Scalar isoscalar form factors for energies above 1 GeV

. We present a new parametrization for scattering amplitudes and form factors, which is consistent with high-accuracy dispersive representations at low energies but at the same time allows for a data description of higher mass resonances such as the f 0 (1500) and f 0 (2020). The formalism is general and thus can be applied to many decay processes. As an example we discuss the decay of ¯ B 0 s → J /ψππ ( KK ). From the amplitude ﬁxed in a ﬁt to the experimental data pole positions and residues are extracted via Padé approximants.


Introduction
Dispersion theory is a powerful tool in order to rigorously describe final-state interactions of the lightest mesons, where scattering phase shifts are known with high precision. Since it is model independent it can even be applied to the decays of heavy mesons into pions and kaons [1][2][3]. Not only does it ensure analyticity and unitarity, but it is also consistent by construction with the high-precision analyses using Roy-and Roy-like equations [4][5][6][7][8][9].
However due to additional channels as well as resonances above 1 GeV its application is limited to a energy regime below said threshold. In order to extend the description to a larger energy range we need to leave the safe grounds of dispersion theory to find an effective parametrization. This model should not spoil unitarity and analyticity, but it should also be able to describe effects coming from additional channels and resonances. Furthermore, it should be consistent with the constraints given by dispersion theory. Such a parametrization was provided for the vector pion form factor by Ref. [10] and generalized for the scalar pion form factors in this work [11].

Formalism
In this paragraph we present the formalism as it was derived in Refs. [10,11]. Although it is general, to be concrete we here proceed along the example of an isoscalar scalar partial wave and include the channels ππ (channel 1), KK (channel 2), and 4π (channel 3), which we model by either ρρ or σσ.
In order to describe the unitary partial wave-amplitude T i f from the initial state i to the final state f we use a Bethe-Salpeter equation of the form Here V in denotes the scattering potential from channel i into the intermediate state n. The loop function G nn describes the propagation of the intermediate state.
Similar to the two-potential formalism [12] we proceed by splitting the scattering kernel into two parts V = V 0 + V R . As seen in experiment inelasticities at higher energies are typically accompanied by resonances. Thus we define the resonance exchange potential V R such that it primarily contributes at higher energies, is suppressed at lower ones, and provides the coupling to all considered channels. This separation also allows us to rewrite the full scattering amplitude into T = T 0 + T R . We assume that T 0 fulfills a Bethe-Salpeter equation with V 0 by itself and incorporates the most relevant crossed channel interactions. Thus we can parametrize it with a unitary representation only using phases and inelasticities. As in our case the dipion system interacts strongly in the S -wave with the dikaon system at a cms energy of √ s = 1 GeV, we assume a coupled-channel description of them via V 0 . By construction T 0 contains the physics of f 0 (500) (σ) as well as f 0 (980). The corresponding scattering matrix is given by The ππ scattering phase shift δ and inelasticity η as well as the ππ → KK scattering phase ψ and modulus g were subject to many rigorous dispersive studies [4][5][6][7][8][9] and thus serve as possible input for our parametrization. In this work we explicitly used the coupled channel results of Ref. [13].
As T 0 is unitary by itself the resonance exchange amplitude T R is not independent of T 0 . The full solution can be written as where Ω denotes the vertex function, which can be calculated numerically as an Omnès solution [14,15] of the integral equation Similarly we also need to introduce the self energy Σ, which we write as a once-subtracted dispersion integral The subtraction constant can be absorbed in the not yet defined resonance potential V R . Note that for Σ 33 only the four-pion phase space contributes, which we model through the decay of two ρ or σ mesons. See Refs. [11,16] for details. As V 0 as described above alone fails to describe effects above 1 GeV, we include them via the resonance exchange potential V R . It includes s-channel particle exchanges of N R resonances, which are not covered by V 0 . In order to reduce its impact at low energies we subtract it at s = 0, which leaves us with     Here we introduced the bare mass m r as well as the bare resonance-channel coupling constant g r i , which need to be determined by a fit.
In order to describe a transition form factor Γ f of some source to the final state f we use the P-vector formalism [17], which gives us after some algebraic manipulation The direct transition between the source and the intermediate channel n is given by M n . We parametrize it by The first term is a polynomial in s with free coefficients, which need to be determined by a fit to data. The normalization Γ f (0) corresponds to P f (0), which can be determined by using the Feynman-Hellmann theorem with anss source. Thus the ππ channel has a normalization of Γ π (0) = 0 and the KK Γ K (0) = 2/ √ 3 [1]. Accordingly we also fix Γ 4π = 0, which is OZI suppressed. The second term describes the mixing of the resonance r with the source, which is described by the additional parameter α r . This term cancels potential zeros at the bare mass positions.

Fits
In this section we discuss for illustration a fit to the decaysB 0 s → J/ψπ + π − (K + K − ) measured at LHCb [18,19]. For more details on the free parameters we refer to Ref. [11].
We consider the angular moment averages defined by the partial-wave-expanded helicity amplitudes H L λ . Here λ = 0, , ⊥ refers to the helicity of the J/ψ, and L denotes the angular momentum of the dipion or dikaon system. The two most relevant angular moments for us are given by Here p ψ denotes the momentum of the J/ψ in theB 0 s rest frame and p m the pion (kaon) momentum in the dipion (dikaon) rest frame. For more information on the angular moments we refer to Refs. [1,20].
In the following we assume that the S -wave H 0 0 is proportional to the scalar isoscalar pion (kaon) form factor and can be described by Eq. (7), while we parametrize P-and D-waves as Breit-Wigner functions according to the LHCb studies [18,19].
An example fit with χ 2 /ndf = 376.2/(384 − 30 − 1) ≈ 1.07 using N R = 2 in Eqs. (6) and (8) and a ρρ channel is shown in Fig. 1. The first thing we see is that the event distribution Y 0 0 as well as the interference term Y 0 2 are well described for bothB 0 s → J/ψπ + π − as well asB 0 s → J/ψK + K − . Furthermore we notice that the decay into pions is indeed Swave dominated, while the decay into kaons also shows strong contributions of P-and Dwaves. Therefore the former decay indeed seems to be particularly suitable in order to extract information about pion and kaon dynamics at higher energies.
Interestingly we see that the peak structure at 1.45 GeV in the event distribution ofB 0 s → J/ψππ, corresponding to the f 0 (1500) resonance, is in line with the analysis performed by the LHCb collaboration. However at higher energies our fit seems to prefer a broader resonance with a higher mass. Because of the interplay of many resonances with different angular momenta it would be interesting to reanalyze the energy region above 1.5 GeV with higher statistics data sets and additional final channels such as an exclusive 4π channel, which is not yet available in partial-wave decomposed form [21].
In Fig. 2 we show the extracted pion and kaon isoscalar scalar form factors using a ρρ channel. We consider N R = 2 with a constant polynomial in Eq. (8) (Fit 1), a linear polynomial (Fit 2), and N R = 3 with a constant polynomial (Fit 3). We see that the three fits are consistent with each other below 1.5 GeV. Above this energy the form factors are not as strongly constrained and show more significant deviations. We can also compare our results to a two-channel Omnès solution and notice that below 1 GeV they seem to be consistent with each other.

Pole positions
As we utilized a fully unitary and analytical amplitude to extract the S -wave it is possible to extract resonance poles. For this purpose we used Padé approximants [22][23][24]. With a series of different fits, which are further explained in Ref. [11], we extracted the resonance poles of four different S -wave resonances on the nearest unphysical sheet.
The first resonance corresponds to the f 0 (500). We obtain a pole with a mass of (442 ± 2) MeV and a width of (512 ± 10) MeV. By construction the pole position should be close to the one defined by the input scattering amplitude T 0 [13], which lies at (441 − i 544/2) GeV. While the real parts are consistent with each other, we see some deviation in the imaginary parts. This shows that even at lower energies small shifts of our parametrization to the dispersive results are possible. These deviations might be improved upon by subtracting higher orders of the resonance exchange potential V R .
A similar picture can be seen for the f 0 (980) resonance, which can still be covered by the dispersive solutions [8]. As a benchmark we again use the input [13], which prefers a pole position of (998 − i 42/2) MeV. Compared to that we obtain a resonance with mass (996 ± 6) MeV and width (57 ± 11) MeV. As overall we could reproduce these two resonance poles we assume that our extraction with Padé approximants is sensible and also applicable for the two heavier resonances.
The first additional resonance that we obtain is the f 0 (1500) with a mass of (1465 ± 18) GeV and width (100 ± 19) MeV. This result is perfectly consistent with the extraction of LHCb given by a mass of (1465.9 ± 3.1) MeV and width (115 ± 7) MeV [18]. In contrast, our fit prefers to shift the second resonance to a higher mass and width. While LHCb found a resonance with a mass (1809 ± 22) MeV and width (263 ± 30) MeV [18], quoted as f 0 (1790),