A new global fit of the $L^r_i$ at next-to-next-to-leading order in Chiral Perturbation Theory

A new fit is done to obtain numerical values for the order $p^4$ low-energy-constants $L_i^r$ in Chiral Perturbation Theory. This includes both new data and new calculated observables. We take into account masses, decay constants, $K_{\ell4}$, $\pi\pi$ and $\pi K$ scattering lengths and slopes and the slope of the pion scalar formfactor. We compare in detail where the changes w.r.t. to the 10 year old"fit 10"come from. We discuss several scenarios for estimating the order $p^6$ constants $C_i^r$ and search for possible values of them that provide a good convergence for the ChPT series. We present two such sets. One big change is that the fits do not have the expected behaviour in the limit of large $N_c$ as well as before.


Introduction
Since its very beginning Chiral Perturbation Theory (ChPT) [1,2,3], the effective field theory of QCD at low energies, has been successful in the description of several hadronic observables. Unfortunately when one tries to perform loop calculations to improve the precision of the predictions, one faces a problem. The couplings appearing in the L 4 Lagrangian are many, i.e. 10, and they must be determined from phenomenology. One of the first determinations of such couplings was done already in [3]. There most of the next-to-leading order (NLO) couplings were inferred both from phenomenology and from considerations lead by large N c estimates (where N c is the number of colours).
Considering such good results it is important to decide whether ChPT is a suitable theory to achieve precise determinations of the hadronic observables. It urged then to carry on a program and perform next-to-next-to-leading order (NNLO) calculations [4]. In the last 10 years many two-loop calculations in three-flavour ChPT have been done, see [5] for a review.
Notice however that going to higher orders raises a serious issue: the number of the unknown couplings increases rapidly. If on the one side adding loop diagrams should allow us to include better corrections and improve our descriptions, on the other side, many unknown parameters contribute and this seriously threatens the predictivity of the theory. Furthermore without knowing the values of such constants the convergence of the chiral expansions are difficult to test, although feasible with the method described in [6].
The two-loop expressions now available can be used to perform a new global fit at NNLO of some of the next-to-leading-order (NLO) low-energy-constants (LECs) into the game, the L r i . A first attempt was done [7] when some experimental information was available for the K ℓ4 decay and by estimating the NNLO contributions using dispersive analysis. The fit was refined later on, when the full NNLO calculation for this process was performed [8,9]. After that many other observables have been calculated at NNLO and many of them are also better known experimentally. Therefore the time has come to perform a new fit of the L r i couplings at a NNLO precision. In this paper we present results for such a fit. Some studies using the extra observables but without performing a full new fit were reported in [10,11,12]. Notice that in a preliminary phase we converted all the FORTRAN programs used to evaluate the amplitudes up to NNLO into C++ code. Our fits are all performed using such programs.
The paper is organized as follows. In Section 2 we sketch out the ChPT formalism and its main underlying ideas. In Section 3 we review the phenomenological input we used in our results. We also show which are the L r i that give the largest contributions for each observable we included. Notice that now much more input is present compared to the past fits [8,9]. In Section 4 we present the main model we used to estimate the C r i , i.e. the coupling constants appearing in the NNLO Lagrangian. Such an estimate is usually called the resonance estimate. In Section 5 we summarize the status of the main existing NNLO fit so far: fit 10 of [9]. In Section 6 we show our main findings using the C r i estimates of Section 4. We quote different fits so to show how each observable we include affects our findings. The best fit we get we call fit All and should be considered the main output of this work. This fit exhibits several differences with fit 10. One especially striking feature is that it does not respect any longer the large N c relation 2L r 1 ≈ L r 2 . We also show for fit All the convergence of the expansions for masses and decay constants, which is much improved compared to the one of fit 10. In Section 6.1 we perform a fit of the L r i using as input different experimental results for the K ℓ4 amplitude. We show that with this input the large N c relation 2L r 1 ≈ L r 2 is better satisfied, although the resulting fit is not as good as fit All in convergence for the masses. In Section 6.2 we try to justify and test our estimate of the C r i . In Section 6.3 we compare further the fits obtained using their predictions for the two-flavour LECsl i . Fit All again results as the most convincing one. In Section 6.4 we show results for fits based on a different estimate of the C r i couplings that can be found in [13], this is essentially a chiral quark model estimate (CQM). The fits are not as good as fit All, nor for the χ 2 nor for the convergence of the expansions. Also in this case the large N c relation is not satisfied. In Section 6.5 we provide a short comparison with the recent determination of L r 5 of [14]. In Section 7 we show results for an another treatment of the C r i . We let the C r i couplings to take random values, although they are forced to keep the size 1/(16π 2 ) 2 . These fits have been done requiring extra constraints of convergence for mass and decay constant expansions, as explained in Section 7.1. In this way it is easier to select only credible fits.
The results of such a study are finally shown in Section 7.2. We found very many good fits that correctly predict all the observables used as input and with low χ 2 . These fits are unfortunately different looking from each other. Therefore we can only show which are the ranges where we found the L r i to vary. For some of the NLO constants such ranges are very wide. This method shows however that it is possible to fit the NNLO expressions to the observables with C r i of the expected size and it also allows to study well the strong correlations between the couplings. Finally in the appendix we present a table where we quote our estimates for the NNLO couplings.

Chiral Perturbation Theory
We devote this section to a brief description of the formalism of three-flavour ChPT [1,2,3]. Introductory references are [15,16]. The notation in the following is the same as in [4]. ChPT relies on the assumption that the flavour symmetry of QCD is spontaneously broken to the diagonal subgroup, SU(3) L × SU(3) R → SU(3) V . According to the Goldstone theorem, 8 pseudo Goldstone bosons then arise. These are identified with the low lying pseudoscalar mesons and are organized in a unitary 3 × 3 matrix where φ is a hermitian 3 × 3 matrix: The Lagrangian describing the low-momentum strong interactions of the light mesons must be invariant under SU(3) L × SU(3) R local transformations. The most general lowest order Lagrangian is with The fields s, p, l µ = v µ −a µ and r µ = v µ +a µ are the standard external scalar, pseudoscalar, left-and right-handed vector fields introduced by Gasser and Leutwyler [2,3]. The constants F 0 and B 0 are instead the leading-order (LO) LECs. The notation X stands for trace over up, down and strange quark flavour.
Starting from this Lagrangian we can then build up an effective field theory by including loop diagrams and higher order Lagrangians, where operators of higher dimensions are included. Their coupling constants are then counter-terms, i.e. their infinities absorb the UV divergences coming from the loop diagrams. In this way one obtains a theory renormalized order by order. Unfortunately going to higher orders the number of operators allowed by the symmetries increase and therefore also the number of unknown coupling constants. At NLO there are 10 LECs, called L r i , while at NNLO there are as many as 94, called C r i . We will always quote the renormalized versions where the C r i are made dimensionless by using the physical value of F π . The renormalization scale is chosen to be µ = 770 MeV. While there is in principle enough phenomenological information to fit the first ones, we still need to rely on theoretical models or on some other method for the latter ones, as those described in Sections 4, 6.4 and 7.

Fitting procedure and input observables
In this section we first show how we perform the fits and then we review shortly the observables we use as input and their values.

χ 2
The fit is performed using MINUIT in its C++ version [17,18]. MINUIT is a routine to find the minimum value of a multi-parameter function. The procedure to perform the fits is very similar to the one explained in [9]. The function to be minimized is the χ 2 of the fit. It is obtained summing up the partial contributions χ 2 i(part) for each input observables where x i(meas) are the physical values for each input observables and ∆x i their associated errors. x i(calc) are the results as calculated by ChPT up to NNLO. The rest of this section is devoted to list the values and the uncertainties used for each x i .

Masses and decay constants
The masses and decay constants of the light pseudoscalar mesons have been calculated at NNLO in [19]. We use them as physical parameters, namely as input to calculate the several observables, similarly to what was done in [9]. Their values are given in [20] and are m π + = 139.57018 MeV, m π 0 = 134.9766 MeV, m η = 547.853 MeV, The measurements in (5) and (6) differ slightly from the ones used in the latest full fit [9].
Notice that preliminary results of our work have been reported in the proceedings [21,22] and several unpublished talks. Those results were based on the masses of [23], that differ slightly from both (5), (6) and (7), (8).
All the new fits shown in this paper have been produced using the values in (5) and (6). However we have not observed any substantial modification of the outputs when using the old masses (7) and pion decay constant (8). As discussed below other changes in experimental input are behind the changes of central values.
The masses depend at LO on B 0m (withm = (m u + m d )/2) and on B 0 m s . L r 4 , L r 6 , L r 5 and L r 8 appear in the expression at NLO. In m 2 η there is also a NLO contribution from L r 7 . The decay constant F π depends instead on F 0 , as an overall factor, and on L r 4 and L r 5 .
3.3 F K /F π As input observable for our fits we will use the ratio F K /F π to eliminate the dependence on the unknown constant F 0 , since it contributes as an overall factor for F K as well. In the end the value of F π then determines the value for F 0 . The ratio takes the value [20] F K F π = 1.197 ± 0.007, that is also in full agreement with several lattice estimates as reported in [24]. Using F K /F π at NLO we are sensitive to L r 5 . To perform the fit we expand the ratio as so that we can keep track of the exact contributions from the different orders. However we also check that the same quantity estimated as gives approximately the same value 1 . Notice the experimental result for the ratio F K /F π (9) differs substantially from the one used in [9]. Ref. [9] used F K /F π = 1.22 ± 0.01. The change in F K /F π is one of the major sources of difference with [9] as will be shown later in Section 6.

The quark-mass ratio m s /m
For the masses we have a similar problem, they depend on the quark masses and on B 0 . We thus use as was done in [8,9] the ratio of the strange quark mass over the isospin doublet quark massm as input observable. The two following relations involving the light pseudoscalar meson masses hold at LO in ChPT where with m 0 we indicate the meson masses at LO. They are calculated subtracting from the physical values the NLO and NNLO expressions. We include both relations in (12) in the fits. For the pion mass we use the neutral pion mass m π 0 . In the kaon case we need to correct the physical value for the mass since its electromagnetic contribution is sizeable. We take the average between m K + and m K 0 and then we subtract the electromagnetic contribution as stated by the Dashen's theorem and an estimate of its violation: The factor 1.8 in (13) is due to the corrections to Dashen's theorem where we use the value of [25]. The value of the quark mass ratio has been calculated by several lattice collaborations. The authors of [9] used as standard input m s /m = 24 with a 10% uncertainty, but they also checked that m s /m = 26 was compatible. Here instead we use m s /m = 27.8 as obtained by the Flavianet Lattice Averaging Group in [24], and we again adopt a 10% uncertainty for the error to be used in the fits when comparing with the theoretical values (12).
In the end the calculated NLO and NNLO masses are used to determine the lowest order mass or alternatively B 0m .

K ℓ4 formfactors
The decay K + (p) → π + (p 1 )π − (p 2 )e + (p ℓ )ν(p ν ) is given by the amplitude [7] In (14) V µ and A µ can be parametrized in terms of four formfactors: F , G, H and R. However the R-formfactor is negligible in decays with an electron in the final state. Using partial wave expansion and neglecting d wave terms one obtains for the F , G and the H formfactors [26]: F (s π , s ℓ , cos θ) = f s (s π , s ℓ )e iδs + f p e iδp cos θ + . . . , G(s π , s ℓ , cos θ) = g p (s π , s ℓ )e iδp + . . . , H(s π , s ℓ , cos θ) = h p (s π , s ℓ )e iδp + . . . , where we also assumed that the p phase is the same for the three formfactors. In (15) s π (s ℓ ) is the invariant mass of dipion (dilepton) system, θ is the angle of the pion in their rest frame w.r.t. the kaon momentum. The F and G formfactors were calculated at NNLO in [8]. The quantities are especially sensitive to L r 1 , L r 2 and L r 3 . Also the H formfactor is known at order p 6 [27] but we do not use it as input observable since it depends on a different set of LECs, those from the anomalous intrinsic parity sector.
Now more precise results from the NA48/2 experiment are available [29] and their second order fit of the formfactors read Notice that in [29] no measure of f s is reported, therefore we always use the f s = 5.75±0.097 from the E865 collaboration [30] in our fits. Multiplying by f s and combining the errors in quadrature we obtain as measures In [29] there is also a linear fit which gives f ′ s fs = 0.073±0.004 and thus f ′ s = 0.420±0.024. Deciding which of the two fits for the F s formfactor should be used is a relevant issue. The problem is how much we can rely on the curvature of the formfactor F s . As a matter of fact it is difficult for NNLO ChPT to reproduce the large negative curvature f ′′ s , as was also noted in [9,6]. A dispersive analysis approach combined to two loops ChPT, similar to the one done for ππ scattering [31], might clarify the situation. An indication of this is given by Figure 7 of [9]. It is visible there that the dispersive result for K ℓ4 decay [7] has a larger curvature than the two-loop result [9].
Let us conclude with a cautionary remark about the K ℓ4 data. In [32] it was made clear that isospin breaking effects at threshold give important corrections. These have not been taken into account in the NA48/2 analysis [29], thus they might affect significantly their findings.

ππ scattering
The ππ scattering amplitude can be written as a function A(s, t, u) which is symmetric in t, u: where s, t, u are the usual Mandelstam variables. The three flavour ChPT calculation of A(s, t, u) was done in [11]. The isospin amplitudes T I (s, t) (I = 0, 1, 2) are T 0 (s, t) = 3A(s, t, u) + A(t, u, s) + A(u, s, t) , and are expanded in partial waves where t and u have been written as t = − 1 2 (s −4m 2 π )(1 −cos θ), u = − 1 2 (s −4m 2 π )(1 + cos θ). In (22) we indicate with P ℓ (cos θ) the Legendre polynomials. Near threshold the t I ℓ are further expanded in terms of the threshold parameters t I ℓ (s) = q 2ℓ (a I ℓ + b I ℓ q 2 + O(q 4 )),  Table 1: The values of the scattering lengths and slopes as found in [31] and our fitting uncertainties. In the third column the normalization factors are given. We quote here only those scattering parameters added as input in our fits.
where a I ℓ , b I ℓ . . . are the scattering lengths, slopes,. . .. These thresholds parameters constitute our observables. Currently a very precise determination of these parameters exists. It is based on a dispersive analysis approach and on two-flavour ChPT and can be found in [31]. In Table 1 we quote the values of the threshold parameters we use in our fits and their corresponding uncertainties, which we took to be double the ones in [31]. For most fits we used only a 0 0 and a 2 0 but we have checked that the others listed in Table 1 are also well within the uncertainties quoted. Notice also that the NA48/2 experiment in [29] obtained compatible values for a 0 0 and a 2 0 from the measurement of the δ = δ p − δ s phase shift in K ℓ4 decays.

πK scattering
The πK scattering process has amplitudes T I (s, t, u) in the isospin channels I = 1/2, 3/2. They have been calculated at NNLO in ChPT in [12]. As for ππ scattering, it is possible to define scattering lengths and slopes a I ℓ , b I ℓ . So we introduce the partial wave expansion of the isospin amplitudes where P ℓ (cos θ) are the Legendre polynomials. Then we expand the t I ℓ (s) near threshold where Table 2: The values of the scattering lengths and slopes as found in [33]. The uncertainties quoted here are those used in our fits and are double the ones of [33]. In the third column the normalization factors are given. We quote only those scattering parameters added as input in our fits.
is the magnitude of the three-momentum in the center of mass system. The Mandelstam variables are given in terms of the scattering angle θ by (25) defines the πK scattering parameters a I ℓ and b I ℓ that are our input observables. These have been computed from Roy and Steiner type equations in [33]. The results for the s and p waves scattering parameters we use are reported in Table 2. For most numerical results we used only a 1/2 0 and a 3/2 0 but we have checked that all the others agree within uncertainties.

Scalar formfactor
The scalar formfactor for the pion is defined as The observables r 2 π S and c π S are used as input in our fits. The NNLO ChPT calculation for these quantities was performed in [10]. The scalar formfactor cannot be measured experimentally. Measuring the ππ phase shifts and using a dispersive representation it is possible to infer its energy behaviour and therefore the values of r 2 π S and c π S [34,35,36] Notice that the result for r 2 π S is also compatible with the lattice result of [37]. This is all the information we can extract from the scalar formfactors. Currently, there are basically no results available for F π s (0) and for the energy behaviour of the kaon scalar formfactors or of the strange contribution to the pion formfactor.
3.9 L r 9 and L r 10 We do not attempt to fit the remaining NLO LECs, L r 9 and L r 10 . Those LECs we fit are independent of L r 9 and L r 10 , or alternatively, none of the observables 2 we discuss depend on them. One needs to include additional information to constrain their values.
L r 9 appears alone at NLO in the electromagnetic radius of the pion vector formfactor. The NNLO contribution dependent on the other L r i is rather small [38]. It was therefore possible to fit that constant almost independently from the other couplings [38]. Furthermore it never appears at NLO in any of the observables used here as input thus it does not affect much our fits. We always set L r 9 = (0.593 ± 0.43) × 10 −2 for µ = 0.77 GeV. L r 10 can be estimated using τ decays data on the V − A spectral function [39]. Its value was found to be L r 10 = (−4.06 ± 0.39) × 10 −3 at µ = 0.77 GeV. However this constant never appears in the observables under study not even at NNLO. Therefore it does not have any influence on our fits. For this reason we always set such constant to zero in our fits.

Resonance estimates for the C r i
The many unknown coupling constants that appear in the p 6 Lagrangian, the C r i , represent the major problem for performing the fit with a O(p 6 ) precision. A lot of effort went into trying to estimate them using different models and treatments. The one we present here, also used in [8], is the resonance saturation model [40,41]. It is based on the idea that the LECs encode the information from physics above Λ ChPT ≈ 1 GeV, and that they are dominated by the physics just above this scale, i.e. the physics of low-lying resonances. Therefore we need a Lagrangian that describes these new particles and their interactions with the pseudoscalar mesons of the octet. We include only vector, scalar and the η ′ fields. We use the same estimate described in [8], thus we refer the reader to that paper for further details, including the Lagrangians used at the resonance level.
The model is used then to estimate the p 6 contributions depending on the C r i . In [8], the heavier mesons were integrated out producing p 6 Lagrangians for the pseudo-Goldstone boson. The heavy resonance fields for the vector mesons produce and the scalar mesons While for the η ′ they obtained In (31) and (32) In [8] the above Lagrangians were not rewritten in the standard form of the Lagrangian at p 6 . That work has since been done using more general resonance lagrangians in [42,43]. We have checked that the results using the Lagrangians (31,32) directly agrees with the same inputs using the C r i directly in terms of resonance parameters as derived in [42,43]. The η ′ contribution was rewritten in the C r i in [44]. The values we choose for the different couplings are the same as in [8] and the masses are the experimental ones [20].
In Table 18 in the appendix we quote the C r i as estimated through the resonance model. We did not include more sophisticated resonance models because this would have again increased strongly the number of free parameters to be fitted. As discussed below we also have indications that terms suppressed by 1/N c , N c the number of colours, might be important. These cannot at present be estimated using this type of approach.

Existing fits
In this section we describe a bit more in detail the earlier fits. The main full fit done is fit 10 in [9] 3 . Earlier determination of the L r i did not fully include NNLO effects and we thus do not discuss them here. The values for the L r i obtained in fit 10 are reproduced in Table 3 in the column labelled fit 10. This is a full NNLO fit of the L r i and it was done including the quantities and the L r i whose value they influence most: fit 10 fit 10 iso The results for fit10 of [8] and for a similar fit done without including isospin breaking corrections for the masses (fit10 iso) and also using the masses in (5) and decay constant F π as in (6). The uncertainties are those calculated by MINUIT. The two fits reported are in agreement within uncertainties. Table 4: The convergence of the expansion for the meson masses and the decay constants for fit 10 iso. A similar behaviour holds for fit10. The masses quoted are normalized to the physical masses, while the decay constants to F 0 (F 0 = 0.0869 GeV). (7) and (8) 2. the K ℓ4 formfactor parameters:

masses and pion decay constant with the old values as in
4. m s /m = 24 constrains L r 7 , L r 8 via the masses in (12).
The C r i contributions were estimated using resonance saturation as described in Section 4. They also included there the axial-vector resonances, although their contribution was rather small. The scale of saturation was set to µ ≡ 0.77 GeV, but µ = 0.5, 1 GeV were within errors. In fit 10 isospin breaking corrections in the masses and decay constants were also included, though the authors of [9] noticed that the neglect of isospin violation was a good approximation. Indeed the fits performed including or not these effects are in agreement within errors as can be seen from Table 3 comparing the columns fit 10 and fit 10 iso.
Fit 10 has been so far a quite successful fit. Not only because it already included many quantities at order p 6 , but also because the resulting L r i nicely confirmed the estimates from resonance models. These are lead by the large N c expansion which predicts e.g. 2L r 1 ≈ L r 2 and L r 4 ≈ L r 6 ≈ 0. While the second relation was imposed, the first one was found to be well satisfied. This added credibility to the fit itself even though it relied on the resonance estimate for the tree-level p 6 contributions.
However the convergence of the perturbative expansion for this fit is not as expected. The different orders for the masses and decay constants are reported in Table 4 for fit 10 iso 4 . The O(p 4 ) order of the masses turns out to be tiny, far less than the expected 30%. On the other hand the NNLO contribution is definitely too large. The sources of this bad convergence are basically two. First the constraint L r 4 ≡ L r 6 ≡ 0 that clearly sends to zero many contributions coming from the NLO tree-level diagrams. Secondly most of the C r i appearing in the masses expressions are estimated to be zero as well. Therefore they cannot help in canceling large two-loop contributions. On the other hand the convergence for the decay constants is quite satisfying.
After fit 10 was performed many other observables have been calculated at O(p 6 ) in SU(3) ChPT such as the ππ and πK scattering threshold parameters. Of course it is very important to compare the pure ChPT predictions obtained using fit 10 with the values of Tables 1 and 2. These comparisons have been done and can be found in Table 1 of [11] and in Table 4 of [12]. Fit 10 is mostly in agreement within errors, although there are small discrepancies in some of the threshold parameters. Some comparison with scalar formfactors was done in [10]. The last three papers used the same inputs as fit 10 and tried to vary L r 4 and L r 6 to see if some preferred regions could be found. Here we redo the fit from the beginning with all inputs. fit 10 iso

New Fits
The aim of this section is to show how the new measurements and observables included in our global fits, change the results compared to fit 10. We have rewritten as mentioned above all programs into C++ and are using the isospin symmetric versions of the calculations. We therefore first redid the fit using the same inputs as fit 10. The outputs are given in Table 3 in the column labelled fit 10 iso. This also shows that the minor changes in masses and F π as well the isospin breaking corrections do not affect the fit values appreciably. We will now add the effects of the changed experimental inputs and of the additional inputs to see how they change the fitted values of the L r i . In Table 5 we present several fits. These have all been performed using the resonance estimate of the C r i of Section 4 and Table 18 in the appendix, setting the scale of saturation µ = 0.77 GeV. Furthermore, we used the new values of the masses and decay constant of (5) and (6). We remind the reader that the use of these new parameter-values affects the output only within the uncertainties. Hereafter we summarize the steps in which we have included the new information.
NA48/2 The input observables and their values are the same as for fit 10 iso, but we use the new measurements in (19) for the K ℓ4 decay from the NA48/2 collaboration [29].
The new measurements lead immediately to a striking feature: the large N c relation 2L r 1 ≈ L r 2 does not hold any longer. It even turns out that L r 2 L r 1 . Notice that, as explained in Section 3.5, the slope f ′ s comes from a second order fit of the f s formfactor and therefore it differs from the one used in fit 10. In Section 6.1 we will present also results for the linear fit of the f s formfactor.
F K /F π Same as fit NA48/2 but with the new value in (9) for F K /F π . L r 5 is mainly affected and becomes smaller than in fit 10. As a consequence also the convergence of the decay constants expansion is worsened, e.g. F π /F 0 | p 4 ≈ 0.134 while F π /F 0 | p 6 ≈ −0.126.
All⋆ In this fit we include a few more observables, i.e. the ππ scattering parameters a 0 0 and a 0 2 , the πK scattering parameters a 1 2 0 and a 3 2 0 and the pion scalar radius r 2 S . We also release the constraint L r 4 = L r 6 = 0 because now we have many more observables included. Unfortunately since none of the new observables involve information on the masses (and therefore on those two couplings), we still cannot achieve the precise values of L r 4 and L r 6 . They are highly correlated and MINUIT gives a very large uncertainty for those. Furthermore since now the O(p 4 ) contributions to the masses due to L r 4 and L r 6 are not zero, L r 5 and L r 8 diminish.
All This fit is very similar to fit All⋆. Here we adopt the new value for the quark mass ratio m s /m = 27.8. However we find that using values for the quark mass ratio between 27 and 29 does not change the results considerably. The constants L r 7 , that appears in the η mass, and L r 8 are strongly affected by this change. This is also relatively true for L r 4 and L r 6 . We also tried to perform the same fit but setting L r 4 ≡ L r 6 ≡ 0. The resulting fit is very similar to fit NA48/2 but it has a huge χ 2 (χ 2 = 45).
C r i ≡ 0 In the last column of the table we quote the fit obtained including the same input as for fit All, but setting all the C r i ≡ 0. This fit has been done to show how the different C r i can affect the L r i fit. Notice that the constants L r 1 , L r 2 and L r 3 change a lot, while the others stay in the same area as in fit All. This is not surprising: the last few constants are indeed primarly fitted from quantities where many contributing C r i are large-N c suppressed and those which are not are set to zero also in the simple resonance estimate used.
All p 4 Same fit as All but all expressions are now at NLO. Use this fit for one-loop ChPT results. Note that this produces very high values for L r 4 and L r 6 . The underlying reason is that the lower value of F K /F π requires a smaller L r 5 than before and the pion scalar radius then requires at this order a larger L r 4 . This effect is also visible in fit All but is reduced when including the NNLO corrections.
Fit All is what we consider as the present best fit for NNLO ChPT calculations, it thus superseeds fit 10 of [9].
Let us discuss how the ChPT expansion is affected by the new values for the L r i . The various terms of the mass expansions read for fit All  Table 6: The results for f + (0) in the two K ℓ3 decays. This is an update of Table 3 in [45] while those for the decay constants are In (36) and (37) we used the same normalizations as in Table 4, although now F 0 = 0.065 GeV, this is due to the larger value of L r 4 which comes however with a large error. Notice that the convergence of the mass expansions in (36) is improved compared to the one of fit 10 in Table 4. However (36) looks strange: the LO masses are larger than the physical ones and there are significant cancellations between NLO and NNLO. Furthermore, even if the convergence is improved, it is still quite different from the one expected. E.g. the m 2 π | p 4 contribution is much smaller than the expected 30% and it is of the same size as the p 6 order. The convergence for the decay constants is a bit worsened compared to the one of fit 10, due to the low value of L r 5 , but it is still acceptable. Notice also that when the ratio F K /F π is calculated with (11) the resulting value is 1.168, which is 3% smaller than the expected 1.197 . This can be due to higher order corrections that are included in the ratio of (11), but not in (10).
We performed more fits than those quoted in Table 5. We included more ππ scattering parameters and πK scattering parameters. We found that these fits are compatible with fit All of Table 5 within uncertainties. The same is true when we add the quantity c π s . For completeness we quote the value for f + (0), the formfactor of K ℓ3 decay at zero momentum transfer. This quantity was calculated at NNLO in [45]. We check how much the new L r i of fit All would affect its value. Notice that at zero momentum the dependence on L r 9 drops out. The other L r i appear only at NNLO. We discuss both the case of the charged kaon, the K + ℓ3 decay, and the neutral one K 0 ℓ3 . The results we obtain using the masses of the particles involved in the decay are given in Table 6. These numbers can be seen as an update of those in Table 3 of [45]. The numbers for the pure loops are changed w.r.t. [45] mainly because of the change in F π . Fit 10 iso is essentially the same as fit 10 used in [45] but has a small (1%) difference. The L r i -dependent contribution changes but since it stays small the total result is very similar to [45]. The total value for f + (0) for K + ℓ3 is instead (38) and for f + (0) for K 0 ℓ3 is instead By comparing the two results in (38) and in (39) one realizes that the differences due to the L r i have a very small effect on f + (0). The main uncertainty still remains the value of the contribution of the C i . The estimate for the relevant constants used in this paper leads to f + (0)| C i ≈ −0.045 but the fitting inputs used here do not strongly constrain the relevant combination.

Linear fit for K ℓ4 decays
One of the most striking features of the results presented in Table 5 is that as soon as the new results from the quadratic fit of the NA48/2 collaboration [29] are included, the constants L r 1 and L r 2 take unexpected values. Indeed, as was noted above, they do not respect any longer the large-N c relation 2L r 1 ≈ L r 2 , but already in fit NA48/2 they are L r 1 ≈ L r 2 while when also the ππ and the πK scattering lengths are included (fit All) we even obtain L r 1 > L r 2 . On the other hand when we calculate the curvature f ′′ s using the L r i as obtained in fit All we obtain f ′′ s = −0.124 to be contrasted with the experimental value f ′′ s = −0.437. Furthermore, whenever we include as input also f ′′ s we again obtain fits compatible to the ones in Table 5, but with much larger χ 2 (e.g. χ 2 ∼ 35 for fit All) and the largest contribution comes exactly from f ′′ s . These results confirm what was already stated at the end of Section 3.5, i.e. the state-of-art ChPT does not reproduce such a large negative bend. Since f ′ s and f ′′ s are highly correlated, the linear and the quadratic fit of the F s formfactor present rather different slopes.
For such reasons we perform fits using the slope of the linear fit f ′ s /f s = 0.073 [29] as well. The resulting fit, analogous to fit All, is reported in Table 7. By inspection one can see that the large N c relation 2L r 1 = L r 2 still does not precisely hold, but at least 1.4L r 1 ≈ L r 2 . On the other hand L r 4 and L r 6 are again not suppressed, while L r 7 and L r 8 are unexpectedly small. Also all the constants have large uncertainties.
The convergence of the chiral expansions is worse than the one for fit All. The various terms of the mass expansions read while those for the decay constants are In the light of these results it is rather difficult to draw a conclusion. The very different predictions for L r 1 and L r 2 obtained in fit All and this fit confirm that the picture of ChPT for K ℓ4 decays is still incomplete. As mentioned earlier, we expect a dispersive analysis to produce a larger curvature.

Some small variations on fit All
In the resonance estimate described in Section 4 there is at least an assumption not entirely justified. We assume the scale at which the saturation happens to be 0.77 GeV, i.e. the mass of the lowest lying resonance. Nothing prevents us to choose a larger or smaller scale, although this is still expected to be in the same range of energy. To check whether this assumption is safe we try to fit from data the saturation scale parameter as well. The results are rather reassuring. Fit All of Table 5 is completely unaffected by this procedure. The fitted saturation scale is 0.77 ± 0.45 GeV.
The fit in Table 7 shows a little difference. The fitted saturation scale is now 0.71 ±0.31 GeV. However the L r i do not change that much and the look of the fit is pretty much the same as before.
We also attempt to find better estimates of the C r i constants releasing the values of the couplings g V , c d and c m . Again we try to fit them using both the input of fit All and of the fit in Table 7 (linear fit of NA48/2 instead of quadratic). In the first case we find in fact g V and c m close to the ones in (34). They read g V = 0.097±0.123 and c m = 0.045±0.049 GeV. For c d we find instead a value larger than expected, i.e. c d = 0.093 ± 0.100 GeV. Anyway they are all affected by large uncertainties. The L r i fit is somewhat compatible with the one fit 10 iso All All linear [31,24] Table 8: The values of the scale-independent SU(2) LECsl i . In the first three columns we show the values as predicted by fit 10, fit All and fit All linear using the NNLO matching conditions of [46]. The numbers between parenthesis are the NLO results. In the last column we quote the known values from [31,24]. Notice that the uncertainty overl 4 is double the one quoted in [31] due to the still unclear situation for the lattice results [24].
of Table 5, fit All, within uncertainities because of the large ranges allowed for g V , c m , c d . The fits are in a very broad minimum here with only one degree of freedom. If we apply the same procedure but with the same K ℓ4 input as for Table 7 (NA48/2 linear fit) we arrive to similar conclusions: the values of g V and c d are similar to the ones in (34), while c d is larger. Again all the resonance couplings present large uncertainties. The L r i are here rather well compatible with those in Table 7. We also try to multiply the C r i by an overall constant α and include it as a fitting parameter. It is encouraging to see that the result is α ≈ 1.03, namely the best fit is reached with basically the same values of the C r i from resonance estimate. Obviously the fit obtained is very similar to fit All. When we apply the same procedure to the fit in Table 7 the constant α takes the value 0.90. This affects the fit of Table 7, but still within uncertanties.
From this we conclude that fit All is stable against small changes in the resonance estimate of the C r i .

Adding input:l i constants in two-flavour ChPT
The authors of [46,47] study three flavour ChPT in the limit where the m s is assumed to be much larger thanm and the external momenta. In this case they can integrate out the strange quarks and SU(3) × SU(3) ChPT reduces to SU(2) × SU(2) ChPT. Matching the results from the two frameworks they calculate explicitely the dependence of the twoflavour LECs (the scale-independentl i and the c r i ) on the strange quark mass and on the three-flavour LECs. These relations have been worked out using two different methods at order p 6 in [46,47].
There exist different evaluations of thel i .l 3 has been estimated rather well using lattice results [24].l 1 ,l 2 ,l 4 andl 6 have been obtained by matching two-flavour ChPT with dispersive results [31], but (contradictory) lattice results exist for those too [24]. We have increased the error onl 4 because of this.
In Table 8 we summarize all the values of these constants and the results obtained by plugging the L r i and C r i of fit 10 iso, fit All and fit All linear in both the NNLO and NLO relations of [46,47]. By comparison of the first three columns with the last one of Table 8 it is easy to see that none of the fits correctly reproduces all thel i values. A similar conclusion holds also for the fit in the next-to-last column of Table 5, where all the C r i are zero. The disagreement with thel i in this case is actually even stronger. These fits encounter particular trouble in fittingl 2 .
We tried to fit thel i whose values appear on the last column of Table 8 in addition to the inputs used for fit All. Not surprisingly it is not possible to accommodate all those inputs at the same time. The resulting χ 2 is approximately 22 and its largest contributions come exactly from thel i . Excluding from the fitl 2 but including the others improves the situation. The resulting L r i values are very close to the ones of fit All, the most important deviation being 10 3 L r 3 = −3.18. The χ 2 takes the value 3.15 with 7 degrees of freedom. Also in this case the value forl 2 is still far from the expected one. This is not surprising. In [46] it was found that the constantl 2 depends on the couplings L r 2 , L r 3 and on the combination 2C r 13 − C r 11 . The authors of [46] observed there that to find agreement with the determined value ofl 2 the combination of C r i must not be zero. Unfortunately those are two large-N c suppressed couplings and therefore they are set to zero in our resonance estimate (see Table 18). We also try to fit those two C r i using alsō ℓ 2 as input observable, but this has been unsuccessful as well. In this way we manage to accommodate the value forl 2 , but thenl 1 is off, since it contains also a different combination of C r 11 , C r 13 and of C r 6 . This last coupling is also N c -suppressed and thus estimated to be zero. In the end there is no way out: when we try to fit C r 6 too, there are other quantities taking very different values. The C r i are too correlated to be able to fit only a few of them.
As far as regards the fit in Table 7 the results for thel i are even less clear. Its predictions are reported in the third column of Table 8. It is straightforward to see that now even the predictedl 1 is off. Of course when we try to fit all thel i the χ 2 is very large (χ 2 ≈ 37.7). Contrary to what happened for fit All, the situation does not improve that much when we excludel 2 . For this fit seems to be very hard to reach the correct value forl 1 too. The resulting χ 2 in this second case is 5.82 withl 1 = −1.4.
Finally an extra cautionary remark. Requiring that the SU(3) ChPT constants predict the values for the SU(2) ones might not be a very safe assumption. What it assumes is that both SU(2) and SU(3) ChPT work well for the same quantities. For the ππ scattering quantities, which are very much determined by loop parts, relatively small differences can become amplified in the resulting values of the LECs.

A chiral quark model estimate for the C r i
We discussed above a simple resonance saturation estimate for the NNLO LECs C r i . There are other attempts at predicting these values as well from chiral quark models. As a representative of this we choose [13]. It also is a large N c approximation but with a somewhat different pattern than our resonance saturation. Their method is based on a study of the relation between the chiral Lagrangian up to order p 6 and QCD, they find as expected that the LECs can be given in terms of some Green functions of QCD. In  Table 9: The results as obtained using the C r i estimates of [13]. Both the fits include the same observables as fit All of Table 5. In the second column the coefficient α multiplied by the C r i has been included as fitting parameter. The L r i are given at µ = 0.77 GeV.  Table 10: The results as obtained using the C r i estimates of [13] showing the convergence for the fit where α is left free. The normalizations are the same as explained in Table 4. Now F 0 = 0.065 GeV.
the evaluation of these Green functions, several assumptions and approximations are made such that it is not a full derivation but something like a chiral quark model. Their results are presented in Table IV of [13].
We also use their estimate to perform the fits. The results can be found in Table 9. There are results for two different fits. They have been obtained including all the observables as for fit All of Table 5. In the first column we use the C r i as quoted in Table IV of [13], whereas in the second column we multiply them by an overall constant α that is also fitted. This second fit was done because we observed that the values for the C r i of [13] are somewhat larger than the ones of the resonance estimates of Table 18. The fit confirms this observation and finds as best value for α = 0.27, i.e. C r i considerably smaller than the ones in [13]. The value of the χ 2 for the two fits of Table 9 is somewhat worse than for fit All. Indeed it seems that it is now very difficult to fit the slope g ′ p of the G formfactor for K ℓ4 decay. From the second fit of Table 9 one can notice that when the C r i are allowed to C r i [13] α × C r i [13]   The results as obtained using the C r i estimates of [13]. Both the fits include the same observables as the fit in Table 7, i.e. as fit All but with the linear fit from NA48/2. In the second column the coefficient α multiplied by the C r i has been included as fitting parameter. The L r i are given at µ = 0.77 GeV.  Table 12: The convergence for the fit where α is left free is shown. The normalizations are the same as explained in Table 4. Now F 0 = 0.062 GeV. Fit as in rightmost column of Table 11.
take smaller values the L r 2 constant compensates for that. This allows the fit to reach a better value for the g ′ p . As can be seen in Table 10, even the convergence for the masses and decay constants is worse than the one for fit All reported in (36) and (37) respectively. Notice that we have not quoted the convergence for the fit obtained without multiplying the C r i by the coefficient α. In fact this is found to be even worse than the one of Table 9, the p 4 terms being constantly larger than, although comparable in size to, the p 6 ones.
In Table 11 and 12 we quote the results obtained with the C r i of [13], but fitting the slope of the linear fit for the F s formfactor as in the fit of Table 7. Conclusions similar to the ones drawn for Table 9 hold here too. [14] fit All best reso best rand 10 3 L r 1.10 ± 0.14 0 0.02 2.01 Table 13: The values of the couplings L r 5 , C r 14 + C r 15 and C r 15 + 2C r 17 as obtained from a fit to lattice data [14] (second column) and from our fits. Notice that the resonance model used for fit All estimates the combinations of C r i occurring here as zero. The columns best reso and best rand are taken from Tables 16 and 18 and are discussed in Section 7. All the values are at the scale µ = 0.77 GeV.

Comparison with a recent L r 5 determination
The authors of [14] propose a simplification of the NNLO predictions of ChPT to perform fits to lattice data points. As an example they fit the lattice results [48] for F K /F π with their approximation to the two-loop ChPT prediction. Since too many couplings appear at NNLO they are forced to fit only a few of them and fix the others. They decide to fit L r 5 and the only two C r i -combinations contributing: C r 14 + C r 15 and C r 15 + 2C r 17 . C r 15 is 1/N c suppressed and therefore set to 0 in the resonance saturation model. C r 14 and C r 17 also do not get contributions from the resonances we included. In other models [14] C r 14 and C r 17 are estimated to be negative, but are very small in absolute value. In [14] the other L r i , appearing at NNLO in F K /F π , are set to the values of fit 10 [9]. The results of the fit to lattice data are quoted in the first column of Table 13. For comparison we quote in the same table the values as obtained by our best fits, including those for the random C r i search described in Section 7 and quoted in Table 18 in the appendix. The table shows that the value of L r 5 in fit All is compatible with the result of [14] within uncertanties and even more compatible when we look at the fit F K /F π in Table 5 where we required L r 4 = 0 as in [14]. The fits best reso and best rand instead are very different. For these last two fits we can also compare the values of the C r i -combinations. As far as regards the combination C r 14 + C r 15 our random sets acquire both a negative value in contrast with [14]. The second combination instead agrees upon the sign, but are rather different in value.
For further comparison we can do also a fit similar to fit All but setting in addition 10 5 C r 14 = 0.31 and 10 5 C r 17 = 0.55. The results are given in Table 14. The fit obtained is different from fit All, in particular we find an even lower L r 5 and the χ 2 is somewhat larger. The value of L r 5 has decreased quite a bit while the others are compatible with fit All but the central values for L r 4 and L r 6 are rather large. Finally notice that the L r i of fit All are very different from the ones of fit 10 and might therefore affect the findings of [14]. We cannot draw any further conclusions at present. In future work we intend to include more lattice results which should clarify this issue.  The fit as obtained using the same input as fit All and the C r i as in the resonance estimate of Table 18 with the exception of C r 14 and C r 17 set respectively to 0.31 × 10 −5 and 0.55 × 10 −5 . The L r i are given at µ = 0.77 GeV.
7 Releasing the C r i All the fits presented in the previous section have unusual NNLO corrections to the masses and many also to the decay constants. In addition, if we included the requirement that thel i were also fitted well we could not find a simple good fit. An additional reason to go beyond what we have is that all the estimates used above with the exception of the singlet η contribution only contribute to the NNLO LECs 5 that are leading in N c . In the masses and decay constants in addition the estimates from the resonance exchange give no contribution at leading order in N c at all. This is an unsatisfying situation, we do expect that the masses and decay constants should get some contribution from the NNLO constants. Inspection of the relations between thel i and the SU(3) LECs [46] shows that only combinations of the C r i appear that are suppressed by N c . Thus especially the problem withl 2 above requires some nonzero values for the N c suppressed constants.
We could in principle allow all the C r i to be free and include them in the fit as well. However, from our earlier work in [6] it is clear that with the inputs used at present there are enough free combinations of the C r i to fit all physical inputs directly. For this reason we also have explored another technique of C r i estimate, based on a random walk 6 method. Hereafter we describe the main features of the algorithm used. See also the flowchart in Figure 1. The algorithm is a version of simulated annealing.
We first start with an initial set C , as explained in the text. In the bottom decision square R is a random number selected with a uniform distribution in the interval (0,1), while T is a parameter set such that it is of the same order of magnitude of the χ 2 . More details can be found in the text.
2. all zero 3. as obtained by resonance estimate (see Table 18) 4. as obtained by multiplying the constants of [13] by 0.27 (see Table 18).
Then we perform the fit on the L r i using those C r(old) i . After this we take a random step according to the formula where r i in (42) is a random number generated through a uniform distribution in the interval (−1, 1) and we have used ǫ = 0.01 and 0.001. For those C r i that are N c suppressed we further multiply C r(step) i by 1/3. In this way the random generated C r i set still respects the large-N c suppressions to a certain extent. We perform the fit of the L r i using the new set C r(new) i and we check the χ 2 obtained. If the χ 2 decreases then we substitute the C . Sometimes we also allow C r(new) i to be selected even though the corresponding χ 2 is not smaller than the previous one (see last step in the flow diagram of Figure 1). This is done so to let the C r i to take quite different values and thus to test as many different sets as possible. It is also needed for our algorithm to be able to move out of local minima. We find that, when we let our algorithm run long enough, we cover quite many different sets of C r i . We chose different initial C r i to widen their range of variability. In addition we have started the random walk from the same starting point several times including different random starting points. We chose as random starting points 1/3/(16π 2 ) 2 with the extra factor of 1/3 since without the extra 1/3 we never reached a χ 2 smaller than one.
The fits are performed including the same input as for fit All but with a few extra requirements. We add as input the curvature of the scalar formfactor c π S in (30) and all thel i of the last column of Table 8. We do not instead demand to fit all the ππ and πK scattering parameters, since it costs in terms of computing time. So, as done for fit All, we include only a 0 0 , a 0 2 , a 1/2 0 and a 3/2 0 . Notice that we do not find any large discrepancies when we add more scattering parameters in fit All, as was remarked at the end of Section 6.
We also require a good convergence of the masses and decay constants expansions. The reason is that in this way we have the possibility to "select" those C r i granting us convergence for those quantities. This also allows to keep under control the quality of the fits. Otherwise too much freedom would be left to the C r i constants, and many different fits with a low χ 2 but with very bad convergences, can be reached. Clearly such convergence constraints have a strong effect on the L r i constants as described in the next Section 7.1.

Convergence constraints
We devote this section to a discussion of the convergence constraints imposed on the masses and decay constants in the fits with random C r i . Let us first show the case of the decay constants.
We performed the fits constraining the NNLO contributions to F π , F K and F K /F π constants to be small, i.e. less than the 10% of the LO ones. Remember that the expansions for the F π and F K decay constants are and that in our fits, as explained in Section 3.2, we include their ratio as Specifically the convergence constraints are included through the following partial χ 2 i(part) : With (45) we have been able to find many fits with a good convergence. On the other hand the NLO contribution for F π turns out to be smaller than the expected 30%. The reason resides in the third of the relations in (45). Requiring all the NNLO pieces for all the decay constants to be small implies that the single contributions F π | p 6 and F K | p 6 are small. But also the term (F π /F 0 )| 2 p 4 must be small, otherwise the NNLO contribution of F K /F π is allowed to be large. This leads to small NLO corrections for F π and thus a F 0 ≈ F π .
We apply similar restrictions also to the masses where M stands for π, K and η mesons and m M 0 are the leading order contributions to the masses. We have kept the value of m s /m fixed at 27.8 as for fit All. We conclude this section with a final remark. One might wonder why we have not imposed similar constraints also for fit All, since these could improve the convergence of the expansions. The reason is that when we require them, we obtain a reasonably good fit (χ 2 ≈ 8 with 10 degrees of freedom ) and with better constrained L r 4 and L r 6 . But it also causes much worse predictions for thel i , e.g.l 3 ≈ 6.5.

Results
Now we are ready to show the outcomes of our studies when the C r i are set to random values using the procedure of Figure 1 and with the constraints listed in Section 7.1 above.  First of all we must point out that due to the freedom we allow to the C r i many different fits of the L r i have a low χ 2 . We set initial C r i equal to zero, resonance exchange or chiral quark model estimates as well more random starts. We easily reach χ 2 < 1, and we found many fits with χ 2 ≈ 0.5. Reducing the steps of the random walk we can even find smaller values. However once the χ 2 reaches a reasonably low value, e.g. 1, there is no apparent reason why one should prefer one specific fit to another. Due to the several different sets of C r i under study, we can only quote the ranges where the L r i vary and where we obtain a χ 2 < 1. Such ranges are quoted in Table 15 and are obtained, starting from different initial sets C r(in) i . Keep in mind that these ranges depend on the C r i chosen. Plugging in a L r i fit without the corresponding C r i will not produce any sensible results. The way we determined those numbers is shown in Figure 2 on the example for L r 1 where we have plotted a number of fits that gave χ 2 < 1 for the different starting points. We have typically stopped the fits when a χ 2 of about 0.4 or below was found and the tails at low χ 2 are an artefact, they were done with runs with a very low ǫ and a very low T .
The results are rather cumbersome. L r 1 , L r 2 and L r 3 look quite free to vary in large ranges. Notice also that the large N c relation 2L r 1 ≈ L r 2 is still not recovered. On the other hand, due to the converge constraints of Section 7.1 we narrow the intervals for the other constants. Especially L r 5 takes a large value and L r 4 a small one. As explained in Section 7.1, we essentially require that F K /F π , F K and F π have small NNLO corrections which give that F K /F π is given by the NLO and thus determines L r 5 at a fairly large value. That F π has small corrections at all then in turn requires a fairly small L r 4 . The choice to constrain the convergence of those quantities is dictated by the lack of information to constrain more the C r i . When we release such constraints indeed we find different looking fits, but affected by a bad convergence. Somewhat more surprising is that the L r 6 typically takes on values that are smaller than L r 8 . As you can see no clear final conclusion can be drawn with such results. When we  Table 16: The results as obtained using the C r i from the best χ 2 found starting from the resonance or from a completely random one as described in the text. The L r i are given at µ = 0.77 GeV. The corresponding C r i sets can be found in Table 18 in appendix.  Table 17: The convergence for the best χ 2 found starting from the resonance estimate, here F 0 = 0.079 GeV. Fit as in left column of Table 16. The convergence for the best χ 2 found starting from the fully random estimate, here F 0 = 0.078 GeV. Fit as in left column of Table 16.
performed this study we were hoping not to find as many good fits and smaller ranges for the L r i . The study shows instead that it is very difficult, if not impossible, to narrow the ranges for the L r i with such a poor knowledge of the C r i . On the other hand it also shows that fits of the L r i with good convergence do exist, if the C r i are changed. In Table 16 we show the L r i obtained for the smallest χ 2 found starting from the resonance estimate and from fully random C r i as described above and we show the convergence for some quantities in those two fits in Table 17. The fits with very low χ 2 we have obtained, such as the two shown here, tend to have similar expansions for the masses and the decay constants. In order to see how the various C r i look like we have added the values for these two fits in the appendix.
We can draw some conclusions by studying correlations. The effect ofl 2 is very visible if we plot for the various fits with χ 2 < 1 L r 2 , which is the NLO dependence ofl 2 on the LECs,  Figure 3: The correlations between L r 2 and 2C r 13 − C r 11 . This results from including the value ofl 2 in the fit. versus 2C r 13 − C r 11 , which is the dependence on the NNLO LECs. The tight correlation shows that the NNLO contribution depends very little on the other L r i . More precisely, this shows the NLO LEC combination at NLO versus the NNLO LEC combination at NNLO and that the NNLO contribution depends fairly little on the value of the L r i . We get more of those constraints directly. A very similar one is from the value ofl 1 shown in Figure 4. The correlations for other observables tend to be weaker indicating that the NNLO contributions are more dependent on the value of the L r i for these cases. We show examples with a weaker but still existing correlation in Figure 5 where the correlations resulting from F K /F π are shown and in Figure 6 for r 2 π S . In both cases we have plotted on the horizontal axis the combination of L r i the quantity depends on at NLO and on the vertical axis the combination of C r i the quantity depends on at NNLO. There are also correlation between the fitted values of the L r i . L r 1 , L r 2 and L r 3 show a reasonable correlation among themselves but are essentially not correlated with the others. There are weaker correlations between L r 4 and L r 6 and between L r 7 and L r 8 . These correlations are shown in Figures 7, 8, 9, 10, and 11. We have shown a curve in all plots guiding the eye as well and given it in the caption of the figure.
Note that throughout this section we have considered all fits with a χ 2 < 1 to be essentially possible.  Figure 4: The correlations between 2L r 1 + L r 3 and C r 6 − C r 11 + 4C r 13 . This results from including the value ofl 1 in the fit.   Figure 6: The correlations resulting from the pion scalar radius r 2 π S .

Conclusions
In this paper we have shown the results for a new global fit of the L r i at NNLO, with techniques similar to the ones in [8,9]. Different treatments of the p 6 coupling constants have been considered: the resonance estimate of [8], the results of [13] and the use of randomly selected C r i . The results are difficult to interpret and unexpected. All the fits that have been performed using the NNLO couplings from [8] or [13] show both strong and weak points. The fits obtained from the randomly selected C r i are too different from each other to draw a final conclusion, although they give a rough indication on where we can expect to find the values of L r i . They also provide a proof of principle that with reasonable values of the C r i a reasonably convergent series for SU(3) ChPT can be obtained for many quantities.
The fit that presents the least discrepancies and best convergence of the chiral expansion is fit All in Table 5, which has been obtained with the resonance estimate of the C r i . It succeeds in fitting many observables like the ππ and πK scattering parameters and the slope of the scalar formfactor of the pion. It also reproduces quite well the experimental results for the f s and g p K ℓ4 formfactors although it does not predict the curvature of f s . The perturbative expansions for masses and decay constants reported in (36) and (37) look suspicious but are acceptable. On the other hand it does not satisfy the large N c relation 2L r 1 ≈ L r 2 and it fails in well constraining the L r 4 and L r 6 values. Finally its prediction for ℓ 2 is far from the current estimate of that constant. We have at present not included many results from lattice QCD. We expect that this should improve in a few years allowing for another step forward in confronting ChPT with data. Table 11 is a little smaller but essentially the same. The last two columns are from the random walk/simulated annealing estimates for the C r i reported in Section 7 from the best χ 2 found starting from the resonance estimate and a fully random starting point. These are the C r i for the fits reported in Table 16. The main purpose is to show the typical sizes of the C r i we obtained for the fits and that the pattern for the good fits can be quite different.