The sigma and f_0(980) from Ke4 + pi-pi scatterings data

We systematically reconsider, within an improved"analytic K-matrix model", the extraction of the sigma = f_0(600) and f_0(980) masses, widths and hadronic couplings using new Ke4 = K-->pi-pi e nu_e data on pi-pi phase shift below 390 MeV and different sets of pi-pi-->pi-pi / K-K scatterings data from 400 MeV to 1.4 GeV. Our results are summarized in Tables 1, 2 and 5. In units of MeV, the complex poles are: M_sigma=452(12) - i 260(15) and M_f=981(34) -i 18(11), which are comparable with some recent high-precision determinations and with PDG values. Besides some other results, we find: |g_{sigma K+K-}|/|g_{sigma pi+pi-}|=0.37(6) which confirms a sizeable g_{sigma K+K-} coupling found earlier, and which disfavours a large pi-pi molecule or four-quark component of the sigma, while its broad pi-pi width (relative to the one of the rho-meson) cannot be explained within a \bar qq scenario. The narrow pi-pi width of the f_0(980) and the large value: |g_{f K+K-}|/|g_{f pi+pi-}|=2.59(1.34), excludes its pure (\bar uu+\bar dd) content. A significant gluonium component eventually mixed with \bar qq appears to be necessary for evading the previous difficulties.

2. The analytic K-matrix model for ππ → ππ/KK In this approach, the strong processes are described by a K matrix model representing the amplitudes by a set of resonance poles [55] 4 . In that case, the dispersion relations in the multi-channel case can be solved explicitly, which is not possible otherwise. The model can be reproduced by a set of Feynman diagrams, including resonance (bare) couplings to ππ and KK and (in the original model [55]) 4-point ππ and KK interaction vertices which we shall omit for simplicity in [47] and here. A subclass of bubble pion loop diagrams including resonance poles in the s-channel are resummed (unitarized Born). In this letter, we discuss the approach for the case of : 1 channel ⊕ 1 "bare" resonance (K-matrix pole) and 2 channels ⊕ 2 "bare" resonances and we restrict to the SU (3) symmetric shape function. In the present analysis, the introduction of a real analytic form factor shape function, which takes explicitly into account left-handed cut singularities for the strong interaction amplitude, allows a more flexible parametrisation of the ππ → ππ/KK data. In our low energy approach, it can be conveniently approximated by: which multiplies the scalar meson couplings to ππ/KK. In this form, the shape function allows for an Adler zero at s = s AP and a pole at σ DP > 0 simulating the left hand cut. 1 channel ⊕ 1 "bare" resonancē Let's first illustrate the method in this simple case. The unitary P P amplitude is then written as: where T P P = e iδP sin δ P /ρ P (s) with ρ P (s) = (1 − 4m 2 P /s) 1/2 ; G P = g 2 P σ,B are the bare coupling squared and : with: (θρ P )(s) = 0 below and (θρ P )(s) = ρ P (s) above threshold s = 4m 2 P . The "physical" couplings are defined from the residues, with the normalization: The amplitude near the pole s 0 where D P (s 0 ) = 0 and D P (s) ≈ D ′ P (s 0 )(s − s 0 ) is: The real part of D P is obtained from a dispersion relation with subtraction at s = 0 and one obtains: The generalization to this case is conceptually straightforward though cumbersome. Let us consider two 2body channels coupled to 2 "bare" resonances labelled a and b, with bare masses squared s Ra and s Rb : -Let f πa (s) , f πb (s) ,f Ka (s) , f Kb (s) be four shape functions, real analytic in the s-plane, with left cut, and f πaa (s),f πbb (s) ,f πab (s),f Kaa (s),f Kbb (s) ,f Kab (s), six functions, real analytic in the s-plane, with right cut. Their imaginary parts on the cut for s ≥ 4m 2 P are: Imf πbb (s + iǫ) = (θρ π f 2 πb )(s) , Imf πab (s + iǫ) = (θρ π f πa f πb )(s) , and analogous for the 2ndKK channel. -Let's define the bare inverse propagators: and the "bare" couplings g πa , g πb , g Ka , g Kb of the resonances to the channels, through the pure 1-resonance inverse propagators: -Let's define the full denominator function D(s), analytic in the s-plane, with right cut s ≥ 4m 2 π : and the partial propagators -Then is a set of unitary elastic amplitudes.
-The inelasticity η is related to the amplitudes or Smatrix as: where the sum of pion and kaon phase shifts is: -In the following, we shall work in the minimal case with one shape function: where:

Phenomenology of elastic ππ → ππ scattering Data input
The only data input used in this process is the pion phase shift δ π well measured experimentally. We shall use the new precise data from NA48/2 on Ke4 ≡ K → ππeν e for the ππ-phase shift below 390 MeV [49] and use from 400 to 900 MeV the CERN-Munich [50] and Hyams et al. [51] ππ-phase from ππ → ππ which agree each others above 400 MeV. These data are shown in Fig. 1. 0 "bare" resonance ≡ λφ 4 model Let's first fit the elastic ππ data by using a λφ 4 model without any "bare" resonance. In this old version of the model [55], one can introduce the shape function f P [47]: where σ D1 ≡ σ Dπ and: with : σ N 1 , σ N 2 in Eq. (17) are the residues of f P (s) at σ D1 ≡ σ Dπ , σ D2 . In fitting the "bare" parameters, we look for a minimum of χ 2 ≡ χ 2 min by varying the range of the interval [4m 2 π , s] inside which we perform the fit. Here, this is obtained for √ s=0.7 GeV where: χ 2 min /ndf=12.04/14=0.86. The fitted values of the "bare" parameters and the resulting values of the physical pole parameters are given in Table 1 5 . The quoted errors of the "bare" parameters come from the fit program MINUIT. The errors induced by each of these "bare" parameters on the physical poles can be added (as currently done) linearly or quadratically 6 . These results indicate that, though not accurate, this original version of the model gives a reasonnable value of the physical parameters. 1 "bare" resonancē This analysis has been done in [47] using the CGL parametrization based on Roy equations with constraints from chiral symmetry [58]. In the following, we shall use instead the new precise data from NA48/2 on Ke4 for the ππ-phase below 390 MeV [49] and use from 400 to 900 MeV the CERN-Munich [50] and Hyams et al. [51] ππ-phase from ππ → ππ which agree each others above 400 MeV. We extract the "bare" parameters from these data: -In the first step, we leave all "bare" parameters free and find a minimum χ 2 : χ 2 min /ndf=9.43/17=0.55 for √ s = 0.75 GeV. The fitted value of the Adler zero is: Values in GeV d (d = 1, 2) of the bare parameters of the K-matrix model for different "bare" resonances input for ππ → ππ elastic scattering. The fit has been performed until √ s ≃ 0.7 GeV (0 res. and 1 res.) and 0.75 GeV (2 res.), where the χ 2 /ndf is minimal. The correlated errors of the "bare" parameters come from the fit of the data using MINUIT. The ones of the physical poles are the quadratic sum of the errors induced by each "bare" parameters (a linear sum would lead to 2-3 times more accurate values due to cancellations of some of the errors in this case.). An average is given in the last column.
1res. 2 res. Average  (16) 2.58 (14) 2.64(10) Table 2 Mass and 1/2 width in MeV of the σ meson in the complex plane. Processes which is relatively bad compared with the theoretical expectation 7 : -Then in the second step, we fix the Adler zero at the value in Eq. 21 and deduce the results in Table 1.
The fit is shown in Fig. 1. These "bare" parameters lead to the physical poles in Table 1, which we consider as improvements of the previous results in [47]. This result is comparable in size and errors with the precise determinations from recent analyses of the analogous ππ → ππ/KK scatterings data using different ap-proaches (Roy equations ⊕ chiral symmetry constraints [58], Roy equations ⊕ control of the high-energy behaviour of the amplitude [59])( Table 2) 8 , which have been obtained before the last Ke4 NA48/2 precise data [49].

"bare" resonances
We repeat the previous analysis by working instead with 2 "bare" resonances. We fix the Adler zero at the value in Eq. 21 and fit the other "bare" parameters. We obtain the results quoted in Table 1 for a χ 2 min /ndf=12.71/16=0.794 at √ s = 0.75 GeV. The fit is shown in Fig. 1. Comments and final results from ππ → ππ From previous studies, we conclude that: -The results from different forms of the model in Table  1 are very stable. The final results from elastic ππ → ππ are the average of the ones from 0, 1 and 2 "bare" input resonances quoted in this Table 1, which are: (16) , -The results, from the 0 "bare" resonance or λφ 4 model show that the existence of the σ pole is not an artifact of the "bare" resonance entering in the parametrization of the ππ amplitude T P P .
Noting that the concavity of the fit curve in Fig. 1 around the ρ-meson mass region has raised some doubts on the data of the phase shift δ π [62], we have redone the fit by assuming that the data increases linearly from the Ke4 one. Using 1 or 2 resonances, we still find, in this extreme case, a σ pole : where a similar value has been obtained earlier [62]. This result may indicate that the existence and the dynamics of the σ is mainly due to the low-energy behaviour of the ππ phase shift δ π data, which are accurately determined from Ke4 by NA48/2 [49].
4. Phenomenology of inelastic ππ → ππ/KK 2 "bare" resonances ⊕ 2 channels parameters In so doing, we take in Table 3 three representatives sets of ππ → ππ/KK data in the existing literature: Table 3 Different data used for each different sets for determining the "bare" parameters in Table 4: δπis the ππ phase shift, η is the inelasticity and δπK is the sum of the π and K phases.   [52] provide the largest one: η min ≈ 0.7 [see Fig. 2 b)].

Input
-For the sum of π and K phase δ πK , we use the one from Cohen et al and from Etkin-Martin [53], which represent the two extreme cases [see Fig. 2 c)]. With these choices, we expect to span all possible re-  gions of the space of parameters, and then to extract results which do not only come from a single experiment. We have not used the data of Kaminski et al. [54] due to the large errors, which, however, agree within the errors with the other data sets used here. GeV, where the χ 2 /ndf is minimal (see Fig. 3). The correlated errors come from the fiitting procedure using the program MINUIT. In the following, we shall use: Letting all "bare" parameters free, we study in Fig. 3, using the fitting program MINUIT, the variation of χ 2 /ndf versus √ s until 1.4 GeV where the data are available. In the fitting procedure, we have chosen the same initial conditions for the 3 sets, where a good convergence with a good χ 2 /ndf of the solutions has been obtained for Set 2 and Set 3. A minimum value χ 2 min /ndf is reached for √ s ≃ (1.225 − 1.250) GeV at which we extract the optimal outputs given in Table 4. At each Table 5 σ and f0(980) meson parameters from ππ → ππ/KK scatterings using the bare parameters in Table 4 : the mass and width are in MeV, while the couplings are in GeV. The errors are the quadratic sum of the ones induced by the "bare" parameters in  (55) corresponding value of χ 2 min /ndf, the fits for different sets of data are shown in Fig. 2. All three sets give good values of χ 2 min /ndf less than one. Poles from 2 "bare" resonances ⊕ 2 channels We use the results of the "bare" parameters in Table 4 obtained at χ 2 min /ndf for deducing the ones of the complex poles in Table 5. The errors on the physical poles are induced by the ones of the "bare" parameters in Table 4 and have been added quadratically. The iteration of solutions from Set 1 has only a local minimum in χ 2 such that, in order to be more conservative, we have multiplied by a factor 2 the related uncertainties of the results coming from the fit. The last column gives the mean value from the three different determinations. We have taken (as is usual in the literature) the weighted average, where the corresponding error is more weighted by the most accurate predictions 9 . One can see in Table 5 that the results from different sets of data are unexpectedly stable for both σ and f 0 (980) parameters, which increase our confidence on their independence on the input data sets.
-For the σ, we obtain the average of the complex pole mass and width given in Table 5: This result is in perfect agreement with the mean value from elastic ππ → ππ scattering in Eq. 22 and comparable in size and errors with the ones in Table 2. Averaging the two predictions in Eqs. 22 and 25, we deduce our final value: Averaging the result in Table 5 and Eq. 22, we deduce: |g σπ + π − | = 2.65(10) GeV, which improves and confirms our previous rough findings in [48] and which is comparable with some other determinations in Table 6 from [48] 10 : the sizeable coupling of the σ toKK disfavours the usual ππ molecule and four-quark assignement of the σ, where this coupling is expected to be negligible.
-For the f 0 (980), we obtain the mean value: which is comparable with the PDG range of values [64]: From Table 5, we also find: |g f π + π − | = 1.12 (31) GeV, in agreement with the determinations in the existing literature (see Table 6). The large value of this ratio of coupling and the relative narrowness of the f 0 (980) width (compared to e.g. the ρ-meson) does not favour the pure (ūu +dd) content of the f 0 (980) where r f πK is expected to be about 1/2 and the width of about 120 MeV [6,9]. This feature has been used as an indication of the four-quark nature of the f 0 (980) (see e.g. [63]) or alternatively of its large gluonium component via a maximal mixing with aqq state (see e.g. [5,6]). Table 6 Modulus of the π + π − and K + K − complex couplings in GeV of the σ and of f0(980) from S-and K-matrix models for ππ → ππ/KK scatterings compared with the ones from φ and J/ψ decays. rSπK ≡ |g SK + K − |/|g Sπ + π − |: S ≡ σ, f .
Processes |g σπ + π − | r σπK |g f π + π − | r f πK Models This work ππ → ππ/KK 2.65(10) 0.37 (6)  Model with 1 "bare" resonance ⊕ 2 channels We have further studied the influence of some other configurations of the model on the fit of the σ by analyzing the minimal case: 1 "bare" resonance ⊕ 2 channels and using for instance Set 3 of data. Letting all "bare" parameters free, the χ 2 min /ndf ≃ 10 is very bad which is obtained by doing the fit from 2m π to √ s = 1.2 GeV.
10 Similar values of r σπK are also found from some fits in [63] but the results obtained there are unstable.
As (intuitively) expected, the previous results for the σ parameters are approximately reproduced: and |g f π + π − | ≈ 2.13 GeV, r σπK ≈ 0.42 , while the f 0 mass is pushed far away from theKK threshold: Due to the bad quality of χ 2 /ndf, the result from this version of the model will not be retained.

5.
On-shell mass, width and couplings of the σ Due to the large width of the σ, a direct comparison of the previous results with the ones obtained from QSSR or some other theoretical predictions in the real axis is questionable. For better comparing the results obtained in the complex plane with the theoretical predictions obtained in the real axis, it is more appropriate to introduce like in [47] the on-shell meson [68] masses and hadronic widths, where the amplitude is purely imaginary at the phase 90 0 : In the same way as for the mass, one can also define an "on-shell width" [47] from Eqs. (3) and (5) evaluated at s = (M os σ ) 2 : which are comparable with the Breit-Wigner mass and width [51,69,70]: These values lead to the on-shell coupling: 6. Comparison with QSSR ⊕ LET predictions -One on hand, the corresponding on-shell (or Breit-Wigner) mass and coupling of the σ are comparable in size with the predictions from combined QSSR ⊕ LET analysis [6,30,32] for a glueball with a large OZI violation for its coupling to ππ andKK 11 : implying: The existence of the σ is necessary for a consistency between the subtracted and unsubtracted QSSR [32], where the gluonium two-point correlator subtraction constant [35]: plays a crucial role (β 1 = −11/2 + n/3 for n flavours), and where the value of the gluon condensate is: α s G 2 = (6.8 ± 1.3) × 10 −2 GeV 4 [71,72,73,74].
-On the other hand, QSSR predicts for a S 2 ≡ 1/ √ 2(ūu +dd) I=0 scalar meson [6,9]: and These results indicate that: -The S 2 is narrower and much higher in mass than the complex σ pole often identified with aqq state in the current literature.
-The f 0 (980) cannot be a pure (ūu +dd) state due to the large ratio of itsKK over itsππ couplings r f πK [Eq. 30] and to its relative (compared to the ρ-meson) small ππ width. It cannot be also a puress orKK molecule due to its non-negligible width into ππ.

Summary and conclusions
We have used new Ke4 ≡ K → ππeν e on ππ phase shift below 390 MeV ⊕ different ππ → ππ/KK scatterings data above 400 MeV, for extracting the σ ≡ f 0 (600) and f 0 (980) masses, widths and hadronic couplings, within an improved "analytic K-matrix model". Ū sing a λφ 4 version of the model, we have noticed from different analysis that the existence of the σ in the complex plane and having a mass of about 452 MeV is not an artifact of "bare" resonances used in the analytic Kmatrix model. W e have also seen that our predictions are very stable vesus the different forms (number of "bare" resonances) of the models. Our results are summarized in Table 5.
T he masses and widths [ Table 2 and Eqs. 26] of the σ are comparable in size and errors with the most accurate determinations in the existing literature [58,59] (see also Table 2), while the ones of the f 0 (980) in Eq. 28 are comparable with the PDG values [64]. The small uncertainties in our determinations can be mainly due to the new accurate data on Ke4 from NA48/2. T he values of the couplings confirm and improve our previous results in [48] and are comparable with the ones from some other processes given in Table 6: -The (unexpected) sizeable coupling of the σ toKK: r σπK ≃ 0.37 (6) [Eq. 27] is a strong indication against a pureππ molecule and four-quark substructure of the σ, whilst its large width cannot be explained (using the QSSR results in a previous section) from a simple (ūu +dd) assignement.
-The large value: r f πK ≃ 2.59(1.34) [Eq. 30] of the ratio of the f 0 (980) couplings toKK over the one toππ and of the f 0 (980) relative narrow width compared e.g. with the one of the ρ-meson does not favour the pure (ūu +dd) assignement of the f 0 (980). In this scheme one would predict a ratio of coupling of about 1/2 and a width of about 120 MeV [Eq. 41]. T he four-quark scenario can explain the largeKK coupling of the f 0 (980) but it fails to explain the large coupling of the σ toKK. T he simpleqq scheme cannot explain the largeKK coupling and the narrowness of the f 0 (980) as well as the broad width of the σ. Ā large gluonium component eventually mixed with aqq state of the σ and f 0 (980) can be advocated [2,5,6,22,32,47,48] for evading these above-mentioned difficulties.