) + log µ 2 µ 2 0 , and therefore the loop function is scale independent.
The unitarized amplitudes are obtained solving the algebraic equation (15). In particular, it was shown in [16] that t 0 KKπη has a pole corresponding to the a 0 (980) resonance which manifests as a peak in the squared amplitude as shown in Fig. (3).
Concerning the meson electromagnetic form factors, the high virtuality of the intermediate photon involved in our process requires to work out the complete γKK vertex functions.
The calculation of the isovector kaon form factor (F K * K ) follows from [2], where an appropriate characterization of this form factor at the energy of the reaction is done. As this form factor is physically dominated by resonances, in [2] it is described with the lowest order terms obtained in RχPT complemented with the exchange of a ρ ′ F (1) with the energy dependent width used in [19] Γ where The values for the constants appearing here are extracted from the central values of Table   XV in [19] as B ρ ′ 4π = 0.65, m ρ ′ = 1504 MeV and Γ ρ ′ = 438 MeV (see also [20]). The parameter b 1 is then fitted to the experimental data for the isovector cross sections reported for e + e − → K * K at √ s = 1400 − 3000 MeV in table VII of [19]. The fit to the isovector cross section within 1σ yields b 1 = −(0.255 +0.030 −0.040 ) × 10 −3 MeV −1 . We use these results to calculate the π 0 η spectrum, obtaining where Here (ω, Q) stands for the momentum of the φ in the center of momentum system of the reaction and p denotes the momentum of the final pion in the π 0 η center of momentum with

III. NUMERICAL RESULTS
We evaluate numerically the integrals and the differential cross section. The mechanisms studied here were shown to be responsible for the production of most of the events in the case of the φππ final state except for the resonant events due to the X(2175) [1]. On the other hand, in the case of the φK + K − final state studied in [2], within the limitations due to the extraction of the K * K isovector transition form factor from data, a good description of experimental points is obtained but there seems to be room for additional contributions around 2200 MeV. In this case, the φK + K − system can be in both isoscalar and isovector states hence the natural candidate for additional contributions is an isovector companion of the X(2175). Interestingly, in the three-body description of the X(2175) proposed in [13], no peak is generated in the isovector channel.
Our calculation of the pure isovector channel e + e − → φa 0 (980) under the mechanisms studied in [1] yields a cross section of the same size as that of e + e − → φf 0 and thus within the reach of present e + e − machines. If there is no such a thing as an isovector companion of the X(2175) as expected from [13], then our result is a concrete theoretical prediction for the e + e − → φa 0 (980) cross section; otherwise, the isovector companion must contribute to this process yielding valuable information on the nature of mesons at this energy. Hence we encourage experimentalists to measure the e + e − → φπ 0 η channel.