# A new parametrization for the scalar pion form factors

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## Abstract

We derive a new parametrization for the scalar pion form factors that allows us to analyze data over a large energy range via the inclusion of resonances, and at the same time to ensure consistency with the high-accuracy dispersive representations available at low energies. As an application the formalism is used to extract resonance properties of excited scalar mesons from data for \({\bar{B}}^0_s\rightarrow J/\psi \pi \pi \). In particular we find for the pole positions of \(f_0(1500)\) and \(f_0(2020)\) \(1465\pm 18 - i (50\pm 9)\,\text {MeV}\) and \(1910\pm 50 - i(199\pm 40)\,\text {MeV}\), respectively. In addition, from their residues we calculate the respective branching ratios into \(\pi \pi \) to be \((58\pm 31)\%\) and \((1.3\pm 1.8)\%\).

## 1 Introduction

The scalar isoscalar sector of the QCD spectrum up to \(2\,\text {GeV}\) has been of high theoretical and experimental interest for many years. One of the main motivations for these investigations is the hunt for glueballs: their lightest representatives are predicted to occur in the mass range between 1600 and \(1700\,\text {MeV}\) with quantum numbers \(0^{++}\) [1, 2, 3, 4]. The most straightforward way to identify glueball candidates is to count states with and without flavor quantum number and see if there are supernumerary isoscalar states; see, e.g., the minireview on non-\({\bar{q}}q\) states provided by the Particle Data Group (PDG) [5] or the reviews Refs. [6, 7]. Unfortunately, regardless of the year-long efforts, the scalar isoscalar spectrum is still not fully resolved: e.g. there is still an ongoing debate whether the \(f_0(1370)\) exists or not [6]. One problem might be that most analyses of experimental data performed so far are based on fitting sums of Breit–Wigner functions, which can lead to reaction-dependent results. To make further progress, it therefore appears compulsory to employ parametrizations that allow one to extract pole parameters, for those by definition do not depend on the production mechanism. This requires amplitudes that are consistent with the general principles of analyticity and unitarity. In this paper we present a new parametrization for the scalar pion form factors that has these features built in, and in addition maps smoothly onto well constrained low-energy amplitudes.

The two-pion system at low energies is well understood from sophisticated investigations based on dispersion theory—in particular the \(\pi \pi \)–\(K{\bar{K}}\) phase shifts and inelasticities can be assumed as known from threshold up to an energy of about \(s=(1.1\,\text {GeV})^2\) [8, 9, 10, 11, 12, 13]. From this information, quantities like the scalar non-strange and strange form factors for both pions and kaons can be constructed, again employing dispersion theory [14, 15, 16, 17, 18, 19, 20, 21]. The resulting amplitudes, which capture the physics of the \(f_0(500)\) (or \(\sigma \)) and the \(f_0(980)\), were already applied successfully to analyze various meson decays, see, e.g., Ref. [20]. In particular the non-Breit–Wigner shape of these low-lying resonances [22] is taken care of automatically. However, to also include higher energies in the analysis, where additional inelastic channels become non-negligible and higher resonances need to be included, one is forced to leave the safe grounds of fully model-independent dispersion theory and to employ a model. Ideally this is done in a way that the amplitudes match smoothly onto those constructed rigorously from dispersion relations. Moreover, to allow for an extraction of resonance properties, the extension needs to be performed in a way consistent with analyticity.

A formalism that has all of these features was introduced for the pion vector form factor in Ref. [23]. In that case, the low-energy \(\pi \pi \) interaction can safely be treated as a single-channel problem in the full energy range where high-accuracy phase shifts are available, since the two-kaon contribution to the isovector *P*-wave inelasticity is very small [12, 24].^{1} However, this is not true for the isoscalar *S*-wave, clearly testified by the presence of the \(f_0(980)\) basically at the \(K{\bar{K}}\) threshold with a large coupling to this channel [26, 27]. Thus, in order to apply the formalism of Ref. [23] to the scalar isoscalar channel it needs to be generalized. This is the main objective of the present article. As an application we test the amplitudes on data for \({\bar{B}}^0_s\rightarrow J/\psi \pi \pi /K{\bar{K}}\) recently measured with high accuracy at LHCb [28, 29], which allows us to extract the strange scalar form factor of pions and kaons up to about \(2\,\text {GeV}\) and to constrain pole parameters and branching fractions of two of the heavier \(f_0\) resonances in that energy range.

This paper is organized as follows. In Sect. 2, we derive the unitary and analytic scalar form factor parametrization to be used. In Sect. 3 we illustrate its application in a coupled-channel analysis of the decays \(\bar{B}_s^0\rightarrow J/\psi \pi \pi \) and \(\bar{B}_s^0\rightarrow J/\psi K{\bar{K}}\). Specifically, we discuss the stability of our fits under changing assumptions for the parametrization concerning the number of resonances, the degree of certain polynomials, as well as the approximation in the description of the effective four-pion channel. In addition, in Sect. 4 we extract pole parameters, in particular for both the \(f_0(1500)\) and the \(f_0(2020)\), via the method of Padé approximants for the analytic continuation to the unphysical sheets. The paper ends with a summary and an outlook in Sect. 5.

## 2 Formalism

*i*and

*f*denote the initial- and final-state channels. To implement unitarity and analyticity we use the Bethe–Salpeter equation, which reads

*i*into the final channel

*f*. The loop operator

*G*is diagonal in channel space and provides the free propagation of the particles of channel

*m*. For example, at the one-loop level the above equation generates an expression of the form

*P*being the total 4-momentum of the system such that \(P^2=s\). For \(m=1,\,2\), the discontinuity of the loop operator element \(G_{mm}\) reads

*k*, \(\rho _k(m^2)\), is given as

*T*-matrix input. Its explicit form is needed at no point; one may think of it as the driving term of a Bethe–Salpeter equation

*T*-matrix \(T_R\), related to the full

*T*-matrix via \(T=T_R+T_0\). Since \(T_0\) is unitary by itself, \(T_R\) cannot be independent of \(T_0\) in order to respect the Bethe–Salpeter equation (2). Solving for \(T_R\) we obtain

*T*-matrix \(T_0\) [38, 39]. Therefore it can be constructed from dispersion theory:

*T*-matrix of Ref. [40] are shown in Fig. 1. One observes in particular the signature of the \(f_0(500)\) or \(\sigma \)-meson, i.e. the broad bump in the imaginary part of \(\varOmega _{11}(s)\) below \(1\,\text {GeV}\), accompanied by a quick variation of the real part, which clearly cannot be parametrized by a Breit–Wigner form. For an earlier discussion about this fact see Ref. [22].

*G*, describing the free propagation of the two-meson states, needs to be replaced by the dressed loop operator \((G\varOmega )\), which describes the propagation of the two-meson state in the presence of the interaction \(T_0\), in order to preserve unitarity. The discontinuity of this self-energy matrix \(\varSigma =G\varOmega \) is given by

*S*-wave, while the \(\rho \) decays in a

*P*-wave. On the other hand, since the discontinuity of \(G_{33}\) enters in the expression for \(\varSigma _{33}\) only as the integrand, this component of the self energy is not very sensitive to the details of the concrete parametrizations employed for the spectral functions.

*P*-vector formalism [41] to the system at hand. The isoscalar scalar form factor \(\varGamma ^{s}_i\) is written as

*i*. Inserting the parametrization of Eq. (19) we obtain, after some straightforward algebra,

*M*reads

This completely defines the formalism. Clearly, the number of inelastic channels can be extended in a straightforward way, however, for the concrete application studied in the following section, three channels turn out to be sufficient as long as no exclusive data for additional channels become available.

## 3 Application: \(\bar{B}_s^0\rightarrow J/\psi \,\pi ^+\pi ^-\) and \(\bar{B}_s^0\rightarrow J/\psi \,K^+K^-\)

### 3.1 Parametrization of the decay amplitudes

It has been argued previously [20, 42] that the *S*-wave projection of the appropriate helicity-0 amplitude for \(\bar{B}_s^0\rightarrow J/\psi \,M_1M_2\) transitions are proportional to the corresponding strange scalar form factors of the light dimeson system \(M_1M_2\); in particular, there are chiral symmetry relations between the different dimeson channels that fix the *relative* strengths to be equal to those of the matrix elements in Eq. (1) at leading order in a chiral expansion [42]. We conjecture here that the same will still hold true for the inclusion of the effective third (\(4\pi \)) channel. In this sense, the \(\bar{B}_s^0\) decays allow to test the pion and kaon strange scalar form factors, up to a common overall normalization.

*L*denotes the angular momentum of the pion or kaon pair, and \(\lambda = 0,\parallel ,\perp \) refers to the helicity of the \(J/\psi \). The angular moments are then given as

*i*denotes the relevant channel. For the form factors we use the parametrization introduced in Sect. 2. Since the main focus of our analysis lies on the

*S*-waves, we approximate the

*P*- and

*D*-waves as Breit–Wigner functions [43],

*R*with helicity \(\lambda \), its phase \(\phi _\lambda ^R\), and a total rescaling factor \(w_\lambda ^L\) for the helicity amplitude \(\mathcal {H}_\lambda ^L\). The factors \(F_B^{(J)}\) and \(F_R^{(L)}\) are the Blatt–Weisskopf factors of Eq. (8). Two different scales are employed therein: while \(F_B^{(J)}\) depends on the argument \(z=r_B^2 \, p_\psi ^2\) with \(r_B=5.0\,\text {GeV}^{-1}\), for \(F_R^{(L)}\) we use \(z=r_R^2 \,p_\pi ^2\) with \(r_R=1.5\,\text {GeV}^{-1}\) as in Eq. (8) [36]. The position as well as width of the corresponding resonance is then included in the Breit–Wigner function

*S*-

*D*-wave interference in \(\left<Y_2^0\right>\), our fits are only sensitive to the relative phase motion of \(\mathcal {H}_0^0\) and \(\mathcal {H}_0^2\). To reduce the total number of free parameters for all partial waves except the

*S*-wave, we fix the resonance masses \(m_R\) as well as their respective widths to the central values found in Refs. [28, 29]. Furthermore we fix both \(h_\lambda ^R\) as well as \(\phi _\lambda ^R\) with \(\lambda =\parallel ,\perp \) to the central values of the LHCb fits. However, since the phase motion of our

*S*-wave will be different from the one of the LHCb parametrization [20], we allow \(w_\lambda ^R\) to vary. For the helicity amplitude \(\mathcal {H}_0^2\) we keep both \(h_0^R\) as well as \(\phi _0^R\) flexible. To avoid unnecessary parameters we set \(w_0^2=1\). The number of free parameters is discussed in more detail in Sect. 3.2.

### 3.2 Fits to the decay data

*S*-wave contains in total up to \((N_c+N_s+1)N_R+2N_cN_s\) parameters, where \(N_c\) (\(N_s\)) denotes the number of channels (sources) included; in this study \(N_s=1\), \(N_c=3\), and \(N_R\) is either 2 or 3, depending on the fit. The last term in the sum above comes from the non-resonant couplings of the system to the source. The number of those parameters can be reduced from the observation that the normalizations of the pion and the kaon form factors can be fixed to \(c_1=0\) and \(c_2=1\) [20]. Since the four-pion channel is expected to couple similarly weakly to an \(\bar{s}s\) source as the two-pion one (given OZI suppression at \(s=0\)), we also set \(c_3=0\). Thus the only free parameter from the constant terms in the sources \(M_i\) can be absorbed into the overall normalization \(\mathcal {N}\) introduced in Eq. (28). Below we present fits without (\(\gamma _i=0\), resulting in \(5N_R\) parameters) as well as with linear terms in the production vertex defined in Eq. (23) (\(\gamma _i\ne 0\), providing three more free constants).

For the decay \(\bar{B}_s^0\rightarrow J/\psi \,\pi \pi \) the dipion system is in an isoscalar configuration; due to Bose symmetry the pions can therefore only emerge in even partial waves. Since we restrict ourselves to a precision analysis of the *S*-wave, we adopt the *D*-waves of Ref. [28] and accordingly include two resonances, namely \(f_2(1270)\) and \(f_2^\prime (1525)\). For the 0 polarization we introduce four new parameters given by the amplitude \(h_0^R\) and \(\phi _0^R\), while we fix \(w_0^0=1\). For the other two helicity amplitudes we constrain \(h_\lambda ^R\) and \(\phi _\lambda ^R\) while keeping \(w_\lambda ^0\) variable. This gives another two free parameters. In total we obtain six additional free parameters.

Since \(K^+\) and \(K^-\) do not belong to the same isospin multiplet, they do not follow the Bose symmetry restrictions. Thus the *P*-wave in the decay \(\bar{B}_s^0\rightarrow J/\psi \,K^+K^-\) is non-negligible and, in fact, dominant. It shows large contributions of the \(\phi (1020)\) as well as of the \(\phi (1680)\). Since the *P*-wave does not interfere with *S*- or *D*-waves in the angular moments \(\left<Y_0^0\right>\) and \(\left<Y_2^0\right>\), we adopt the parameters of LHCb [29]. In order to allow for some flexibility, we also fit \(w_\lambda ^1\), resulting in three parameters. The *D*-wave includes the resonances \(f_2(1270)\), \(f_2^\prime (1525)\), \(f_2(1750)\), and \(f_2(1950)\). For \(\lambda =0\) we fit both \(h_0^R\) as well as \(\phi _0^R\) with fixed \(w_0^2=1\), resulting in eight free parameters. For the other helicity amplitudes we stick to the LHCb parametrization and keep \(w_\lambda ^2\) free, which results in two additional fit parameters. Therefore in total we have 13 additional free parameters for this channel.

All in all we have \(5N_R+20(+3)\) free parameters for \(\gamma _i=0\) (\(\gamma _i\ne 0\)). Clearly this number is larger than the number of parameters of two single-channel Breit–Wigner analyses, however, the advantage of the approach advocated here is that it allows for a combined analysis of all channels in a way that preserves unitarity, and for a straightforward inclusion of the \(4\pi \) channel in the analysis. Note that the scalar resonances studied here are known to have prominent decays into four pions [5]; cf. also theoretical approaches modeling some of them as dynamically generated \(\rho \rho \) resonances [31, 32, 33, 34].

*S*-wave resonances from their data [28], namely \(f_0(1500)\) and \(f_0(1790)\). Since there is no \(f_0(1790)\) in the listings of the Review of Particle Physics by the PDG [5], we use the name \(f_0(2020)\) for the higher state, in particular since the parameters we extract below are close to those reported for that resonance. The first fit includes our parametrization with \(N_R=2\) and \(\gamma _i=0\) (Fit 1). To test the stability of this solution, we also include a fit with \(N_R=2\) and \(\gamma _i\ne 0\) (Fit 2) as well as a fit with \(N_R=3\) and \(\gamma _i=0\) (Fit 3). In order to obtain an estimate of the systematic uncertainty, we repeat each fit with two different assumptions about the third channel, which we take to be dominated by either \(\sigma \sigma \) or \(\rho \rho \). The respective reduced \(\chi ^2\) of the best fit results are listed in Table 1. We show the corresponding angular moments in Figs. 4 (\(\rho \rho \)) and 5 (\(\sigma \sigma \)). In principle we could have also investigated mixtures of \(\sigma \sigma \) and \(\rho \rho \) intermediate states or parametrizations representing the channels \(\pi (1300)\pi \) or \(a_1(1260)\pi \) reported to be relevant for the \(f_0(1500)\) [5], however, since with the given choices we already find excellent fits to the data although the corresponding two-point function \(\varSigma _{33}\) look vastly different for the \(\sigma \sigma \) and the \(\rho \rho \) case (cf. the lower right panel of Fig. 2), studying other possible decays will be postponed until data for further exclusive final states become available.

Reduced \(\chi ^2\) for the best fits. See main text for details

\({\chi ^2}/{\mathrm {ndf}}\) | \(\sigma \sigma \) | \(\rho \rho \) |
---|---|---|

Fit 1 | \(\frac{429.9}{384-30-1}=1.22\) | \(\frac{376.2}{384-30-1}=1.07\) |

Fit 2 | \(\frac{413.3}{384-33-1}=1.18\) | \(\frac{361.4}{384-33-1}=1.03\) |

Fit 3 | \(\frac{366.9}{384-35-1}=1.05\) | \(\frac{335.4}{384-35-1}=0.96\) |

We note first of all that the \(\rho \rho \) fits have a lower reduced \(\chi ^2\) compared to the \(\sigma \sigma \) fits. Allowing for a linear term in the source further improves the data description, as witnessed by the differences of Fits 1 and 2. The overall best reduced \(\chi ^2\) is obtained by including another, third, resonance.

For the \(\rho \rho \) fit (see Fig. 4) we see that Fit 2 improves the description of \(\left<Y_0^0\right>_{\pi \pi }\) in the energy region between 1.6 and \(2.0\,\text {GeV}\). The biggest change between Fit 3 and the other ones is given by the better description of the high-energy tail in the decay \(\bar{B}_s^0\rightarrow J/\psi K^+K^-\).

In Fig. 8 we compare the form factor of the additional, effective \(4\pi \), channel \(\varGamma _3^s\). We see that the results of the fits with the \(4\pi \) channel parametrized as \(\rho \rho \) differ significantly from the ones employing the \(\sigma \sigma \) variant. Moreover, also Fits 1–3 differ strongly from each other, even in the kinematic regime that can be reached in \({\bar{B}}_s^0\) decays. To further constrain these amplitudes it is compulsory to include data on \(\bar{B}_s^0 \rightarrow J/ \psi 4\pi \) in the analysis, which is so far unavailable in partial-wave-decomposed form [44].

In Fig. 9 we also show a comparison of our phases and inelasticities to those extracted in Ref. [45] (plotted as purple dashed lines) and the preferred solution [7] of the CERN–Munich \(\pi \pi \) experiment [46] (data points with error bars). As one can see in the phase shifts, all analyses agree up to about \(1.5\,\text {GeV}\). However, the effect of the \(f_0(1500)\), present in all analyses, is very different. Also for the inelasticity there is no agreement between our solution and those from the two other sources, but here the deviation starts basically with the onset of the \({\bar{K}}K\) channel; for a more detailed discussion of the current understanding of the inelasticity in the scalar isoscalar channel, we refer to Ref. [11]. Note that there is also no agreement between the amplitudes of Ref. [7] and Ref. [45]. Thus, at this time one is to conclude that \(T_{11}\) above \(1.1\,\text {GeV}\) is not yet known.

In a similar way, we can also compare the extracted \(\pi \pi \rightarrow K\bar{K}\) amplitude \(T_{12}\) with its absolute value *g* as well as its phase \(\psi \), which are both shown in Fig. 10. While the resonance effects of the \(f_0(1500)\) look qualitatively well-described by our high-energy extension, we see some differences to the actual data [47, 48]]. Note that the shown results are a prediction based solely on the \(\bar{B}_s^0\) decay data and could be improved upon by explicitly taking the phase motion into account in the fit.

## 4 Extraction of resonance poles

*s*-plane from the parametrizations discussed above. Traditionally those are given in terms of a mass

*M*and a width \(\varGamma \), connected to the pole position \(s_p\) via [5]

*T*-matrix as input, which, due to left-hand cuts induced by crossing symmetry, has a complicated analytic structure that cannot be deduced from the phase shifts straightforwardly, we use the framework of Padé approximants to search for the poles on the nearest unphysical sheets. For a thorough introduction into this topic, see e.g. Refs. [49, 50, 51].

*s*-plane of the first Riemann sheet to the lower complex

*s*-plane of the neighboring unphysical sheet, we may expand both around some properly chosen expansion point \(s_0\) according to

*M*resonance poles lying on the unphysical Riemann sheet. In the following we set

*M*to 1, allowing for the extraction of the resonance that lies closest to the expansion point \(s_0\). The numerator ensures the convergence of the series to the form factor or the scattering matrix for \(N\rightarrow \infty \). In order to obtain the complex parameters \(a_n\) and \(b_n\), we fit Padé approximants to both the form factor and the scattering matrix simultaneously. As both \(T_{11}\) and \(\varGamma ^{s}_1\) have the same poles, the parameters \(b_n\) are the same for both, however, the \(a_n\) are different. Note furthermore that the \(a_0\) parameters are constrained by \(\varGamma _1^s(s_0)\) or \(T_{11}(s_0)\), respectively.

*s*, but in the conformal variable

*s*-plane of the first Riemann sheet to the inner upper half of a unit circle in the complex

*w*plane, without introducing any unphysical discontinuities. The lower half of the second Riemann sheet is then mapped onto the lower half of the unit circle in the complex

*w*-plane. This allows us to search for the two lowest poles within a circle around the expansion point \(s_0\), without being limited by the proximity of the \(\pi \pi \) and \(K\bar{K}\) thresholds, which are automatically taken care of.

As in principle the results still depend on the expansion point \(s_0\), we proceed as follows. We first calculate Padé approximants for a varying \(s_0\); near the true pole position, the extracted Padé pole stabilizes. Finally we choose the \(s_0\) that minimizes \(\varDelta ^N\) for the maximum order of *N* employed.

*R*to \(\pi \pi \) and the coupling \(g_{Rss}\) of the \(\bar{s}s\) source to the resonance

*R*. They are defined by the near-pole expansions [27, 52]

Padé poles for \(f_0(500)\), \(f_0(980)\), and \(f_0(1500)\) for \(N=5\), as well as \(f_0(2020)\) for \(N=6\). The error is the uncorrelated sum of statistical and systematic uncertainty

Fit | \(\frac{\sqrt{s_0}}{\text {GeV}}\) | \(\text {Re}\sqrt{s_p}/\text {MeV}\) | \(-2\times \text {Im}\sqrt{s_p}/\text {MeV}\) | \(|r_T|/\text {GeV}^2\) | \(\arg (r_T)\) | \(|r_\varGamma |/\text {GeV}^2\) | \(\arg (r_\varGamma )\) | ||
---|---|---|---|---|---|---|---|---|---|

\(f_0(500)\) | \(\rho \rho \) | 1 | 0.481 | \(441\pm 1\) | \(504\pm 2\) | \(0.204\pm 0.002\) | \(-145\pm 1\) | \(0.0309\pm 0.0028\) | \(-160\pm 3\) |

\(f_0(500)\) | \(\sigma \sigma \) | 1 | 0.466 | \(440\pm 1\) | \( 521\pm 1\) | \(0.205\pm 0.001\) | \(-149\pm 1\) | \(0.0254\pm 0.0010\) | \(-169\pm 2\) |

\(f_0(500)\) | \(\rho \rho \) | 2 | 0.483 | \(441\pm 1\) | \(503\pm 1\) | \(0.204\pm 0.001\) | \(-145\pm 1\) | \(0.0275\pm 0.0010\) | \(-159\pm 2\) |

\(f_0(500)\) | \(\sigma \sigma \) | 2 | 0.486 | \(443\pm 1\) | \(521\pm 2\) | \(0.205\pm 0.002\) | \(-147\pm 1\) | \(0.0279\pm 0.0032\) | \(-161\pm 4\) |

\(f_0(500)\) | \(\rho \rho \) | 3 | 0.481 | \(441\pm 2\) | \(505\pm 3\) | \(0.202\pm 0.002\) | \(-145\pm 2\) | \(0.0279\pm 0.0039\) | \(-159\pm 4\) |

\(f_0(500)\) | \(\sigma \sigma \) | 3 | 0.485 | \(442\pm 1\) | \(510\pm 1\) | \( 0.203\pm 0.001\) | \(-146\pm 1\) | \(0.0284\pm 0.0023\) | \(-161\pm 3\) |

\(f_0(980)\) | \(\rho \rho \) | 1 | 0.941 | \(998\pm 2\) | \(65\pm 3\) | \(0.099\pm 0.006\) | \(-164\pm 3\) | \(0.258\pm 0.016\) | \(107\pm 4\) |

\(f_0(980)\) | \(\sigma \sigma \) | 1 | 0.941 | \(998\pm 1\) | \(48\pm 2\) | \(0.082\pm 0.007\) | \(-164\pm 5\) | \(0.258\pm 0.019\) | \(109\pm 5\) |

\(f_0(980)\) | \(\rho \rho \) | 2 | 0.941 | \(1001\pm 2\) | \(65\pm 3\) | \(0.114\pm 0.011\) | \(-160\pm 6\) | \(0.270\pm 0.020\) | \(109\pm 5\) |

\(f_0(980)\) | \(\sigma \sigma \) | 2 | 0.941 | \(998\pm 1\) | \(50\pm 2\) | \(0.082\pm 0.006\) | \(-166\pm 5\) | \(0.249\pm 0.014\) | \(108\pm 4\) |

\(f_0(980)\) | \(\rho \rho \) | 3 | 0.941 | \(993\pm 3\) | \(65\pm 3\) | \(0.094\pm 0.005\) | \(-168\pm 3\) | \(0.261\pm 0.012\) | \(103\pm 3\) |

\(f_0(980)\) | \(\sigma \sigma \) | 3 | 0.941 | \(998\pm 2\) | \(60\pm 2\) | \(0.099\pm 0.007\) | \(-163\pm 5\) | \(0.281\pm 0.016\) | \(109\pm 4\) |

\(f_0(1500)\) | \(\rho \rho \) | 1 | 1.459 | \(1460\pm 6\) | \(109\pm 7\) | \(0.131\pm 0.017\) | \(-82\pm 3\) | \(0.18\pm 0.03\) | \(-53\pm 5\) |

\(f_0(1500)\) | \(\sigma \sigma \) | 1 | 1.449 | \(1456\pm 4\) | \(107\pm 8\) | \(0.047\pm 0.005\) | \(-86\pm 3\) | \(0.23\pm 0.02\) | \(-74\pm 4\) |

\(f_0(1500)\) | \(\rho \rho \) | 2 | 1.517 | \(1465\pm 4\) | \(116\pm 4\) | \(0.115\pm 0.007\) | \(-86\pm 2\) | \(0.18\pm 0.02\) | \(-50\pm 2\) |

\(f_0(1500)\) | \(\sigma \sigma \) | 2 | 1.449 | \(1452\pm 5\) | \(103\pm 8\) | \(0.045\pm 0.005\) | \(-82\pm 6\) | \(0.23\pm 0.02\) | \(-54\pm 6\) |

\(f_0(1500)\) | \(\rho \rho \) | 3 | 1.466 | \(1465\pm 5\) | \(105\pm 7\) | \(0.097\pm 0.018\) | \(-87\pm 3\) | \(0.18\pm 0.03\) | \(-57\pm 4\) |

\(f_0(1500)\) | \(\sigma \sigma \) | 3 | 1.476 | \(1477\pm 6\) | \(90\pm 9\) | \(0.097\pm 0.010\) | \(-86\pm 7\) | \(0.12\pm 0.04\) | \(-51\pm 16\) |

\(f_0(2020)\) | \(\rho \rho \) | 1 | 2.145 | \(1996\pm 67\) | \(998\pm 163\) | \(0.215\pm 0.407\) | \(4\pm 82\) | \(2.23\pm 0.62\) | \(18\pm 15\) |

\(f_0(2020)\) | \(\sigma \sigma \) | 1 | 1.900 | \(1888\pm 9\) | \(344\pm 12\) | \(0.005\pm 0.002\) | \(-104\pm 24\) | \(0.48\pm 0.04\) | \(106\pm 4\) |

\(f_0(2020)\) | \(\rho \rho \) | 2 | 1.949 | \(1869\pm 9\) | \(461\pm 15\) | \(0.026\pm 0.013\) | \(31\pm 33\) | \(0.51\pm 0.06\) | \(-10\pm 11\) |

\(f_0(2020)\) | \(\sigma \sigma \) | 2 | 1.900 | \(1908\pm 10\) | \(344\pm 19\) | \(0.008\pm 0.006\) | \(-101\pm 64\) | \(0.41\pm 0.10\) | \(103\pm 13\) |

\(f_0(2020)\) | \(\rho \rho \) | 3 | 1.949 | \(1919\pm 23\) | \(366\pm 47\) | \(0.011\pm 0.006\) | \(77\pm 51\) | \(0.45\pm 0.11\) | \(32\pm 15\) |

\(f_0(2020)\) | \(\sigma \sigma \) | 3 | 1.900 | \(1910\pm 50\) | \(414\pm 42\) | \(0.014\pm 0.016\) | \(82\pm 69\) | \(0.72\pm 0.34\) | \(66\pm 34\) |

As we did not include any variation of the input phases, we see that the statistical uncertainty coming from the fit parameters of the higher-mass resonances has only a small impact on the poles of \(f_0(500)\) and \(f_0(980)\). In fact the uncertainty is dominated by the systematic error coming from the Padé expansion. At higher energies the statistical uncertainty becomes more significant.

In order to see whether the pole extraction leads to sensible results, we first compare our findings for the \(f_0(500)\) and \(f_0(980)\) to the literature [27, 40, 52, 53]. In our parametrization the \(f_0(500)\) has a mass of \(\left( 442\pm 2\right) \text {MeV}\) with a width of \(\left( 512\pm 10\right) \text {MeV}\). For the \(f_0(980)\) we find a mass of \(\left( 996\pm 6\right) \text {MeV}\) and a width of \(\left( 57\pm 11\right) \text {MeV}\). As Ref. [40] serves as our input below \(1\,\text {GeV}\), their pole positions are taken as a benchmark, which lie at \(\left( 441-i\,544/2\right) \text {MeV}\) and \(\left( 998-i\,42/2\right) \text {MeV}\), respectively. While the real parts are therefore perfectly consistent, we see that our parametrization slightly shifts the imaginary parts of the poles with respect to the input.

The last resonance identified by LHCb as the \(f_0(1790)\) has a mass of \(\left( 1809\pm 22\right) \text {MeV}\) with a width of \(\left( 263\pm 30\right) \text {MeV}\). As we do not impose a Breit–Wigner line shape, our fits seem to prefer a significantly heavier and much broader resonance with mass \(\left( 1910\pm 50\right) \text {MeV}\) and a width of \(\left( 398\pm 79\right) \text {MeV}\). Note that for the average we neglected the pole extracted from Fit 1 with the \(\rho \rho \) parametrization, since this fit describes the prominent resonance structure in the \(\pi \pi \) spectrum less accurately than the rest of the fits. As the pole position of the higher pole extracted in our analysis is in better agreement with the \(f_0(2020)\) of the PDG (which quotes a mass of \(\left( 1992\pm 16\right) \text {MeV}\) and a width of \(\left( 442\pm 60\right) \text {MeV}\) [5]), we will refer to it as such in the following. Furthermore we see that this pole allows for a stronger variance in the different fits. As its line shape does not only depend on the interference with other resonances, but also on further inelasticities, additional information about these channels would be appreciable.

Since the bare resonance coupling strengths \(g_i^r\) as well as the bare resonance masses \(m_r\) are source-independent, we can use the same parameters for any decay with \(\pi \pi \) *S*-wave final-state interactions and negligible left-hand cuts. Therefore a simultaneous study of \(\bar{B}_s^0\rightarrow J/\psi \pi \pi \) and \(\bar{B}_d^0\rightarrow J/\psi \pi \pi \) [56] should be useful to constrain the resonances in the scalar isoscalar channel further.

## 5 Summary and outlook

In this article, we have shown that the parametrization of Ref. [23] for the pion vector form factor can be adapted to the scalar form factors of pions and kaons, marrying the advantages of a rigorous dispersive description at low energies with the phenomenological success of a unitary and analytic isobar model beyond. For the scalar isoscalar channel, the low-energy part must already be provided in terms of a dispersively constructed coupled-channel Omnès matrix. We rely on the conjecture that the resulting strange scalar form factors can be tested in a simultaneous study of the *S*-waves in the helicity amplitudes for the decays \(\bar{B}_s^0\rightarrow J/\psi \pi \pi \) and \(\bar{B}_s^0\rightarrow J/\psi K\bar{K}\), whose leading angular moments we can describe successfully. In this way, we have in fact determined the corresponding strange scalar form factors up to \(\sqrt{s} \approx 2\,\text {GeV}\), in particular for the pion with rather good accuracy. To quantify the uncertainties of the method, we compared fits based on different assumptions, such as different numbers of resonances as well as different final-state channels. Although they describe the data almost equally well, we see a significant systematic uncertainty at higher energies, which should be reduced significantly, however, once further information about the inelastic channels becomes available. For now, we only included an effective \(4\pi \) channel modeled either by \(\rho \rho \) or \(\sigma \sigma \) intermediate states; for a more detailed description of the branching ratios of the heavier scalar isoscalar resonances, we might need to include further inelastic channels such as \(a_1\pi \), \(\eta \eta \), or \(\eta \eta ^\prime \).

As the parametrization developed is fully unitary and analytic, we extracted resonance parameters as pole positions and residues in the complex energy plane, employing Padé approximants. In particular, we determined resonance poles as well as coupling constants for \(f_0(1500)\) and \(f_0(2020)\). While the pole location for the \(f_0(1500)\) is consistent with the one derived from the LHCb Breit–Wigner extraction, we find a significantly shifted pole for the \(f_0(2020)\). This shift ought to be tested experimentally in other processes with prominent *S*-wave pion–pion final-state interactions. Alternatively – or in addition – we might also include scattering data at higher energies in the fits explicitly [45, 57].

## Footnotes

## Notes

### Acknowledgements

We thank T. Isken, B. Moussallam, J. Niecknig, W. Ochs, J. Ruiz de Elvira, and A. Sarantsev for useful discussions. Financial support by DFG and NSFC through funds provided to the Sino–German CRC 110 “Symmetries and the Emergence of Structure in QCD” (DFG Grant No. TRR110 and NSFC Grant No. 11621131001) is gratefully acknowledged.

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