at energies up to 2-3 GeV in the many-channel approach

Using the field-theory-inspired expression for the pion electromagnetic form factor Fπ, a good description of the data in the range −10 < s < 1 GeV is obtained upon taking into account the pseudoscalar-pseudoscalar (PP) loops. When the vector-pseudoscalar (VP) and the axial vectorpseudoscalar (AP) loops are taken into account in addition to the PP ones, a good description of the BABAR data on the reaction ee → ππ is obtained at energies up to 3 GeV. The inclusion of the VP and AP loops demands the treatment of the reactions ee → ωπ and ee → ππππ. This task is performed with the SND data on ωπ production and the BABAR data on ππππ production, both in ee annihilation, by taking into account ρ(770) and the heavier ρ(1450), ρ(1700), and ρ(2100) resonances. The problems arising from including of the VP and AP loops are pointed out and discussed.


I. INTRODUCTION
Some time ago we suggested a new expression for the electromagnetic form factor of the pion F π [1][2][3], which describes the data on the reaction e + e − → π + π − [4][5][6][7] restricted to the time-like region 4m 2 π < s ≤ 1 GeV 2 . The expression takes into account the strong resonance mixing via common decay modes and the ρω mixing. It has both the correct analytical properties and the normalization condition F π (0) = 1, and can be represented in the form: F π (s) = 1 ∆ (g γρ1 , g γρ2 , g γρ3 , .. where i (i = 1, 2, 3, ...) counts the ρ-like resonance states ρ 1 ≡ ρ(770), ρ 2 ≡ ρ(1450), ρ 3 ≡ ρ(1700), ..., the quantity (V = ρ 1,2,3,... , ω) is introduced in such a way that eg γV is the γV transition amplitude, where e is the electric charge. As usual, the coupling constant g V is calculated from the electronic width 3) * achasov@math.nsc.ru † kozhev@math.nsc.ru of the resonance V . The quantities g ij /∆ are the matrix elements of the matrix G −1 given by Eq. (3.7) below, and ∆ = detG. Ellipses mean additional states like ρ(2100) etc. It is assumed that the direct G-parity-violating decay ω → π + π − is absent, that is, g ωππ = 0. The quantity Π ρ1ω is responsible for the ρω mixing. See Ref. [1] for more details concerning Eq. (1.1). We note that an expression similar to Eq. (1.1) was used earlier [8] for the description of data in the time-like domain, but it had a disadvantage in that the normalization condition F π (0) = 1 was satisfied only within an accuracy of 20%.
Using the resonance parameters found from fitting the data [4][5][6][7], the continuation to the space-like region s < 0 was made, and the curve describing the behavior of F π (s) in the range −0.2 GeV 2 < s < 0 GeV 2 was obtained [1] and compared with the data [9] in this interval of the momentum transfer squared. The space-like interval was further expanded to s = −10 GeV 2 in a subsequent work [3], and a comparison was made with the data [10][11][12] existing in that interval. The basic ingredient in the above treatment is the inclusion of the pseudoscalarpseudoscalar (PP) loops, specifically, the π + π − and KK ones. These contributions are dominant at the center-ofmass energy √ s ≤ 1 GeV. Going to higher energies (up to 3 GeV) of the reaction e + e − → π + π − [7] -requires the inclusion of the vector-pseudoscalar (VP) and axial vector-pseudoscalar (AP) intermediate states. This is the aim of the present work. The particular VP state ωπ 0 produced in e + e − annihilation was studied by the SND team in Ref. [13], while the AP state of the type a 1 π is the intermediate state in the reaction e + e − → π + π − π + π − studied by BABAR [14]. An attempt to describe these reactions in the framework of the three-channel approach, taking into account the PP, VP, and AP intermediate states, is also undertaken in the present work. Furthermore, a suitable scheme with three subtractions for the nondiagonal polarization operators is used in the present work as opposed to Refs. [1][2][3] where the scheme with two subtractions was used.
Of course, there are many works in the current literature devoted to analyzing of the pion form factor in models that are different from that proposed here and in Refs. [1][2][3]. In particular, a model with a broken hidden local symmetry added to the π + π − and KK loops at energies √ s ≤ 1 GeV was used in Refs. [15][16][17], without attempting to extend the analysis to higher energies. A subtraction scheme different from ours was used there for the calculation of the pseudoscalar-loop contribution. The task of extending the energy region above 1 GeV was undertaken in Ref. [18], taking into account the contributions of heavier rho-like resonances. However, the mixing among these resonances -necessarily arising due to their common decay modes -was not taken into account in that work. A model similar to the K-matrix approach but with improved analytical properties was proposed in Ref. [19]. As opposed to the above works, the present work uses the field-theory-inspired approach to the problem which takes into account relevant PP-, VP-, and AP-loop contributions and the strong mixing of the ρ-like resonances arising via their common decay modes.
The paper is organized as follows. The polarization operators arising due to the vector -pseudoscalar and axial vector -pseudoscalar loops (the diagonal and nondiagonal) and the nondiagonal pseudoscalar -pseudoscalar polarization operator, are calculated in Sec. II. The expression for the pseudoscalar-pseudoscalar diagonal polarization operator [1][2][3] is reviewed in the same section. The quantities for comparison with experimental data are discussed in Sec. III. The results of the data fitting are represented in Sec. IV. This section also contains a discussion of the problems that arise when including the VP and AP loops. Section V contains the conclusions drawn from the present study. DUE TO  PSEUDOSCALAR-PSEUDOSCALAR,  VECTOR-PSEUDOSCALAR, AND AXIAL VECTOR-PSEUDOSCALAR LOOPS

II. POLARIZATION OPERATORS
The final states π + π − , ωπ 0 , and π + π − π + π − considered in the present work, have the isotopic spin I = 1. Hence, they are produced in e + e − annihilation via the unit spin ρ-like intermediate states ρ 1 ≡ ρ(770), ρ 2 ≡ ρ(1450), ρ 3 ≡ ρ(1700), etc. These states have rather large widths and are mixed via their common decay modes. The finite width and mixing effects are taken into account by means of the diagonal and nondiagonal polarization operators Π ρiρj [1]. In particular, the effects of finite width appear in the inverse propagator of the resonance ρ i via the replacement Indeed, according to unitarity relation, the particular contribution to the imaginary part of the diagonal polarization operator is due to the real intermediate state ab: Hereafter, the quantity s is the energy squared. As explained earlier [1], the dispersion relation written for the polarization operator divided by s, automatically guarantees the correct normalization of the form factor F π (0) = 1. In the present work, the states which are taken into account are the PP states π + π − , K + K − + K 0K 0 of the pair of pseudoscalar mesons, the VP states ωπ 0 , K * + K − +K * 0K 0 +K * − K + +K * 0 K 0 , and the AP states a + 1 (1260)π − + a − 1 (1260)π + , K 1 (1270)K+ c.c. The polarization operators due to the PP loops are considered in detail elsewhere [1].
The following subtraction scheme is used in the present work. The diagonal polarization operators Π ρiρi (s) are regularized by making two subtractions, at s = 0 and at the respective mass squared s = m 2 ρi , i = 1, 2, ...: The nondiagonal polarization operators Π ρiρj (s), i = j, are regularized by making three subtractions, at s = 0 at s = m 2 ρi , and at s = m 2 ρj i, j = 1, 2, ...: The corresponding expression, in the case of two-particle state ab, is Π (ab) ρiρj (s) = g ρiab g ρj ab Π ρiρj (s, m ρi , m ρj , m a , m b ), (2.5) where (2.7) with g ρiab and Γ ρiab being the coupling constant and the partial width of the decay ρ i → ab,respectively. The specific expressions for G (ab) and other necessary quantities are given below. Note that a different scheme with two subtractions for the nondiagonal PP polarization operators was used in Ref. [1][2][3].

A. Pseudoscalar-pseudoscalar loop
The diagonal polarization operators due to the PP loop are represented in the form The function Π (P P ) is θ is the step function. The function G (P P ) (s) [Eq. (2.7)] necessary for the evaluation of the nondiagonal polarization operator due to the PP loop is (2.11)

B. Vector-pseudoscalar loop
The diagonal polarization operators due to the VP loop are represented in the form where the function Π (V P ) ≡ Π (V P ) (s, m ρi , m V , m P ) is calculated from the dispersion relation The notations are as follows. The quantity is the momentum of the particle a or b in the rest frame of the decaying particle with the invariant mass √ s; m ρi , m V and m P are, respectively, the masses of the resonance ρ i , and the vector V and pseudoscalar P mesons propagating in the loop, g ρiV P is the coupling constant of the resonance ρ i with the VP state. It is well known that the partial width of the decay ρ i → V P , grows as the energy increases. This growth spoils the convergence of the integral (2.13). This is the reason for the appearance of the function (s 0 + m 2 ρi )/(s ′ + m 2 ρi ), in the integrand of Eq. (2.13). It suppresses the fast growth of the partial width and improves the convergence of the above integral at large s ′ . However, the integral still remains logarithmically divergent, and one should perform the subtraction of the real part ReΠ ρi . The expression for Π (V P ) resulting from Eq. (2.13) can be represented in the form while m ± = m V ± m P , and θ is the usual step function. The dependence of ReΠ ρ1ρ1 (s) on energy squared at s 0 = 0.09 GeV 2 is shown in Fig. 1.
] necessary for the evaluation of the nondiagonal polarization operator due to the VP loop is where The dependence of ReΠ (ωπ) ρ1ρ2 /g ωρ1π g ωρ2π on s is shown in Fig. 2.

C. Axial vector-pseudoscalar loop
The axial vector-pseudoscalar meson state AP= a 1 (1260)π is considered to be one of the states contributing to the four-pion production amplitude [21]. For soft pions, when taking into account the requirements of chiral symmetry, this amplitude and the corresponding partial width, are very complicated [22][23][24][25][26][27][28]. This prevents one from using the dispersion relation to obtain the contribution of the four-pion state to the polarization operator of the state ρ i . Hence, in the present work, the simplest a 1 π dominance model of the four-pion [29] production is used: e + e − → ρ i → a 1 π → 4π. The amplitude of the transition ρ i → a 1 π is chosen in the simplest form where q, k, and p are, respectively, the four-momenta of the mesons ρ i , a 1 , and π, while ǫ a1 and ǫ ρi denote the polarization four-vectors of a 1 and ρ i . The expression (2.19) is chosen on the grounds that it is explicitly transverse in the ρ i leg. The diagonal polarization operators due to the AP loop are represented in the form where the expression The function f (s) looks as and The dependence of ReΠ (2.28) The dependence of ReΠ (a1π) ρ1ρ2 /g a1ρ1π g a1ρ2π on s is shown in Fig. 4.

D. Polarization operators used in fits
Although the nature of the higher resonances ρ(1450), ρ(1700), ... is the subject of current and future studies, the quark-antiquark model relations between their cou-pling constants are assumed: Polarization operators which take into account three channels described above are the following. The full diagonal polarization operators are In the present work, we take into account the following analytically calculated loops. First, we use the PP π + π − and K + K − + K 0K 0 loops, with Π (P P ) given by Eq. (2.9). Second, we use the VP ωπ 0 and K * K +K * K loops, Similar expressions are used for the nondiagonal polarization operators, where Π (P P ) ρiρj = g ρiππ g ρj ππ Π ρiρj (s, m ρi , m ρj , m π , m π ) +

III. QUANTITIES FOR COMPARISON WITH THE DATA
The following reactions are considered in the present work: and e + e − → π + π − π + π − . (3. 3) The justification of the restriction to these reactions are given in Introduction and below in Sec. IV. Let us turn to the working expressions necessary for the comparison with the experimental data.
A. π + π − production In the present case, the relevant quantity is the socalled bare cross section of the reaction (3.1): where F π (s) is the pion form factor Eq. (1.1), the quantity a(s) takes into account the radiation by the final pions, and α = 1/137 is the fine-structure constant. The necessary discussion concerning the quantities in Eq. (3.4) are given elsewhere [1].

IV. RESULTS OF DATA FITTING
This section is devoted to the presentation of the results of fitting the data on the reactions e + e − → π + π − , e + e − → ωπ 0 , and e + e − → π + π − π + π − . Two possible fitting schemes were used.
The parameters found from the fitting scheme 2 are listed in Table I. Let us comment on each of the mentioned channels.
A. Fitting e + e − → π + π − data When fitting the data on the reaction e + e − → π + π − at energies √ s ≤ 1 GeV in our previous publication [1][2][3], the fitting scheme 1 was used. There, the restriction to the PP loop was justifiable because of rather low energies under consideration. Using the resonance parameters found from fitting the data in the time-like region, the pion form factor F π (s) in the space-like region s < 0 was calculated up to −s = Q 2 = 0.2 GeV 2 and compared  with the NA7 data [9]. A comparison with the data [10][11][12] in the wider range up to −s = Q 2 = 10 GeV 2 was made in Ref. [3]. In the present work, we give the corresponding plot in Fig. 5 for the sake of completeness. The continuation to the space-like domain in the fitting scheme 2 is discussed below. The cross section of the reaction e + e − → π + π − fitted in the scheme 2 is shown in Fig. 6. As for the ρ(770) resonance parameters are concerned, one can observe that in comparison with the fit in the scheme 1 [1][2][3], the bare mass of the resonance ρ 1 determined in the scheme 2 in the sensitive channel e + e − → π + π − is typically lower. Compare Table I here and the Table I in, e.g., Ref. [1]. The same concerns the coupling constant g ρ1 which parametrizes the leptonic decay width (1.3). The TABLE I. The resonance parameters found from fitting the data on the reactions e + e − → π + π − [7], e + e − → π + π − π + π − [14], and e + e − → ωπ 0 [13], in the fitting scheme 2 (see text). The parameter ReΠ ′ ωρ is responsible for ωρ mixing, see Ref.
[1] for more detail. The parameter gρ 4 can be found from the sum rule = 1, which provides the correct normalization Fπ(0) = 1.
parameter e + e − → π + π − e + e − → π + π − π + π − e + e − → ωπ 0 mρ 1 [MeV] 765.6 ± 0. coupling constant g ρ1ππ in the scheme 2 is greater than in the scheme 1. The above distinction can be qualitatively explained by the effect of renormalization of the coupling constants described in Ref. [1]. Indeed, as was shown in Ref. [1], the renormalization results in the substitutions where Equation (4.1) means that the bare g ρ1ππ obtained from the fit is related to the "physical" one obtained from the visible peak, upon multiplying by Z 1/2 ρ , while the opposite is true for g ρ1 . The contributions of the VP loop to dReΠ ρ1ρ1 /ds near s = m 2 ρ1 , as is observed from Fig. 1, is positive and exceed the negative contribution from the PP loop, see Fig. 7 in Ref. [1]. The same is true for the AP loop. As a result, one has Z ρ > 1.
Although the energy behavior of the cross section up to √ s = 1.7 GeV is described in the adopted model, including the dip near 1.5 GeV, one can see that the structure in the interval 2-2.5 GeV demands, in all appearance, additional ρ-like resonances and/or intermediate states in the loops. We tried to include the contribution of K 1 (1400)K+c.c. states coupled solely to the resonance ρ 4 , with the fitted coupling constant g ρ4K1(1400)K and mass m K1(1400) . This slightly improves the agreement in the interval 1.75 < √ s < 2 GeV but occurs at the expense of adding two additional free parameters and does not result in reproducing the peak near √ s = 2.3 GeV.
The continuation to the space-like region s < 0 with the resonance parameters obtained in the region s > 4m 2 π in fitting scheme 2 with the VP and AP loops added, results in unwanted behavior of F π (s), see Fig. 7. Specifically, the curve goes through experimental points [9] up to s = −0.2 GeV 2 , but at larger values of −s = Q 2 one encounters infinities arising from the the Landau poles due to the VP and AP loops. As was pointed out in Ref. [1], the Landau pole is present even in the case of the PP loop, but its position is at Q 2 ≈ 90 GeV, that is, it is far from accessible momentum transfers. In the case of the VP and AP loops the Landau poles appear in the region accessible to existing experiments [10][11][12], because of the large magnitude of the coupling constant g ρ1ωπ = 13.2 GeV −1 [30].
An important feature of the new expression for the pion form factor obtained in Ref. [1] which was not mentioned there is that it does not require [3] the commonly The data are from Ref. [7], and the curve is drawn using the resonance parameters of scheme 2. The ρ−ω resonance region is shown in the insert.
accepted Blatt-Weisskopf centrifugal factor [31] C π (k) = 1 + R 2 π k 2 R 1 + R 2 π k 2 , in the expression for Γ ρππ (s) [21]. Here k is the pion momentum at some arbitrary energy while k R is its value at the resonance energy. The fact is that the usage of R πdependent centrifugal barrier penetration factor in particle physics -for example, in the case of the ρ(770) meson [21], results in the overlooked problem. Indeed, the meaning of R π is that this quantity is the characteristic of the potential (or the t-channel exchange in field theory) resulting in the phase δ bg of the potential scattering in addition to the resonance phase [31]. For example, in case of the P -wave scattering in the potential where the resonance scattering is possible, the background phase is At the usual value of R π ∼ 1 fm, δ bg is not small. However, in the ρ-meson region, the background phase shift δ bg is negligible and the phase shift δ 1 1 is completely determined by the resonance; see Fig. 8 in Ref. [1]. Therefore, the descriptions of the hadronic resonance distributions which invoke the parameter R π , have a dubious character.
cross section is shown in Fig. 8. One can see that at energies √ s > 1.75 GeV the chosen scheme with three heavier rho-like resonances ρ 2,3,4 cannot reproduce the structures in the measured cross section such as the bizarre sharp turn in the energy behavior followed by fluctuations. As in the case of the reaction e + e − → π + π − , the contributions of the AP loops a 1 (1260)π and K 1 (1270)K+c.c. coupled to all ρ i resonances (i = 1, 2, 3, 4) were invoked to explain the features above 1.75 GeV. The structures remain unexplained.
As is seen from Table I, the coupling constants g ρ1 and g ρ1ππ of the ρ(770) meson found from fitting this channel, differ from those found from fitting the π + π − one. Furthermore, the coupling constant g ρ1a1π found from fitting the channel e + e − → π + π − π + π − , is suppressed in comparison with the naive chiral-symmetry estimate g a1→ρ1π ∼ 1/2f π ∼ 5 GeV −1 [32], where f π = 92.4 MeV is the pion decay constant. We believe that this difference is an artifact of the oversimplified a 1 π model and the price that comes with the possibility of using the analytical calculation of the VP and AP loops to simulate the contributions of the multiparticle meson states in polarization operators [33]. In the meantime, the coupling constants g ρ1a1π ≈ 2.4 and 5.6 GeV −1 found from fitting the e + e − → π + π − and e + e − → ωπ 0 channels respectively, look sensible. For comparison, the estimates of g ρ1a1π in the model adopted in the present work are ∼ 6 GeV −1 and ∼ 4 GeV −1 , as extracted from Γ a1 ≈ 0.6 GeV and 0.3 GeV [21], respectively. Note also that Γ a1→ρπ→3π ∼ 1 GeV when evaluated in the generalized hidden local symmetry chiral model for m a1 ≈ 1.2 GeV [32]. The width of the visible peak in Fig. 8 is about 0.44 GeV which should be compared with Γ ρ3 ( √ s = m ρ3 ) = 0.45 GeV evaluated with the π + π − π + π − column of the Table I. C. Fitting e + e − → ωπ 0 data Quite recently, new data on the reaction e + e − → ωπ 0 in the decay mode ω → π 0 γ were published by SND collaboration [13]. They are analyzed with the fitting scheme 2. The resulting curve calculated with the parameters cited in the Table I is shown in Fig. 9.

V. CONCLUSION
The main purpose of the present work is to describe the pion electromagnetic form factor F π (s) up to the J/ψ energy range, using the expression obtained in Ref. [1]. This expression, when restricted to the PP loops in polarization operators, permits a good description of the data of SND, CMD-2, KLOE, and BABAR on π + π − production in e + e − annihilation at √ s < 1 GeV, describes the scattering kinematical domain up to −s = Q 2 = 10 GeV 2 , and does not contradict the data on ππ scattering phase δ 1 1 . The goal of extending the description to the energies up to 3 GeV in the time-like domain was reached by the inclusion of the VP-and AP-loops in addition to the PP ones. These loops contain the couplings of the ρ-like resonances with the VP-and AP-states and generate, in turn, the final states ωπ 0 and π + π − π + π − in e + e − annihilation. Therefore, consistency demands the treatment of these final states as well. As is shown in the present work, the energy behavior of the cross sections of the reactions e + e − → ωπ 0 and e + e − → π + π − π + π − obtained in the adopted simplified model, does not contradict the data. The statistically poor description of the cross section of the reaction e + e − → π + π − π + π − is, probably, an artifact of the oversimplified model for its amplitude which ignores both the requirements of the chiral symmetry at lower energies and a complicated intermediate state at higher energies. The proper treatment of the reaction e + e − → π + π − π + π − is beyond the scope of the present work. Nevertheless, we included this poor description for the consistency of the presentation. FIG. 9. The cross section of the reaction e + e − → ωπ 0 → π 0 π 0 γ. The data are SND13 [13]. The curve is drawn using the resonance parameters of the scheme 2.
One should not wonder at the fact that the masses of heavier ρ-like resonances quoted in Table I differ from the values quoted in Ref. [21]. In fact, the values in Ref. [21] are only educated guesses, and the masses of heavier ρlike resonances quoted by the Particle Data Group fall into wide intervals; for instance, m ρ2 = 1265−1580 MeV, and m ρ3 = 1430 − 1850 MeV [21]. Furthermore, the quoted values are usually obtained from fitting the data with the help of the simplest parametrization such as the sum of the Breit-Wigner amplitudes. In the meantime it is known that the residues of the simple pole contributions not necessarily reveal the true nature of the resonances involved in the process [34][35][36] when the mixings and the dynamical effects like the final-state interaction become essential.
The real problem is that the continuation to the spacelike domain of the expression for F π (s) with the contributions of the VP and AP loops meets the difficulty of encountering the Landau poles. By all appearances, this is the consequence of the chosen parametrization of the vertex form factor which restricts the growth of the partial widths as the energy increase, in a modest way. A stronger suppression could effectively suppress the couplings of rho-like resonances with the VP and AP states and, in turn, push the Landau zeros to higher spacelike momentum transfers. This is the topic of a separate study.
We are grateful to M. N. Achasov for numerous discussions which stimulated the present work. This work is supported in part by the Russian Foundation for Basic Research Grant no. 13-02-00039 and the Interdisciplinary project No 102 of the Siberian Division of the Russian Academy of Sciences.