$\pi \pi \rightarrow K \bar{K}$ scattering up to 1.47 GeV with hyperbolic dispersion relations

In this work we provide a dispersive analysis of $\pi\pi \rightarrow K\bar{K}$ scattering. For this purpose we present a set of partial-wave hyperbolic dispersion relations using a family of hyperbolas that maximizes the applicability range of the hyperbolic dispersive representation, which we have extended up to 1.47 GeV. We then use these equations first to test simple fits to different and often conflicting data sets, also showing that some of these data and some popular parameterizations of these waves fail to satisfy the dispersive analysis. Our main result is obtained after imposing these new relations as constraints on the data fits. We thus provide simple and precise parameterizations for the S, P and D waves that describe the experimental data from $K\bar K$ threshold up to 2 GeV, while being consistent with crossing symmetric partial-wave dispersion relations up to their maximum applicability range of 1.47 GeV. For the $S$-wave we have found that two solutions describing two conflicting data sets are possible. The dispersion relations also provide a representation for $S$, $P$ and $D$ waves in the pseudo-physical region.


I. INTRODUCTION
The scattering of pions and kaons is interesting for several reasons: First, by itself, in order to test and understand the dynamics of these particles, which are the pseudo-Goldstone Bosons of the QCD spontaneous chiral symmetry breaking. Second, because these scattering processes are one of the main sources of information on the existence and parameters of several meson resonances. In particular, this is the case of light scalar mesons, whose very existence, nature and classification are still a matter of debate (see the note on light scalars in the Review of Particle Properties (RPP) [1]). These resonances are very relevant for the identification of glueballs, tetraquaks or molecular states that lie beyond the ordinary meson states of the naive quark model. Finally, being the lightest mesons, final state interactions (FSI) of pions and kaons play an essential role in the description of many hadronic processes. The unprecedented statistical samples obtained in the last years on different hadronic experiments and the even more ambitious plans for future facilities have provoked a renovated interest for precise and rigorous analyses of existing meson-meson scattering data, superseding simple model descriptions.
Unfortunately, most of the data on meson-meson scattering [2][3][4][5][6][7] are extracted indirectly from meson-nucleon to meson-meson-nucleon reactions. This extraction is complicated, relying on some model assumptions, and for this reason it is affected with large systematic uncertainties, which can be estimated from the differences between data sets from different experiments (and for ππ scattering even within data sets from the same experiment [2]). Moreover, the description of these data is frequently done in terms of meson-meson models which can lead to artifacts and unreliable determinations of resonances and their parameters. It is for these reasons that dispersive techniques are required.
Dispersion relations are the mathematical expression of causality and crossing. They relate the amplitude at a given energy to integrals of the amplitude and can be used as consistency tests of the experimental data or as constraints on the fits. We will make both uses here. For dispersive integrals to be evaluated just over the physical region, crossing must be used and two main kinds of dispersion relations appear then: Forward Dispersion Relations (FDRs) and those for partial waves generically know as Roy or Roy-Steiner equations [8,9], depending on whether the scattering among particles with equal or different masses. FDRs are rather simple and easily extended to arbitrary energies. They have been recently applied to constrain ππ [10][11][12][13] and Kπ [14] scattering amplitudes that will be used as input in some stages of the present work. Roy-like equations are a complicated system of coupled equations, limited in practice to energies of O(1 GeV) for meson-meson scattering. However, they provide a rigorous continuation the complex plane that allows for a precise and model independent determination of resonances. Actually, it was only in 2012 that the RPP [1] considered settled the issue of the existence and parameters of the much debated scalar f 0 (500) resonance [15], traditionally known as σ-meson, and to a very large extent this was due to the results of dispersive analyses of ππ scattering amplitudes with versions of Roy equations [16,17]. Similarly, the scalar K * 0 (800) or κ-meson has also been obtained from πK scattering using dispersive methods [18], the most reliable value [19] being the Roy-Steiner method based on hyperbolic dispersion relations [20], but according to the RPP this resonance still "needs confirmation" [1]. Roy-Steiner equations have also been applied recently to πN scattering [21] and for γγ → ππ [22]. For meson-resonances beyond ∼1 GeV, Roy-like equations are not used in practice, but other analytic tools have been recently applied [23] to extract resonance poles from the description of amplitudes in the physical region constrained with dispersion relations, thus minimizing the model-dependence.
The purpose of this paper is to obtain a set of simple ππ → KK scattering parameterizations satisfying Roy-Steiner dispersion relations that can be easily used later on both by theoreticians and experimentalists, as has already been the case of previous works for ππ and πK scattering. The motivations to study ππ → KK are the ones explained above for meson-meson scattering in general: i) a rigorous ππ → KK description is a necessary input for further studies of resonances (like scalars in the 1 to 1.6 GeV range), in particular in order to compare their ππ and KK couplings, ii) it is also an essential ingredient in the Roy-Steiner study of Kπ scattering and the determination of the controversial K * 0 (800)-meson (whose determination is one of the goals of a recent proposal at JLab [24]) iii) the ππ → KK amplitude also influences, via unitarity, the ππ → ππ and ππ → N N amplitudes, and consequently those of KN andKN scattering. Finally ππ → KK is a very relevant ingredient in the FSI of numerous hadron decays. For instance, the role of ππ → KK re-scattering has gained a renewed interest due to the recent observation of a large CP violation in recent studies at LHCb [25], although the amplitude used for such studies has been approximated with simple models and the amplitudes obtained here could be used to avoid such assumptions in further studies which are under way. Finally, lattice calculations of the coupled channel ππ, KK, ηη scattering have appeared very recently [26]. Although these calculations are performed still at relatively high pion masses, the physical point where one can compare with our actual ππ → KK could be accessible soon.
Dispersive analysis of ππ → KK scattering and its relation to πK → πK scattering were first performed in the seventies [27][28][29][30]. It was soon clear that the formalism of fixed-t dispersion relations combined with hyperbolic dispersion relations (HDR) for partial waves [9] was best suited to study the physical regions of both channels simultaneously [28,30]. However, ππ → KK data was scarce and these analyses only allowed for crude checks of low-energy scalar partial waves, frequently focusing on threshold parameters and the non-physical region between the two-pion and the two-kaon thresholds (or at most up to 1100 MeV). For a review of the theoretical and experimental situation until 1978 we refer to [31].
The main experimental results on ππ → KK partialwave, that will be thoroughly analyzed in this work, were obtained in the early eighties [4,5], indirectly from πN → KKN ′ reactions. They extend from energies very close to the KK threshold up to 1.6 GeV. Several models exist in the literature describing these ππ → KK data [32], in particular with unitarized chiral Lagrangians [33]. These works are of relevance for studies of f 0 resonances and glueballs in that range.
A renewed interest on dispersive analysis of ππ → KK at the turn of the century was triggered by the need for precise determinations of threshold parameters and Chiral Perturbation Theory low energy constants. Actually, sum rules for πK were obtained from a Roy-Steiner type of equations from HDR [34] in which the ππ → KK amplitude in the unphysical region was obtained as a solu-tion of a dispersive Mushkelishvili-Omnés problem. The ππ → KK partial-wave data of [4,5] was used as input. However, no dispersive analysis of these data has been carried out beyond the KK threshold, mostly due to the relatively low applicability limit of the HDR along the su = b hyperbolas used in those works. It was nevertheless shown that an extrapolation of the HDR solutions beyond their applicability region was fairly close to the data. Finally, in [20] a Roy-Steiner type of analysis was performed to obtain solutions for the πK elastic amplitudes, using once again as input the ππ → KK amplitudes in the physical region. This study was the basis for establishing the existence of the κ meson through a dispersive analysis [19].
The aim of this work is then to provide a simple set of ππ → KK parameterizations that describe the data up to 2 GeV while also satisfying dispersive constraints in the whole region from ππ threshold up to 1.47 GeV. To this end, we will derive a new set of hyperbolic dispersion relations, along (s−a)(u−a) = b hyperbolas, choosing the a parameter to maximize the applicability range which allows us to use them up to 1.47 GeV. This will also allow us to test different and often conflicting data sets and popular parameterizations.
The plan of the work is as follows: in Sec.II we will introduce the notation, in Sec.III we will present simple unconstrained fits to the different ππ → KK data as well as a Regge formalism for the high energy part, taking particular care on the determination of uncertainties. In Sec.IV we will derive our new set of HDR, i.e. Roy-Steiner like equations for partial waves, and formulate the Mushkelishvili-Omnés problem used for both the unphysical region below KK threshold and the physical region up to 1.47 GeV. In Sec.V we will first use these equations as checks for the unconstrained parameterizations. Finally, in Sec.VI we will impose the new relations on the data fits. This will lead to the desired constrained fits to data satisfying the analyticity requirements, which are the main results of this work. In Sec.VII we will summarize our findings and conclude.

II. KINEMATICS AND NOTATION
Throughout this work we will be working in the isospin limit of equal mass for all pions , m π = 139.57 MeV, and equal mass for all kaons, m K = 496 MeV.
Crossing symmetry relates the ππ → KK amplitudes to those of πK scattering. It is then customary to use the standard Mandelstam variables s, t, u for πK scattering, satisfying s + t + u = 2(m 2 π + m 2 K ) and write where G I are the fixed isospin I = 0, 1 amplitudes of ππ → KK whereas the F ± are the s ↔ u symmetric and antisymmetric πK amplitudes, respectively. The latter are defined as where now F I are the fixed isospin I = 1/2, 3/2 amplitudes of πK scattering. These satisfy: from where the s ↔ u symmetry properties of F ± follow. In this work we will also use the partial-wave decompositions of the πK and ππ − KK scattering amplitudes, defined as follows: where q π = q ππ (t), q K = q KK (t) are the CM momenta of the respective ππ and KK states, namely (5) Note the (q π q K ) ℓ factors in the partial waves of the tchannels, which are customarily introduced to ensure good analytic properties for g ℓ (t) (see [35] in the ππ → NN context). The scattering angles in the s and t channels are given by: where λ s = (s − (m π + m K ) 2 )(s − (m K − m π ) 2 ) = 4s q 2 Kπ (s). It is also convenient to define m ± = m K ± m π , Σ 12 = m 2 1 + m 2 2 and ∆ 12 = m 2 1 − m 2 2 , as well as t π = 4m 2 π , t K = 4m 2 K . In the rest of this work, and unless stated otherwise, m 1 = m K , m 2 = m π , ∆ = ∆ Kπ , Σ = Σ Kπ and q = q Kπ (s). For later use we define the Kπ scattering lengths as follows: and similarly for a ± 0 . Let us recall that in the case when we have two identical particles in the initial state, as it happens with two pions in the isospin limit formalism, we define For later use we also write here the explicit expressions for the ℓ = 0, 1, 2 partial waves: Finally, the relation with the S-matrix partial waves, which allows for straightforward comparison with some experimental works, is:

A. The Data
As we have already emphasized in the introduction we will explicitly choose very simple parameterizations to fit the data, so that they can be used easily later on. In this section we will just describe the data without imposing dispersion relations. These will be called Unconstrained Fits to Data (UFD). In this way the fits to each wave are independent from each other. Later on we will impose the dispersion relations as constraints and obtain the Constrained Fits to Data (CFD). This will correlate different waves.
The data we will fit are of four types. First, we will use data on the phases and modulus of the g 0 0 , g 1 1 partial waves extracted from π − p → K − K + n and π + n → K − K + p at the Argonne National Laboratory [4] and from π − p → K 0 s K 0 s n at the Brookhaven National Laboratory in a series of three works [5][6][7], that we will call Brookhaven-I, Brookhaven-II and Brookhaven-III, respectively. Second, for the tensor g 0 2 wave, data for its modulus was given in Brookhaven-II and Brookhaven-III, although as we will see the old experimental parameterizations are not quite compatible with the present resonance parameters listed in the RPP. Third, for higher partial waves, which play a very minor role in the numerics, we use simple resonance parameterizations with their parameters as quoted in the RPP. Finally, for the high-energy range above 2 GeV we rely on recent updates [13,14,19], of Regge parameterizations [36] based on factorization and the phenomenological observations about Regge trajectories or the Veneziano model [37].
B. Partial wave fits from KK threshold to 2 GeV We now describe our partial-wave parameterizations in the region from KK threshold to 2 GeV. For all of them we define a modulus and a phase t I ℓ = |t I ℓ |e iφ I ℓ . We will start with the waves that have less controversy on the data sets and that, as we will see later, satisfy best our Roy-Steiner-like equations, leaving for the end the most difficult one, which is that with ℓ = 0, I = 0. Note that since in the isospin limit all pions are identical particles, Bose statistics applies and ℓ + I must be even.
1. ℓ = 1, I = 1 partial wave For the g 1 1 partial wave there is only data from the Argonne Collaboration (Cohen et al. [4]), extending up to around 1.6 GeV for both the modulus |g 1 1 | and its phase φ 1 1 . Although there is no data on the 1.6 to 2 GeV region, which is the starting energy of our Regge parameterizations, we will see that a rather simple functional form covering the whole range from ππ threshold up to 2 GeV satisfies fairly well the Roy-Steiner equations even before imposing them as constraints. In particular we will use a phenomenological parameterization similar to that in [20]: where the three vector resonances ρ(770), ρ ′ = ρ(1450), ρ ′′ = ρ(1700) have been parameterized by a combination of three Breit-Wigner-like shapes: and m V , Γ V correspond to the masses and widths of the resonances given in Table I. Note thatq 2 P (t) ≡ q 2 P (t)Θ(t − 4m 2 P ) vanishes below the 2m P threshold. In particular, Eq.(11) below KK threshold is similar to the widely used Kuhn and Santamaría form in [38]. In this region, since the coupling to the 4-pion state is negligible and ππ scattering is elastic, Watson's Theorem implies that φ 0 0 (t) should be equal to the phase shift of the I = 1, ℓ = 1 partial wave of ππ scattering. Since C and r 1 are real, they do not contribute to the phase, nor β 1 nor γ 1 , being multiplied byq 2 K , so that the parameters m ρ , Γ ρ , β, γ are obtained from a fit to the dispersive analysis [13] of the ππ phase shift in the elastic region. Indeed, in the lower panel of Fig.1 it can be seen that our parameterization describes remarkably well the ππ scattering data on the phase below KK threshold.
The parameters of the ρ ′′ resonance are fixed for simplicity to those of the RPP [1], whereas those for the ρ ′ are allowed to vary within 1.5 standard deviations within the values listed in the PDG. Note that the ones determined by the CLEO Collaboration [39] are not compatible with our best fit, if one tries to fix those parameters to reproduce the ππ → KK data the χ 2 is increased by almost a factor of 2. Then we fit the rest of the parameters to describe the data in the physical and pseudophysical regions, the best result is shown in Fig. 1 and the parameters are given in Table I. The fit has a total χ 2 /dof = 1.7, but a slightly larger χ 2 /dof = 2.2 is found in the physical region. Conservatively we use the square root of the latter to rescale the fit parameter uncertainties in the table. The data and the results of our Unconstrained Fit to Data (UFD) are shown in Fig. 1. Note that we plot the modulus from KK threshold and that, as already commented, data only reaches up to 1.57 GeV. The shape above that energy is almost entirely given by the m ρ ′′ resonance. Concerning the phase, from the two-pion threshold to the KK threshold it is indistinguishable from that obtained from the ππ dispersive analysis in [13]. In Fig.1 our result below threshold can be compared to the data from elastic ππ scattering [40,41]. Note also the large uncertainty of both the data and the error bands in the region around 1.5 GeV, which is due to the fact that the modulus almost vanishes there. Fortunately, this will also make the contribution of that region to the dispersive integrals almost negligible. The data in Fig. 2 that we use for this wave in the physical region, were obtained in the Brookhaven-II analysis [6], published 6 years after Brookhaven-I. The Brookhaven-II work was a study of the I = 0, J P C = 2 ++ channel of ππ →KK scattering within a coupled channel formalism, which included data from other reactions. The latest Brookhaven-III re-analysis by some members of that collaboration, including even further information on other processes can be found in [7]. Note that our  Note that the phase below KK follows that of I = 1, ℓ = 1 elastic ππ scattering [13]. The white circles and squares come from the ππ scattering experiments of Protopopescu et al. [40] and Estabrooks et al. [41], respectively. normalization differs from that in the experimental works and this is why we are plotting |ĝ 0 2 |, defined as: Contrary to the previous ℓ = 1, I = 1 case, where the ρ(770) resonance dominates the unphysical region, now the lowest resonance is well above the KK threshold and therefore it does not dominate the unphysical region. Thus our ℓ = 2, I = 0 parameterization will have two pieces: one above KK threshold and another one below.
Concerning the physical region, t ≥ t K , note that there is only data for the modulus |ĝ 0 2 |, Fig 2. Therefore, since we also need to have a phase we use a phenomenological description in terms of resonances similar to that in [7], which is a sum of usual Breit-Wigner shapes, although since they overlap significantly we include some interference phases. We thus use: where D 2 (x) = 9 + 3x 2 + x 4 provides the usual Blatt-Weisskopf barrier factor for ℓ = 2, with a typical r = 5 GeV −1 ≃ 1 fm. In Eq.(15) above, T = 1, 2, 3 stands for the tensor f 2 (1270), f ′ 2 (1525) and f 2 (1810) resonances, respectively. Since they decay predominantly to ππ,KK and ππ, respectively, we have set q 1 (t) = q 3 (t) = q π (t), whereas q 2 (t) = q K (t). The mass M T and width Γ T of each resonance after the fit are given in Table II. As can be seen in the Brookhaven-II and III fits in [6,7], the f ′ 2 (1525) was at odds with the present knowledge about this resonance parameters, moreover, the parameters of the f 2 (1810) vary within a huge range even when using almost the same data. As we have no data for the phase of the partial wave it is not possible to fix the position of the masses with accuracy, however, performing a coupled-channel analysis for the tensor partial wave is out of the scope of this work, mostly because we have no dispersive control over other channels apart from ππ → KK. For that reason we have included the masses of both the f 2 (1270) and the f ′ 2 (1525) as additional data for our fit. In particular, we take as input for the fit m f2 = 1.2755 ± 0.0035 GeV which is the average and standard deviation of the values used in the RPP's own average [1]. This we do to have a more conservative estimate of the systematic uncertainty. For the f ′ 2 we take directly the RPP average m f ′ 2 = 1.525±0.005 GeV. The inclusion of the f 2 (1810) is purely phenomenological, following [6,7], just to describe the final rise seen in the modulus, but this resonance still "needs confirmation" according to the RPP. We could have described this raise equally well with another functional form, although it is also clear that there exist some enhancements of the amplitudes and phases for ππ → ππ and ππ → ηη. Its numerical effect on our dispersive integrals is rather small. In Table II we also provide the phases φ T resulting from the fit to data.
Concerning the unphysical region, t < t K , since the contribution of the four pion state is negligible, we have assumed that ππ scattering is elastic. Hence we can use Watson's Theorem to identify φ 0 2 = δ of ππ scattering data. For this we have used a conformal expansion similar to that in [13] but with one more parameter B 2 fixed to ensure a continuous matching of g 0 2 at threshold. Namely: where has been fixed by continuity with the piece above t K in Eq. (14). In Table II we provide values of B 0 , B 1 after fitting the CFD phase-shift in [13]. With this parameterization we obtain a final χ 2 /dof = 1.4. Thus we rescale our uncertainties by a factor of ∼ 1.2. We have checked that this phase is also compatible within uncertainties with the dispersive analysis of the ππ D-wave using Roy and GKPY equations in [42].
Neither Brookhaven-I nor Argonne provide data for this wave, nor the models they used to parameterize it. Nevertheless Brookhaven-I shows a plot with the central value of their phase for this channel, which is later used to extract the g 0 0 phase. As seen in Fig.3 our phase is fairly compatible with the Brookhaven-I model between 1.25 and 1.54 GeV. However, also in that figure it can be seen that the Brookhaven-I model violates Watson's Theorem at low energies, which our phase fully satisfies. In addition, above 1.6 GeV our phase, obtained by fitting the Brookhaven-II data [6] on the modulus with modern values for the f 2 family of resonances, is rather different from the flat behavior of the Brookhaven-I model [5] up   Comparison between the UFD g 0 2 phase and the one obtained with the Brookhaven-I model. Note that the latter violates Watson's Theorem at KK threshold. Also, the former includes an f0(1810) resonance whereas the latter uses a flat background. As explained in the text, the latter is strongly disfavored when fitting Brookhaven II data on the modulus. to 1.9 GeV. The reason is that the Brookhaven-I model used a simple smooth background to describe the 1.6-1.9 GeV region, instead of the f 2 (1810) used in this work. Actually, we have checked that if we impose the phase of the Brookhaven-I model on our fit to the Brookhaven-II modulus, the resulting χ 2 /dof is ∼ 5, and thus strongly disfavored with respect to our phase. Even by deforming our fits by including more parameters, the best we have been able to achieve when imposing the phase of the Brookhaven-I model above 1.6 GeV, is χ 2 /dof ∼ 3, but at the price of introducing contributions difficult to interpret in terms of resonance parameters. Both the violation of Watson's Theorem and the use of such non-resonant background make the Brookhaven-I solution suspicious.
Unfortunately the Brookhaven-I was used to extract the phase of the g 0 0 , which therefore also becomes suspicious above 1.6 GeV. Nevertheless, and with this caveats in mind we will still study the g 0 0 phase coming from the Brookhaven-I collaboration above 1.6 GeV. The reason is that this region lies outside the applicability range of Roy-Steiner equations, so that for our purposes is just input. Fortunately, the modulus there is very small, so that the contribution from this region to the Roy-Steiner equations below 1.6 GeV is very suppressed. In Appendix A, we have calculated either with our g 0 0 phase or the Brookhaven-I phase, and the difference lies within our uncertainties in the region up to 1.47 GeV, which is the one of interest for this work since it is the one where partial-wave dispersion relations can be applied.
This wave is the most complicated but also the most interesting one for hadron spectroscopy, since here we can find the much debated scalar-isoscalar resonances. For the g 0 0 (t) partial wave there are data in the whole region of interest on both the modulus |g 0 0 | and the phase φ 0 0 , which we show in Fig.4. The data sets extend up to 2.4 GeV, but we do not fit that region because from 2 GeV we will use Regge parameterizations. It is then convenient to split in two regions the data description below 2 GeV:

I) Region I: From
√ t min,I = 2m K up to √ t max,I = 1.47 GeV, where data from Argonne [4] and Brookhaven-I [5] coexist. Note that this region will lie within the applicability of Roy-Steiner equations and will be later constrained to satisfy dispersion relations.
Concerning the phase φ 0 0 , it is clearly seen in Fig.4 that from 2m K up to 1.2 GeV, the Argonne [4] and Brookhaven-I [5] sets are incompatible. Let us now recall that, by Watson's Theorem, φ 0 0 at KK threshold should match the scalar-isoscalar ππ → ππ phase shift δ (0) 0 . However, the ππ scattering analyses with Roy and GKPY equations that extend up to or beyond KK threshold [13,43] find δ (0) 0 > 200 o , which is consistent with the Argonne [4] phase, but much higher than the phase of Brookhaven-I [5]. Therefore, for our fits we have discarded the phase of Brookhaven-I [5] below ∼1.15 GeV, i.e. until it agrees with that of Argonne [4].
Concerning the data on |g 0 0 |, shown in Fig. 4, the Argonne and Brookhaven-I sets are consistent among themselves but not with the Brookhaven-II. However, the latter is consistent up to 1.2 GeV with the dip solution for the inelasticity favored from dispersive analyses of ππ → ππ scattering [13,43] (assuming that only ππ and KK states are relevant). Finally, the "dip" solution from ππ scattering in the 1.2 GeV to 1.47 region has such large uncertainties that is roughly consistent with the three data sets.

II) In the region from
√ t min,II = 1.47 GeV to √ t max,II = 2 GeV Roy-Steiner equations will not be applicable and thus this region will only be used as input for our dispersive calculations for lower energy regions. Note that here all experiments are roughly consistent, although the Argonne set only reaches up to ∼1.5 GeV, Brookhaven-I up to ∼1.7 GeV and only Brookhaven-II reaches up to 2 GeV. Therefore in order to test different data sets independently and to be able to impose later Roy-Steiner equations as constraints below 1.5 GeV using as input the region above, we have decided to parameterize our amplitudes by piecewise functions. Actually, each piece will be parameterized by Chebyshev polynomials, because they are rather simple and, in practice, tend to reduce the correlation between the small number of parameters needed to obtain a good fit. They are given by: Thus we first map each energy region i = I, II into the x ∈ [−1, 1] interval through the lineal transformation Note that for any n, p n (1) = 1 and p n (−1) = (−1) n , which is useful for matching the different pieces smoothly up to the first derivative.
Since for the φ 0 0 phase we have already selected a single set on each region, our Unconstrained Fit to Data (UFD) will be given in just two pieces: Note that we set: in order to impose continuity at KK threshold and between the two energy regions, respectively. In addition, we fix C 1 to have a continuous derivative for the central value of the curve. The rest of the parameters of the fit are given in Table III. The total χ/dof = 1.47, which comes slightly larger than one due to some incompatibilities between data sets. Consequently, the uncertainties of the parameters in Table III    data of Brookhaven-II [6]. ii) A UFD C fitting the "Combined" data of Argonne [4] and Brookhaven-I [5]. Both use the same data in Region II. Thus we will use the following functional form: where we now set: in order to ensure continuity between the two regions and we fix F 1 to ensure a continuous derivative for the central value.
Both the UFD B and UFD C fits, whose parameters are given in Tables IV and V

Partial waves with ℓ > 2
For higher partial waves we just use Breit-Wigner descriptions associated to the poles listed in the PDG. In  particular, for the g 1 3 (t) we include a single ρ 3 (1690) resonance. The ℓ = 4 partial wave is only included in the g 0 2 (t) dispersive calculation due to its negligible contribution below 2 GeV for the g 0 0 (t).

C. Higher energies
There is no high-energy experimental information on ππ →KK nor πK → πK. However, the high energy behavior of both processes can be confidently modeled by applying factorization to Regge amplitudes obtained for other processes. In this work we will use, for the s-channel above 1.74 GeV the Regge model description presented in [36] and updated in [13,14], whereas for the t-channel we will use the asymptotic forms of the Veneziano model [37], with the updated parameters in [20], to describe the process above 2 GeV. The reasons to choose 2 GeV in this work are twofold: on the one hand data for the g 0 0 and g 0 2 waves reaches above that energy, on the other hand, even if the g 1 1 data ends at 1.6 GeV, the ρ ′′ (1720) is well established in the RPP and with its 250 MeV width, reaches well above 2 GeV. Thus we rely on our partial-wave parameterizations up to 2 GeV, but not much more.
In what follows we provide the detail of these descriptions using the notation of this work.
For the symmetric amplitude we have the Pomeron P (s, t) contribution and the f 2 or P ′ (s, t) exchange: where, as explained in [36], f K/π is the factorization that allows to convert one ππ−Reggeon into a KK−Reggeon vertex, whereas r is related to the branching ratio of the f 2 (1270) resonance toKK. In addition In contrast, the antisymmetric amplitude is dominated by just one contribution coming from the exchange of a Reggeized ρ: where now g K/π is the factorization constant to change a ππ → ρ Regge vertex into KK → ρ, and All the parameters in Eqs. (26) and (28) correspond to Regge exchanges without strangeness (the Pomeron, f 2 and ρ) and can be determined [36] from processes that do not involve kaons. Therefore in this work we fix them, both for the unconstrained (UFD) and constrained fits (CFD) here, to their updated values of the CFD fits given in [13], which are listed in Table VI. Let us remark that with these parameters our asymptotic value of the Pomeron πK cross section is ≃ 10.3 mb. This is about twice the ≃ 5 ± 2.5 mb value used in [20]. This value was inspired by the work in [16], which asymptotically yielded 6 ± 5 mb for ππ scattering. However, this ππ value has been revisited recently by members of the same group [44] yielding 12.2 ± 0.1 mb for ππ scattering, thus supporting our larger value for πK rather than 5 ± 2.5 mb.
In contrast, the determination of parameters f K/π , r and g K/π needs input from kaon interactions. In principle all them were determined in [36] from KN factorization and we take the f K/π and r values from that reference. Concerning g K/π we take the updated value from the Forward Dispersion Relation study of πK scattering in [14] [10,13]. Since these could be fixed using reactions other than πK scattering, they will be fixed both in our UFD and CFD parameterizations.

Regge
Used both for 1.47 ±0.14 (we use the value from the CFD there). Their values can be found in Table VI. Since their determination involves kaon interactions, we will allow them to vary when constraining our fits with dispersion relations, i.e. from the UFD to the CFD sets. However, in the tables it is seen that the change is minute. For the t-channel, ππ → KK, we also need the exchange of strange Reggeons, for which we will assume that the dominant trajectories K * 1 (892) and K * 2 (1430) are degenerate, Thus we use for them a common trajectory α K * (s) = α K * + α ′ K * s whose parameters, listed in Table VI, are obtained from the linear Regge trajectories for strange resonances and therefore are kept fixed for both our UFD and CFD sets.
All these features are nicely incorporated in the dualresonance Veneziano-Lovelace model [37,45], which was already used in the Roy-Steiner context for πK scattering [34]. Here we are only interested in the asymptotic behavior [20]: where ψ is the polygamma function. Note that the a, b parameters in the above equation will be those defining the hyperbola (s−a)(u−a) = b along which we will define our hyperbolic dispersion relations in the next section. For a given t, s b is the value of s that lies in the previous hyperbola. In order to compare with the expressions in [20], where a = 0, we have kept one the O(b/t) order in the b/t expansion, although its numerical effect is rather small. We estimate the remaining λ parameter from exact degeneracy between the ρ and K * families. We thus match Eq.(27) at 2 GeV with the expression from the degenerate Veneziano model with its original parameter α V ρ = 0.475. In this way we find which is compatible with the value used in [20], λ = 14 ± 5. Conservatively we also add a 25% uncertainty due to the breaking of degeneracy and thus we arrive to our final estimate which for completeness is also listed in Table VII. Given that it is a crude estimate we will allow this value to vary when constraining our fits to obtain the CFD sets. We will see that after imposing the dispersive constraints we obtain λ = 10.7, which due to the degeneracy between the ρ and K * families, suggests g K/π ∼ 0.55, in perfect agreement with the value used here that comes from a dispersive πK study. A final remark on the size of Regge contributions is in order. As commented in the introduction, in the next sections we will obtain partial-wave dispersion relations by integrating hyperbolic dispersion relations. This is an integral over b for a family of (s − a)(u − a) = b hyperbolas, while a = −10.8M 2 π is fixed to the value that maximizes the applicability region (see Appendix D). This means that the exponent α K * + aα ′ K * < α K * and thus the Regge contribution to ππ →KK in this work, for the same number of subtractions, is suppressed with respect to its size in [20], where a = 0. This will allow us to consider less subtractions without Regge contributions growing large.

IV. HYPERBOLIC DISPERSION RELATIONS AND SUM RULES
Our goal is to calculate a set of parameterizations that describe the data up to 1.47 GeV consistently with hyperbolic dispersion relations (HDR). As already advanced in the introduction, in this work we will consider a set of hyperbolas (s − a)(u − a) = b and use a to maximize the energy domain where the hyperbolic dispersion relations hold. Note that the phenomenology of the ππ → KK a = 0 case has been studied in detail in [20,34]. Moreover, HDR with a = 0 were also used for the study of πK scattering below the inelastic threshold [20].
In addition, we will use the smaller number of subtractions needed for each channel. This has the advantage that our equations for g 0 0 and g 1 1 are independent from one another. In contrast, in [20] they use more subtractions and the subtraction constants are constrained by means of sum rules that mix the dispersive representations of both waves.

A. Hyperbolic Dispersion Relations
For their derivation we basically follow the same steps described in [28] but using a = 0, or more recently the steps in [21] but applied here to for ππ → KK instead of πN scattering. Recall that in this work we use hyperbolas (s − a)(u − a) = b, which with s + t + u = 2Σ, implies that s and u on these hyperbolas are the following functions of t: (32) Let us remark that we do not need any subtraction for the antisymmetric amplitude where Whereas for the symmetric one: With these numbers of subtractions the convergence is fast enough so that the asymptotic amplitude contribution is relatively small (recall it starts at t = 4 GeV 2 in this work). In the above equations s b and u b are the values of s and u that lie in the hyperbola (s − a)(u − a) = b for a given value of t. Now, we want to rewrite the subtraction constant h(b, a) and for this we follow the procedure in [28,34]. We thus introduce the following fixed-t dispersion relation (36) Note that two subtractions are needed to ensure the convergence of this fixed-t dispersion relation, due to the Pomeron contribution. Next, recall that G 0 (t, s, u) = √ 6F + (s, t, u), so that by equating Eq. (35) and (36) at t = 0, b = a 2 − 2Σa + ∆ 2 , the values of c(t) and h(b, a) are determined. Actually, Eq.(35) can be rewritten as: where We have explicitly checked that in the a = 0 case we recover the HDR in [28,30,34]. However, with our HDR above we can now choose the a parameter to maximize the applicability region of the HDR once projected into partial waves, which we will do in the next subsection. Before finishing this subsection, a comment on the high energy region is in order. We have three different kinds of contributions above 2 GeV, the first one is G I (t ′ , s ′ b ), which can be calculated from Eq. (29). The second kind is the evaluation of F ± (s ′ , 0): for the symmetric amplitude we just use Eq.(25), while for the anti-symmetric one we use Eq. (27). The last kind is for F ± (s ′ , t ′ b ), which corresponds to an exotic exchange, so that its contribution is negligible.

B. Partial-wave hyperbolic dispersion relations
In this work we want to obtain parameterizations of the ℓ = 0, 1, 2 partial waves which are consistent with data and the hyperbolic dispersive representation. Thus, we project Eqs. (33) and (37) into partial waves using Eq. (9) to obtain a set of Roy-Steiner-like equations: The explicit expressions of the G I ℓℓ ′ (t, t ′ ), G ± ℓℓ ′ (t, s ′ ) integration kernels are given in Appendix B. Since so far in this work we have left free the a parameter, we can now use it to maximize the applicability of the equations right above. Note there are constraints coming from the applicability of the HDR in Eqs. (33) and (37) as well as from the convergence of the partial-wave expansion. As shown in appendix D, by setting a = −10.8m 2 π the applicability range of these equations is −0.286 GeV 2 ≤ t ≤ 2.19 GeV 2 . In other words, we can study the physical region from the KK threshold ≃ 0.992 GeV up to ≃ 1.47 GeV. In contrast, the usual HDR projected into partial waves are only valid up to ≃ 1.3, GeV. Thus, with our choice of a, the applicability of the dispersive approach in the physical region, where we can test or use data as input, has been extended by 55% in terms of the √ t variable, or 67% in terms of t.
As can be directly seen in Eq.(39) the g 1 1 (t) partial wave does not have any scattering length as input parameter and its dominant contribution to the integral comes from its own imaginary part. Since it is not subtracted, the Regge contribution is not small, but we have already attached a conservatively large uncertainty to its residue and we will see that it barely changes when using the dispersive representation as a constraint on data. In the case of even partial waves, one subtraction is necessary to ensure the convergence, and hence the output is always influenced by the scattering lengths coming from πK scattering. In this work we fix them to the values obtained in [14], which are also compatible with the Roy-Steiner prediction in [20]. As already commented, an important advantage of using HDR with the smaller possible number of subtractions is to decouple odd and even partial waves. For example in [20] the Roy-Steiner equation for g 0 0 uses g 1 1 as input.
Finally, we want to remark that, as usual, the high energy part of the integrals in Eqs. (39) is obtained by projecting into the corresponding partial-wave the highenergy part of the integrals in Eqs. (33) and (37), where Regge theory was used as input as explained in previous sections.

C. The unphysical region and the Muskhelishvili-Omnès problem
As can be observed in Eqs. (39), the integration region actually starts at ππ threshold. This means that the integrals extend over an "unphysical" regime where ππ → KK scattering does not occur and thus cannot be described with data parameterizations. Nevertheless, below KK threshold the inelasticity to more than two-pion states is completely negligible. Since ππ is the only available state in that region Watson's Theorem implies that the g It ℓ phase below KK threshold is just that of ππ scattering and thus we write φ It ℓ (t) = δ It l,ππ→ππ (t). Note that Watson's Theorem does not provide any direct information on |g It ℓ |. But once the phase is known, determining the modulus in the unphysical region is nothing but the standard Muskhelishvili-Omnès problem [46], that we describe next following similar steps as in [20-22, 28, 34]. Recalling that partial waves have a right-and left-hand cut we can re-write Eqs.(39) as follows: where the ∆ I ℓ (t) contain the left-hand cut contributions and subtraction terms. Note that ∆ I ℓ (t) does not depend on g I ℓ itself, but on other g I ℓ ′ with ℓ ′ ≥ ℓ + 2, which in the unphysical region are much more suppressed than g I ℓ , due to the centrifugal barrier. Now we define the Omnès function which satisfies where, in the real axis, Ω I l,R (t) can be written as: .
In the real axis, Ω I l,R is nothing but the modulus of Ω I l and therefore a real function. Note that from 4m 2 π to t m the Omnés function has the same cut as g I ℓ (t). Thus, we can define a function which is analytic except for a right hand cut starting at t m . Hence we can write dispersion relations for f I ℓ (t), which in terms of g I ℓ (t) read: When t lies in the real axis above the ππ threshold, a principal value must be understood on each integral. In addition, between ππ threshold and t m on the left hand sides the amplitude is reduced to its modulus (since by construction the Omnés function removes the phase), whereas above t m it is reduced to its real part. Since in the next sections we will choose t m with φ 0 0 (t m ) ≥ π we have introduced one subtraction for the g 0 0 (t) Omnès solution in order to ensure the convergence when t → t m . The subtraction constant α will be obtained by imposing numerically a no-cusp condition on t m for g 0 0 (t). The interest of these equations is that for a given g I ℓ (t), the integrals in the unphysical region only make use of the phases and the ∆ I ℓ . But thanks to Watson's Theorem the former are known from ππ scattering, which we take from the dispersive analysis of [13], and the latter do not involve g I ℓ (t) itself, but only partial waves with ℓ ′ − ℓ ≥ 2. These higher partial waves are suppressed in the unphysical region with respect to that with ℓ. We also need input from Kπ scattering that is known and we take from our recent dispersive data analysis in [14]. Thus we can directly solve g 1 1 (t) and g 0 2 (t), for which we have explicitly checked that the ℓ = 3 and ℓ = 4 contributions are small and negligible, respectively. Once we have g 0 2 (t) we can use it as input to solve Eq.(45) for g 0 0 (t).
It is worth noticing here that, in purity, for the Regge contributions to ∆ I ℓ (t), one has to subtract the projection of the Regge amplitude itself into the desired I, ℓ partial wave. Fortunately this projection is negligible, and our solutions do not depend on this procedure.
We still have to discuss the choice of t m , which is always above the KK threshold. It is important to recall that the derivation of the above equations implies that g output (t m ) = g input (t m ). This condition will always be forced into the output no matter if the data at that energy is in good or bad agreement with dispersion relations. If the data at that energy region were not close to the dispersive solution, the output would be forced to describe it and the result could be strongly distorted in other regions. In particular the g 0 0 wave is the most sensitive to this instability, the effect is more moderate on the g 0 2 and negligible for the g 1 1 because it is already very consistent for any t m choice. Thus, we have studied what energy region is the most consistent for g 0 0 when changing t m and we have found that there are two regions that yield systematically rather consistent results between input and output: one around √ t m = 1.2 GeV, which is also valid for g 0 2 , and another one around √ t m = 1.47 GeV. However, if we chose the latter, we find that the uncertainty in the dispersive result between between KK and 1.2 GeV is so large that there is no dispersive constraint in practice, having larger uncertainties could even produce both g 0 0 (t) solutions to be compatible between them. Moreover by looking at Eqs. (45), (46) and (47) one can notice that t m marks the energy above which |g I ℓ | is used as input for its own equation. Since we are actually trying to test the data parameterizations, within our approach we would like to maximize that region and choose the smaller possible t m . All in all, we have made the final choice √ t m = 1.2 GeV for all partial waves. This is a point above KK threshold where there are no cusps coming from the two most important inelasticities (KK, ηη). In particular, the g 0 2 is well controlled at this energy since its largest contribution comes from the f 2 (1270), a very well-known resonance very close to t m .

V. CONSISTENCY CHECK OF UNCONSTRAINED FITS
In order to study in a systematic way the consistency of the unconstrained data parameterizations of Sect.III with respect to dispersion relations, we first define a "distancesquare" for each dispersion relation. Note its similarity to a χ 2 function, although we are still not fitting or imposing the dispersion relations. Here d i is the difference between the "input" and "output" of each dispersion relation at the energy √ t i . We use thirty energy points √ t i equally spaced from threshold up to 1.47 GeV. In addition, ∆d i is the uncertainty in the d i difference, which is obtained by varying the parameters of our unconstrained fits to data (UFD) within their errors.
As we explained before, Eqs. (45), (46), (47) yield the modulus of the partial wave below t m and the real part above. However, in order to simplify our plots and calculations, we will just display the modulus. In particular by "input" we will understand the modulus of the partial wave on the left hand side of Eqs. (45), (46), (47), i.e. as obtained directly from our fits. Similarly, by "output" we will always mean the modulus of the dispersive representation. Note that for t < t m this modulus is obtained from the right hand side of those equations with principal values on each integral. However, for t > t m only the real part is obtained from the integrals and the modulus is reconstructed by adding the imaginary part from the direct parameterizations.
With the above definition we can study the consistency of each partial-wave dispersion relation. It will be well satisfied on the average if its corresponding d 2 ≤ 1. In case of disagreement it is also relevant to check whether it comes from a particular energy region and for this we will show figures comparing the input and output.
Let us study first the consistency of g 1 1 . We see in Eq.(46) that its partial wave dispersion relation is decoupled from even partial waves. The highest partial wave we have considered in ∆ 1 1 is the ℓ = 3 contribution. Actually, by using the simple model dominated by the ρ(1690) resonance described in Sect.III B 4, we have explicitly checked that its contribution is very small and barely affects our results for g 1 1 below 1.47 GeV. As can bee seen in Fig.5 the dispersion relation in Eq.(46) is remarkably well satisfied, with a total d 2 = 1. Such a nice agreement was expected since it has a large contribution from the ρ(770) that dominates ππ scattering in this channel below KK threshold, and our input from [14] is already consistent with ππ data and dispersion relations. Let us now recall that the ππ → KK data we use as input show large uncertainties an fluctuations (see Fig.1). Our UFD description does not follow visually all these fluctuations but, roughly speaking, it averages them and rises softly and monotonously. Still, our UFD is remarkably consistent with the dispersive representation. Actually we have checked that parameterizations with more oscillations may describe the central values of the data points better, but satisfy worse the dispersive representation than our UFD fit. In the ππ → KK physical region we had also included resonant shapes for the ρ ′ and ρ ′′ resonances in our UFD. As seen from our results, the parameters and shape of the ρ ′ , which for a good part lies within the applicability region of our equations, are fairly consistent with dispersion relations. As commented in Sect.III B 1 the ρ ′′ was used just as a simple form to parameterize the amplitude at energies beyond the reach of our dispersive representation where scattering data do not exist.
One could also be worried that, since the g 1 1 dispersion relation has no subtractions, it may require some tuning on the Regge asymptotics and the λ parameter we estimated with the Veneziano model and degeneracy in subsection III C. However the nice fulfillment of the dispersion relation yields strong support for our λ estimations.
In the case of the g 0 2 (t) dispersion relation, Eq.(47), it involves even partial waves with ℓ ≥ 4, but they are almost negligible below 2 GeV. As seen in Fig. 6, when using the UFD parameterizations, the g 0 2 (t) dispersion relation is clearly not well satisfied right above KK threshold and this incompatibility fades away near 1.1 GeV. At threshold, the deviation is ≃ 3σ. Very naively one could have expected this region to be dominated by the f 2 (1270) resonance tail, since the threshold is merely 1.5 widths away from the resonance peak. However, if one tries to use a simple Breit-Wigner description instead of our UFD parameterization, then d 2 ≥ 6. Thus, such naive expectation does not hold, which justifies the elaborated form of our parameterization in Eq.14. Nevertheless, there is still room for improvement that will be achieved when imposing the dispersion relations as constraints in Section VI. C. g 0 0 UFD check Finally, for the scalar-isoscalar dispersion relation in Eq.(45), we need both the g 0 0 (t) and g 0 2 (t). In this case, partial waves with ℓ ≥ 4 are totally negligible below 2 GeV. In Fig.7 we show the results of the g 0 0 (t) dispersion relation when using either the UFD B or UFD C parameterizations as input. In both cases the agreement is poor, particularly due to the results in the region 10-20 MeV above KK threshold, where the dispersive solution increases rapidly. This feature is common to both the UFD B and UFD C and is due to the influence of the f 0 (980). The respective d 2 = 5.6 and d 2 = 2.7 are dominated by this near threshold region. There is a clear need for improvement, that we will achieve by imposing dispersion relations as constraints in the next section, although in both cases the disagreement in the region very near threshold will linger on. However, we will see that for both solutions a very good consistency with dispersion relations can be achieved except for the very near threshold region.
Finally, let us remark that the g 0 0 partial-wave dispersion relation in Eq.(45) depends on the πK scattering length a + 0 . We have checked that the dispersion relation would be better satisfied if we used a somewhat lower value of a + 0 than that obtained in our previous work [14] (which was also compatible with Roy-Steiner determinations [20]). Since in this work we are considering πK scattering amplitudes as fixed input, we keep the value from the πK constrained fit, but this result could be relevant for future re-analysis of πK scattering data. In the upper panel we show the results using as input the UFDB parameterization and in the lower panel those from the UFDC. The gray bands cover the uncertainty of the difference between the input and the respective dispersive result.

VI. CONSTRAINED FITS TO DATA
Therefore, we have just seen that the data on the g 0 2 and even more so on the g 0 0 do not satisfy very well the dispersive representation. There is clear room for improvement. Thus, in this section we will impose the dispersion relations in Eqs. (45), (46), (47) as constraints of the fits. In this way we will obtain a set of Constrained Fits to Data (CFD) which fulfillment of the dispersive representation will be much improved. In this section we use the same functional forms for the amplitudes that we used in Sect.III, but the parameters change from the UFD to the CFD sets. In general the difference between the UFD and CFD parameters is small, with a few exceptions. Nevertheless, due to large correlations in the parameters, even if some CFD parameters deviate from the UFD set, the resulting UFD and CFD curves are typically consistent with one another at the 1 or 1.5 σ level. Only for the constrained analysis of the UFD C , the CFD C g 0 0 partial wave deviates by about 2 σ in the region from 1.25 to 1.45 GeV, but it still compatible with the upper error bars of the data. Hence the CFD description of data is still rather good.
To minimize the discrepancy between the fit used as input in the dispersion relation and the output obtained from the dispersion relation, without deviating much from the data, one first defines a χ 2 -like function where |g I ℓ | exp,k , (φ I ℓ ) exp,k are the experimental values of the k-th data point for the modulus and the phase, respectively, and δ|g I ℓ | exp,k , δ(φ I ℓ ) exp,k are their corresponding errors. The weights W 2 1 , W 2 are used to roughly take into account the degrees of freedom needed to parameterize the curves that describe the modulus and the phase. For simplicity we have chosen the same W 2 1 = 5 and W 2 2 = 12 value for all partial waves as an average value of their degrees of freedom. Note that we actually minimize the sum of this function over the three partial waves of interest (I, ℓ) = (0, 0), (1, 1) and (0, 2). In addition, recall that, as explained in Sec. III B 2, we have added two points to the χ 2 -function to take into account the experimental mass of the f 2 and f ′ 2 resonances. Let us remark that in previous works our procedure was slightly different: we defined a similar χ 2 -like function but in terms of the unconstrained fit parameters, which were not allowed to vary much from their unconstrained best values. In contrast, in Eq. (49) we define or χ 2 -like directly in terms of data, not the unconstrained fit parameters. The reason is that in this work the onset of Regge parameterizations is 2 GeV and thus we use our partial-wave parameterizations to describe data from KK threshold up to 2 GeV. However, the dispersion relations are only applicable up to 1.47 GeV. If we constrained only the fit parameters with the dispersion relations, which affect only the lower-energy data, we would obtain large artificial deviations in the description of the higher-energy data. With the procedure we use here, and contrary to what happened in previous works, if there are some strongly correlated parameters, we can see that their constrained values can deviate appreciably from their unconstrained best values but still the constrained and unconstrained curves look very similar. As the uncertainty variation is of second order, and parameters that are not compatible with old values deviate by a small number of sigmas at most, we still maintain their uncertainties as they are a reliable and almost unchanged estimate of the error, as one can see in the final uncer- tainty band plotted in the figures for the CFD parameterizations.
A. Constrained g 1 1 (t) partial wave Let us recall that the UFD I = 1, ℓ = 1 wave from KK threshold up to 1.47 was already consistent with the dispersive representation. By imposing our dispersion relations d 2 decreases just from 1 to 0.6. The difference between the constrained input and dispersive output for the g 1 1 wave can be seen in Fig. 8. Actually, as seen in in Fig. 9 imposing the dispersive constraints barely changes this wave, i.e. the UFD and CFD curves are almost indistinguishable both for the modulus and the phase of g 1 1 . Note also that, as shown in Fig. 10, the dispersive CFD output perfectly describes the data. In that Figure we also show the CFD modulus in the unphysical region and the continuous matching at threshold.
The new CFD parameters can be found in Table I where it can be checked that the CFD values are remarkably consistent with the UFD ones: only one is beyond one standard deviation just at 1.5 σ. As we are using a non-subtracted HDR to study the odd angular momentum partial waves, the small improvement in the description of this partial wave comes mostly from the slight variation of the Regge parameters. Nevertheless, as it can be seen in Table VII, our CFD result for the λ Regge parameter is compatible with its UFD value, thus supporting the degeneracy between the ρ and K * families.
It is worth noticing that, as we are using no subtractions, the value of the ππ → KK amplitude at t = 0, b = ∆ 2 can be related to the a − 0 πK → πK scattering length a − 0 = (a 1/2 − a 3/2 )/3, using Eq. (33), to obtain the following sum rule [20,47]: Note that the scattering length results from the integration over both πK → πK and ππ → KK channels. Using as input for G 1 our constrained parameterizations just calculated and our the CFD parameterizations for Kπ scattering in [14], we find m π (a 1/2 − a 3/2 ) = 0.249 ± 0.032, (sum rule+CFD).
(51) To be compared with m π (a 1/2 − a 3/2 ) = 0.251 ± 0.014, (sum rule in [20]) obtained in [20] using this same sum rule with their unconstrained input from ππ → KK and the Kπ solutions from their Roy-Steiner analysis of Kπ. We obtain a larger uncertainty since we use the Regge asymptotics from 2 GeV instead of 2.5 GeV as in [20] and because, in contrast to [20], we also include uncertainties in all partial-waves.
Those two values obtained using the sum rule can also be compared with direct calculations from the Kπ amplitudes: The first is obtained from our recent dispersive analysis using Forward Dispersion Relations as constraints on fits to Kπ data [14] and the second from the solutions of Roy-Steiner equations in [20].
B. Constrained g 0 2 (t) partial wave For this wave the agreement was not as good as for the I = 1 and ℓ = 1 partial wave, particularly in the threshold region. After minimization the overall agreement has improved considerably, from d 2 = 1.6 down to 1.1. However, as seen in Fig. 11, our CFD parameterization still shows some small discrepancy with its dispersive output near threshold, although the deviation has improved substantially in that region compared to the unconstrained case.
This improvement is achieved without changing much the CFD parameterization with respect to the UFD. The CFD parameters change little from their previous UFD values, as seen in Table II. In addition, in Fig. 12 we can see that the deviations from the UFD to the CFD modulus are almost imperceptible. There are some differences near threshold but, unfortunately, when plotting  the modulus together with data, the resulting curves look almost identical due to a q(s) 5 factor. In contrast, we can see in Fig. 13 some small difference between the UFD and CFD phase φ 0 2 . This change is actually the one mostly responsible for the improvement in the d 2 .
We have also checked that the values obtained at the KK threshold still fulfill Watson's Theorem when using the values ππ scattering values obtained from dispersion relations [13,42]. One should be careful not to force too much the fit in the threshold region because, as commented in the UFD case, this could spoil the f 2 (1270) mass, which is very well established from different experiments, not just scattering. That is why we considered the f 2 and f ′ 2 masses as additional data points when fitting the ππ → KK data. We have also added this extra contribution when minimizing the χ 2 to obtain the CFD set.  We have tried different parameterizations, including additional flexibility upon Breit-Wigner-like parameterizations, but we have not been able to find a solution that satisfies better the dispersion relation near threshold without spoiling severely the data description.
Finally, let us note that this dispersion relation has some sensitivity to πK scattering, in particular to the scalar partial wave. A more thorough study would require allowing the πK scattering amplitude to vary when imposing the hyperbolic dispersion relations as constraints, but that is well beyond the scope of this work dedicated to ππ → KK, where we have taken πK scat-  12: The continuous line is our final CFD parameterization of the data on the modulus ofĝ 0 2 (t) from the Brookhaven-II analysis [6]. The gray band stands for the uncertainty from the CFD parameters.The dashed line is the UFD parameterization. The difference between the UFD and CFD parameterization near threshold is imperceptible due to the q 5 factor. tering as fixed input.
C. Constrained g 0 0 (t) partial wave The scalar partial wave g 0 0 is the most interesting in this work, given that we are dealing with two incompatible sets of experimental data for the modulus and also because neither of them are consistent with the dispersive representation.
As seen in Section III, on the one hand we have the Brookhaven-II [6] data and, on the other hand, the data of Brookhaven-I [5] and Argonne [4]. From these two sets we obtained the UFD B and UFD C parameterizations, respectively. For the phase we had a single UFD parameterization. Let us recall that the overall UFD C agreement with its dispersive output up to 1.47 GeV is poor, with d 2 = 2.7, whereas the UFD B is even more inconsistent with d 2 = 5.6. In that respect the UFD B parameterization may seem disfavored. However, the UFD C modulus is clearly incompatible with the value that would be obtained from the inelasticity of ππ scattering obtained from dispersion relations [13] assuming two coupled channels, ππ and KK. For that reason we will study here both UFD B and UFD C and will obtain a fit to each data set constrained with our dispersion relation in Eq. (45). We will see that after this process both constrained solutions will be equally acceptable with respect to their consistency regarding dispersion relations.
Let us note that we now use as input the g 0 2 CFD parameterization obtained in the previous subsection. The consistency test of the constrained g 0 0 results can be found in Fig. 14. It can be seen that we obtain an equally good consistency for both the CFD B and CFD C parameterizations except for the region very close to threshold. The behavior in this region is controlled by the f 0 (980) shape in the elastic region of ππ scattering and thus is out of the scope of this work, since we consider it input. The rest of the energy region up to 1.47 GeV has values of d 2 below one.
In Fig. 15 we also compare both CFD parameterizations against their respective UFD parameterizations and the data. There one can see that the UFD and CFD phases are almost identical, except in the 1.1 to 1.2 GeV region where the CFD is higher by more than one standard deviation, and in the 1.9 GeV region where the CFD phase is again higher but well within uncertainties. Actually there are two CFD B and CFD C phases but they are totally indistinguishable.
Concerning the modulus, the UFD C and CFD C are compatible, whereas the CFD B is slightly lower than the UFD B in the 1.05 to 1.15 region, but clearly higher in the 1.3 to 1.45 region. These differences go above the 2-σ level, so that they lie still reasonable close to the data, but prefer to cross the top of the experimental uncertainty bars.
Note that the "dip" structure in the inelasticity from ππ scattering occurs around 1.1 GeV, whereas the biggest difference between the in UFD B and the CFD B is found above 1.25 GeV, so that we conclude that such a dip is not the cause of the deviation for the UFD B set. The dip structure favored by ππ scattering dispersive analyses can therefore be accommodated also with the hyperbolic dispersive representation of ππ → KK.
Therefore we conclude that the data most commonly used in the literature (Argonne [4]) is not necessarily the only acceptable solution and that one does not have to ignore the Brookhaven-II data. Actually, we have shown  In the upper panel we show the results using as input the CFDB parameterization and in the lower panel those from the CFDC. The gray bands cover the uncertainty in the difference between the input and dispersive results. By comparing with Fig. 7 we see that the fulfillment of the dispersion relation by the CFD set has improved considerably with respect to the UFD parameterization. Also, there is no significant difference in the consistency of the CFDB and CFDC sets.
that with the CFD B solution the Brookhaven-II data can also be fairly well described being consistent with ππ → KK dispersion relations and with the dispersive determination of the inelasticity in ππ scattering that, in contrast, is not consistent with the Argonne data. In this sense the CFD C is disfavored against the CFD B set. Finally, in Fig. 16 we also show the CFD B and CFD C parameterizations in the unphysical region. There one can observe that their respective pseudo-threshold behaviors are quite different. Namely, the modulus of the CFD B around the f 0 (980) peak is larger than that of the CFD C . Such different behaviors may have a sizable impact for future studies of πK → πK dispersion relations.

VII. CONCLUSIONS AND OUTLOOK
In this work we have performed a dispersive study of ππ → KK scattering by means of partial-wave dispersion relations of the Roy-Steiner type, i.e. based on hyperbolic dispersion relations. While other studies with similar equations used dispersion theory to obtain information on the sub-threshold region, we have also used them for the first time in the physical region. Moreover, we have derived a set of equations based on (s − a)(u − a) = b hyperbolae in which we have obtained the value of a that maximizes the applicability range of these hyperbolic dispersion relations. Compared to the existing a = 0 case we have increased the applicability range of the hyperbolic partial-wave dispersion relations in the physical region by 67% in the t variable. This has allowed us to study dispersively the existing data sets on ππ → KK up to 1.47 GeV.
In particular, on a first step we have obtained a set of unconstrained fits to data (UFD) for each partial wave g I ℓ (t), where ℓ and I are the angular momentum and isospin, respectively. For the case of the scalar-isoscalar wave g 0 0 we have provided two alternative fits, called UFD B and UFD C , to differentiate between fits to two conflicting sets of data. In addition, we have provided high energy parameterizations for ππ → KK scattering, based on factorization and Regge theory, that we need for the high energy part of our dispersive integrals. We have then tested these UFD parameterizations against our dispersion relations. We have found that the P wave UFD is very consistent with dispersion relations. Also, the D wave is crudely consistent with these equations, although there is clear room for improvement. In contrast, we have found that the unconstrained fits to both solutions of the scalar-isoscalar wave show a significant inconsistency with the dispersive representation, particularly, but not only, near threshold. These deviations are not related to the high energy input, and thus they become a first warning to the phenomenological use of simple fits to the existing data.
Next, we have provided a new set of fits to data using the hyperbolic partial-wave dispersion relations as constraints. For the P and D waves, these constrained fits to data (CFD) satisfy the dispersion relation within uncertainties while describing very well the experimental data. There is only some relatively small tension in the D-wave threshold region. In particular we have shown that a simple description of the D-wave threshold region with a simple Breit-Wigner parameterization of the nearby f 2 (1275) resonance is not acceptable.
We have also found that, with the exception of the region very close to threshold, both constrained parameterizations of the g 0 0 wave, labeled CFD B and CFD C , satisfy well the dispersion relations, while still describing reasonably well their respective sets of data. Nevertheless some systematic deviations from the data central values are needed in order to satisfy the dispersive representation, particularly for the UFD B in the region between 1.25 and 1.45 GeV. This becomes a second warning towards considering only the most popular data set described by UFD C : the data on which the UFD C set is based can be also described consistently with hyperbolic partial-wave dispersion relations, and is favored by previous ππ scattering dispersive analyses. This second set should definitely not be discarded, if not directly favored against the most popular one.
In conclusion, our constrained data fits provide reliable, precise and simple parameterizations of data on S, P and D partial waves up to 2 GeV, which are consistent with the hyperbolic dispersive representation up to its maximum applicability limit of 1.47 GeV.
As an outlook for this work, our constrained parameterizations could be used by both the theoretical and experimental hadron communities as input for other processes. Actually, in the near future we plan to use them for further studies. For example: to implement re-scattering effects in CP violating decays involving pions and kaons, or to study the much debated f 0 (1370) and f 0 (1500) resonance by means of model-independent methods based on analyticity, or combined with ππ scattering determinations, to obtain a precise determination of the a ± 0 scattering lengths from sum rules. Finally, we will use them as input for a similar dispersive analyses of Kπ scattering data and the rigorous and precise determination of light-strange resonance parameters. In particular, this input will be very useful for a precise determination of the elusive K * 0 (800), by analyzing data using hyperbolic partial-wave dispersion relations of the type derived here.   However, as a matter of fact that the g 0 0 (t) phase above 1.6 GeV is different if one assumes the presence of the f 2 (1810) in the g 0 2 (t). If one wants to be consistent with that assumption, which at present in the RPP seems to be favored versus the flat solution used in [5], then one should use our "New UFD" rather than the main one in the text. Of course, the difference below 1.47 GeV is negligible.

Appendix C: t-channel numerical solution
In order to calculate numerically the Omnès integrals it is convenient to make a change of variables to facili-tate the integration near t m . For concreteness we explain the g 1 1 (t) dispersion relation, following closely the method explained in [20,22] although in our case it has one less subtraction. The others waves are similar. We start by separating within the integrals the regions above and below t m , .
We now introduce the variable v(t ′ ) = (t ′ −t m )/(t m −t) and write: As shown in [20] this equation also implies the continuity of the partial waves at the matching point t m . Since τ (t m ) = ∞ and using 1 π , inside Eqs. (45), (46), (47) one recovers the matching values |g 0 0 (t m )|, |g 1 1 (t m )|, |g 0 2 (t m )|. In addition, for g 0 0 , and due to the introduction of the free parameter α, one has to impose a smooth continuity condition by fixing α, done numerically in this work. Otherwise spurious cusps would be produced for the modulus of the amplitude at t = t m , spoiling the analytic structure and its behavior at different values of t.