Constraints on low energy QCD parameters from \(\eta \rightarrow 3\pi \) and \(\pi \pi \) scattering
Abstract
The \(\eta \,\rightarrow \,3\pi \) decays are a valuable source of information on low energy QCD. Yet they were not used for an extraction of the three flavor chiral symmetry breaking order parameters until now. We use a Bayesian approach in the framework of resummed chiral perturbation theory to obtain constraints on the quark condensate and pseudoscalar decay constant in the chiral limit. We compare our results with recent CHPT and lattice QCD fits and find some tension, as the \(\eta \,\rightarrow \,3\pi \) data seem to prefer a larger ratio of the chiral order parameters. The results also disfavor a very large value of the pseudoscalar decay constant in the chiral limit, which was found by some recent work. In addition, we present results of a combined analysis including \(\eta \,\rightarrow \,3\pi \) decays and \(\pi \pi \) scattering and though the picture does not changed appreciably, we find some tension between the data we use. We also try to extract information on the mass difference of the light quarks, but the uncertainties prove to be large.
1 Introduction
Chiral perturbation theory (\(\chi \)PT) [3, 4, 5] is constructed as a general low energy parametrization of QCD based on its symmetries and the discussed order parameters appear at the lowest order of the chiral expansion as low energy constants (LECs). Interactions of the light pseudoscalar meson octet, the pseudoGoldstone bosons of the broken symmetry, directly depend on the pattern of SB\(\chi \)S and thus can provide information as regards the values of \(\Sigma (N_f)\) and \(F(N_f)\).
Chosen results for the two flavor order parameters
Chosen results for the three flavor order parameters
Phenomenology  Z(3)  X(3) 

NNLO \(\chi \)PT (BE14) [9]  0.59  0.63 
NNLO \(\chi \)PT (free fit) [9]  0.48  0.45 
NNLO \(\chi \)PT (”fit 10”) [10]  0.89  0.66 
Re\(\chi \)PT \(\pi \pi \)+\(\pi K\) [11]  > 0.2  < 0.8 
Lattice QCD  Z(3)  X(3) 

RBC/UKQCD+Re\(\chi \)PT [12]  0.54 ± 0.06  0.38 ± 0.05 
RBC/UKQCD+large \(N_c\) [13]  0.91 ± 0.08  
MILC 09A [14]  0.72 ± 0.06  0.62 ± 0.07 
Phenomenology  R 

Dashen’s theorem LO [16]  44 
Dashen’s theorem NNLO [16]  37 
\(\eta \rightarrow 3\pi \) NNLO \(\chi \)PT [16]  41.3 
\(\eta \rightarrow 3\pi \) dispersive [17]  37.7 ± 2.2 
\(\eta \rightarrow 3\pi \) dispersive [18]  34.2 ± 2.2 
\(\eta \rightarrow 3\pi \) dispersive [19]  33.0 ± 3.4 
\(\eta \rightarrow 3\pi \) dispersive [20]  32.7 ± 3.0 
Lattice QCD  R 

FLAG 2016 \(N_f\,{=}\,2\) [6]  40.7±4.3 
FLAG 2016 \(N_f\,{=}\,2+1\) [6]  35.7±2.6 
In this paper, we use a Bayesian approach in the framework of resummed chiral perturbation theory to extract information on the three flavor chiral condensate, chiral decay constant and the mass difference of the light quarks. Our experimental input are well known observables connected to \(\eta \,\rightarrow \,3\pi \) decays and \(\pi \pi \) scattering. We assume a reasonable convergence of Green functions connected to these observables and investigate the constraints this assumption can provide for the discussed parameters. The results presented here are a significant update on our initial reports [21, 22, 23]
In Sect. 2 we shortly summarize our theoretical foundation. Section 3 discusses the \(\eta \,\rightarrow \,3\pi \) decays, while Sect. 4 provides an overview of our calculation of these processes. Section 5 introduces \(\pi \pi \) scattering into our analysis. The Bayesian statistical approach is reviewed in Sect. 6 and a discussion of our assumptions can be found in Sect. 7. Section 8 is concerned with investigating the compatibility of our theoretical predictions with the \(\pi \pi \) scattering data. We employ a \(\chi ^2\)based analysis in Sect. 9 to evaluate the quality with which our theoretical predictions reconstruct the experimental data and use the result to choose between several assumptions. Finally, in Sect. 10, the main results of the Bayesian analysis are presented and compared with available literature. We conclude in Sect. 11.
2 Resummed \(\chi \)PT
We use an alternative approach to chiral perturbation theory, dubbed resummed \(\chi \)PT (Re\(\chi \)PT) [24], which was developed in order to accommodate the possibility of irregular convergence of the chiral expansion. This is a typical scenario if the X(3) and Z(3) were indeed suppressed and the leading order was not dominant in the chiral expansion. In such a case one would have to be careful in the way how chiral expansion is defined and dealt with, as reshuffling of chiral orders could lead to unexpectedly large higher orders.

The Standard \(\chi \)PT Lagrangian and power counting [3, 4, 5] is used. In particular, the quark masses \(m_{q}\) are counted as \(O(p^{2})\).

Only expansions of quantities related linearly to Green functions of QCD currents are trusted, these are called safe observables. It is assumed that the nexttonexttoleading and higher orders in these expansions are reasonably small, though not necessary negligible. Leading order terms do not need to be dominant.

Calculations are performed explicitly to nexttoleading order, higher orders are included implicitly in remainders. The first step consists of performing the strict chiral expansion of the safe observables, which is understood as an expansion constructed in terms of the parameters of the chiral Lagrangian, while strictly respecting the chiral orders.

In the next step, the strict expansion is modified in order to correct the location of the branching points of the nonanalytical part of the amplitudes, which need to be placed in their physical position. This can be done either by means of a matching with a dispersive representation or by hand. The bare expansion is obtained.

After that, an algebraically exact nonperturbative reparametrization of the bare expansion is performed. It is obtained by expressing the \(O(p^{4})\) LECs \(L_i\) in terms of physical values of experimentally well established safe observables—the pseudoscalar decay constants and masses. The procedure generates additional higher order remainders. We refer to these as indirect remainders.

The physical amplitude and other relevant observables are then obtained as algebraically exact nonperturbative expressions in terms of the related safe observables and higher order remainders.

The higher order remainders are not neglected, but estimated and treated as sources of error.
3 \(\eta \rightarrow 3\pi \) decays
As we have shown in [27], we argue that the four point Green functions relevant for the \(\eta \,\rightarrow \,3\pi \) amplitude (see (4.1) below) do not necessarily have large contributions beyond nexttoleading order and a reasonably small higher order remainder is not in contradiction with huge corrections to the decay widths. The widths do not seem to be sensitive to the details of the Dalitz plot distribution, but rather to the value of leading order parameters—the chiral decay constant, the chiral condensate and the difference of u and d quark masses, i.e. the magnitude of explicit isospin breaking. Moreover, access to the values of these quantities is not screened by EM effects, as it was shown that the electromagnetic corrections up to NLO are very small [28, 29]. This is the motivation for our effort to extract information as regards the character of the QCD vacuum from this decay.
4 Calculation
The details of the calculations with explicit formulas can be found in [27]. We only summarize the basic steps here, which closely follow the procedure outlined in [35].
5 \(\pi \pi \) scattering
6 Bayesian statistical analysis
\(P(X_i)\) in (6.1) are the prior probability distributions of \(X_i\). We use them to implement the theoretical uncertainties connected with our parameters and remainders. In this way we keep the theoretical assumptions explicit and under control. It also allows us to test various assumptions and formulate ifthen statements as well as implement additional constraints (see below).
7 Assumptions

\(\eta \,\rightarrow \,3\pi \) direct remainders: \(\delta _A\), \(\delta _B\), \(\delta _C\), \(\delta _D\),

\(\pi \pi \) scattering direct remainders: \(\delta _{\alpha _{\pi \pi }}\), \(\delta _{\beta _{\pi \pi }}\),

indirect remainders: \(\delta _{M_\pi }\), \(\delta _{F_\pi }\), \(\delta _{M_K}\), \(\delta _{F_K}\), \(\delta _{M_\eta }\), \(\delta _{F_\eta }\), \(\delta _{M_{38}}\), \(\delta _{Z_{38}}\).
8 Subthreshold parameters of \(\pi \pi \) scattering
As mentioned in Sect. 3, in [27] we tested the compatibility of a reasonable convergence of the chiral series, laid out explicitly in the form of assumptions in the previous section, with the experimental data in the case of the \(\eta \rightarrow 3\pi \) observables. This is an important first step in order to avoid using observables which are problematic from the theoretical or experimental point of view. Analogously, in this section we take a closer look at the subthreshold parameters \(\alpha _{\pi \pi }\) and \(\beta _{\pi \pi }\), which was not done in [24].
We numerically generated a large number of theoretical prediction for \(\alpha _{\pi \pi }\) and \(\beta _{\pi \pi }\), statistically distributed according to the assumptions described in Sect. 7. The parameter Z was fixed in two scenarios (Z = 0.5 and Z = 0.9), while X was varied in the full range \(0<X<1\). Figure 1 displays the obtained theoretical distributions in comparison with the experimental data.
As can be seen, both parameters show a significant dependence on X, while \(\beta _{\pi \pi }\) depends on Z as well. In the case of \(\beta _{\pi \pi }\), a broad range of the generated theoretical predictions is consistent with experimental data. Even though the observable seems sensitive to the values of the chiral order parameters, the amount of available information to be extracted might be limited by the experimental error, which is quite substantial.
The picture seems to be more tricky when considering \(\alpha _{\pi \pi }\). The central experimental value is outside the 2\(\sigma \) band of the theoretical distribution in the whole range of the free parameters. If taken at face value, possibly confirmed by more precise data, it would indicate a very small value of X, i.e. a vanishing chiral condensate. Such a scenario would be, however, inconsistent with any current determination of the order parameter (see Table 2).^{2} The experimental error on the value of \(\alpha \) is very large and our prior expectation therefore is that a large correction of the central value is very possible. It would be thus advisable to be cautious when interpreting the outcomes based on these data in the following.
Our conclusion with regard to the suitability of the subthreshold parameters of \(\pi \pi \) scattering for the purpose of extraction of information as regards the chiral order parameters is hence twofold  both \(\alpha _{\pi \pi }\) and \(\beta _{\pi \pi }\) seem sensitive to the value of one or both of them, but at present available information is limited due to the quality of the experimental data we have at hand.
9 \(\chi ^2\)based analysis
In addition to the Bayesian analysis described in Sect. 6, the results of which will follow in Sect. 10, we also perform a search for the minimum of the \(\chi ^2\) distribution in the Monte Carlo generated set of data points. The aim is to check the quality with which this set of theoretical predictions can reconstruct the experimental data, which is hard to quantify using the Bayesian method. This test also enables us to compare various theoretical approaches, e.g. the alternative ways the dispersive representation can be implemented.
Because we have much more free parameters than experimental inputs, we generally expect a well working theory to be able to reconstruct the experimental data very precisely with a large enough sample of generated data points. This means the minimum of \(\chi ^2/n\), where n is the number of experimental observables employed, should be close to zero. On the other hand, a min.\(\chi ^2/n\approx 1\) means there is typically a 1\(\sigma \) deviation between the best of the generated theoretical predictions and the experimental data, which might signal some tension and would not be fully satisfactory.
As the minimum of the \(\chi ^2\) distribution is subject to fluctuations stemming from the statistical nature of our procedure, we also report the number of points for which \(\chi ^2/n<1\). This metric then reveals how well the region of the parameter space where the experimental data lie is covered by the generated theoretical predictions. A reasonably high number should be considered a necessary condition for a wellfounded analysis, as it demonstrates that the theory has no obvious problem to reconstruct the experimental data and also that we have generated a large enough sample of points.
When comparing different theoretical approaches, one model can overlay the experimental region with fewer points than the other for several reason. It might be that the points originate from a less probable part of the theoretical distribution (e.g., the tail) which means that less likely values of the parameters and remainders are needed in order to reconstruct the experimental data. In that sense, the model is less probable to be true. The \(\chi ^2\)based analysis can thus form a basis for preference when choosing between alternative approaches. However, it is also possible that one model is simply more sensitive to the values of some of the parameters or remainders, which means the theoretical distribution has a larger spread. This is no reason to exclude the model, of course, and one should be aware of this possibility and check for it.
\(\chi ^2\)based comparison of the dispersive representations
Free parameters  Exp. data  Disp. aproach  \(\sqrt{\mathrm {min.}\chi ^2/n}\)  \(N(\chi ^2/n<1)\) 

X, Z  \(\Gamma ^+\), \(\Gamma ^0\), a  (4.6)  0.80  57 
X, Z  \(\Gamma ^+\), \(\Gamma ^0\), a  (4.7)  0.59  1048 
Z, R  \(\Gamma ^+\), \(\Gamma ^0\), a  (4.6)  0.65  225 
Z, R  \(\Gamma ^+\), \(\Gamma ^0\), a  (4.7)  0.46  1068 
\(\chi ^2\)based comparison of the pion mass implementations
Free parameters  Exp. data  Pion mass  \(\sqrt{\mathrm {min.}\chi ^2/n}\)  \(N(\chi ^2/n<1)\) 

X, Z  \(\Gamma ^+\), \(\Gamma ^0\), a  \(\overline{M_\pi }\), \(M_\pi ^0\)  0.59  1048 
X, Z  \(\Gamma ^+\), \(\Gamma ^0\), a  \(\overline{M_\pi }\)  0.93  7 
X, Z  \(\Gamma ^+\), \(\Gamma ^0\), a  \(M_\pi ^0\)  0.92  3 
Z, R  \(\Gamma ^+\), \(\Gamma ^0\), a  \(\overline{M_\pi }\), \(M_\pi ^0\)  0.46  1068 
Z, R  \(\Gamma ^+\), \(\Gamma ^0\), a  \(\overline{M_\pi }\)  0.89  9 
Z, R  \(\Gamma ^+\), \(\Gamma ^0\), a  \(M_\pi ^0\)  0.93  3 
\(\chi ^2\)based comparison for the main analysis used in Sect. 10
Free parameters  Exp. data  \(\sqrt{\mathrm {min.}\chi ^2/n}\)  \(N(\chi ^2/n<1)\) 

X, Z  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\)  0.002  165,874 
X, Z  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\), \(\beta _{\pi \pi }\)  0.03  104,670 
X, Z  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\), \(\alpha _{\pi \pi }\), \(\beta _{\pi \pi }\)  0.28  51,278 
Z, R  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\)  0.003  87,034 
Z, R  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\), \(\beta _{\pi \pi }\)  0.02  40,919 
Y  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\), \(\alpha _{\pi \pi }\), \(\beta _{\pi \pi }\)  0.15  25,041 
Y  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\)  0.002  120,130 
Y  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\), \(\beta _{\pi \pi }\)  0.02  62,203 
Y  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\), \(\alpha _{\pi \pi }\), \(\beta _{\pi \pi }\)  0.28  23,428 
Y, R  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\)  0.002  125,724 
Y, R  \(\Gamma ^+\), \(\Gamma ^0\), \(\delta (aa_{exp})\), \(\alpha _{\pi \pi }\), \(\beta _{\pi \pi }\)  0.18  20,054 
As can be seen in Table 4, the approach (4.7) is able to reconstruct the data more precisely, while the theoretical distributions have a very similar form. This conforms to our intuition from Sect. 4 and thus, in what follows, we use the representation based on (4.7) exclusively.
As we have found, this is really the case in the approach with an identical pion mass in both the charged and the neutral channel amplitudes, where the predicted ratio \(r_\Gamma \) turns out to be too high. This is reflected in both the minimum of \(\chi ^2/n\) not being quite close to zero and in the number of points for which \(\chi ^2/n<1\) being substantially lower in our \(\chi ^2\)based test for any value of the parameters in the allowed range, as can be seen in Table 5. Meanwhile, the form of the distributions do not change considerably, as might be expected when only the numerical value of an input parameters is changed.
In other words, in this case the theory seems to have a harder time to reproduce the experimental data with the required precision. This might look surprising given the number of free parameters in the fit, but one has to realize that, while the decay widths in the two channels are independent from the experimental point of view, they are very strongly correlated on the theoretical side given Eq. (4.3). The experiment in fact shows that this relation is not precisely fulfilled in nature. This is the reason we have chosen to implement distinct values of the pion mass in the two channels for the main analysis, which is the same approach as was taken in [16].
As can be seen, the number of points seems sufficient. The coverage is better in the approach with \(\eta \rightarrow 3\pi \) observables only. We can observe a drop in precision when \(\alpha _{\pi \pi }\) is included, which correlates with the discussion in Sect. 8. As we will see, this corresponds to the fact that the signal in the used experimental value of \(\alpha _{\pi \pi }\) is in some tension with the one contained in the \(\eta \rightarrow 3\pi \) data.
10 Results and discussion
In this section we present the outputs of the Bayesian analysis, i.e. the probability density functions \(P(X_i\mathrm {data})\), where \(X_{i}\in \{X,Z,Y,R\}\) are the chiral symmetry breaking and explicit isospin breaking parameters, respectively, and \(\mathrm {data}\) represent a subset of the set \(\left\{ \Gamma _{+},\Gamma _{0},a,\alpha _{\pi \pi },\beta _{\pi \pi }\right\} \) of the \(\eta \rightarrow 3\pi \) and \(\pi \pi \rightarrow \pi \pi \) observables.
Characteristics of obtained probability distributions, \(R=35.8\pm 2.6\)
\(\overline{x}\)  \(\sigma _{x} \)  Median  \( 1\sigma \) C.L.  \(2\sigma \) C.L.  

\(P(X\Gamma _{+},\Gamma _{0},a)\)  0.56  0.22  0.58  (0.31, 0.80)  (0.11, 0.88) 
\(P(X\Gamma _{+},\Gamma _{0},a,\pi \pi )\)  0.56  0.21  0.58  (0.32, 0.78)  (0.13, 0.87) 
\(P(Z\Gamma _{+},\Gamma _{0},a)\)  0.40  0.18  0.40  (0.22, 0.58)  (0.08, 0.78) 
\(P(Z\Gamma _{+},\Gamma _{0},a,\pi \pi ) \)  0.48  0.19  0.48  (0.28, 0.68)  (0.11, 0.82) 
\(P(Y\Gamma _{+},\Gamma _{0},a)\)  1.44  0.32  1.44  (1.11, 1.76)  (0.78, 2.05) 
\(P(Y\Gamma _{+},\Gamma _{0},a,\pi \pi )\)  1.20  0.30  1.20  (0.90, 1.50)  (0.60, 1.80) 
\(P(Y\alpha _{\pi \pi } )\)  0.55  0.42  0.48  (0, 0.72)  (0, 1.38) 
\(P(Y\alpha _{\pi \pi },\beta _{\pi \pi } )\)  0.53  0.38  0.49  (0, 0.70)  (0, 1.25) 
Characteristics of obtained probability distributions, R free
\(\overline{x}\)  \(\sigma _{x} \)  Median  \(1\sigma \) C.L.  \(2\sigma \) C.L.  

\(P(R\Gamma _{+},\Gamma _{0},a)\)  43.0  13  41.8  (29.7, 56.2)  (20.4, 72.2) 
\(P(R\Gamma _{+},\Gamma _{0},a,\pi \pi )\)  34.4  11.6  32.7  (23.0, 45.9)  (16.7, 62.0) 
\(P(X\Gamma _{+},\Gamma _{0},a)\)  0.56  0.22  0.59  (0.31, 0.80)  (0.10, 0.88) 
\(P(Z\Gamma _{+},\Gamma _{0},a)\)  0.39  0.17  0.38  (0.21, 0.56)  (0.08, 0.77) 
\(P(Y\Gamma _{+},\Gamma _{0},a)\)  1.56  0.46  1.58  (1.09, 1.99)  (0.57, 2.32) 
Let us first discuss the case with the fixed value of \(R=35.8\pm 2.6\), which is the lattice QCD average [40] (see Sect. 7) and include only the \(\eta \rightarrow 3\pi \) observables into the analysis. This is what we consider our main set of results.
As can be seen in Fig. 2, when assuming \(R=35.8\pm 2.6\), there is some tension with several of the previous determinations of the chiral order parameters (Table 2). The \(\eta \,\rightarrow \,3\pi \) data seem to prefer a larger value for the ratio of the order parameters \(Y = X/Z\) than recent \(\chi \)PT and lattice QCD fits. In addition, very large values of the chiral decay constant are excluded at 2\(\sigma \) CL. and a relatively small value is favored. The uncertainties, however, are quite large.
In our approach, \(L_4\) is reparametrized in terms of the remainders and free parameters, including Z. It can thus vary in a wide range, as we have shown in [27]. As the \(\eta \rightarrow 3\pi \) data seem to constrain Z only mildly, as discussed above, we do not get significant information on \(L_4\) either.
The apparent inconsistency with the result of [12] is intriguing. It uses resummed \(\chi \)PT as well, paired with lattice data. One distinction is a different approach to the remainders—the authors use a uniform distribution of the remainders inside a closed interval and thus a sharp cutoff. One could speculate that a normal distribution with unbounded tails, as we use, might lead to larger error bars.
Reference [13], which is based on a large \(N_c\) motivated approximation of the standard \(O\left( p^{6}\right) \) \(\chi \)PT calculations, used on lattice data, reports a very large value of the chiral decay constant and a very low value of \(L_4\). This is consistent with the large \(N_c\) picture assumed in this paper, but quite far away from other determinations and in tension with our limit (10.2). In fact, a large part of the region covered by the fit [13] is excluded by our prior for Z, namely by the constraint stemming from the paramagnetic inequality \(Z<Z(2)\) (7.5).
Let us now add the \(\pi \pi \rightarrow \pi \pi \) data into the analysis. As can be seen in the right panel of Figs. 2 and 4, the picture does not change appreciably when including the subthreshold parameters of \(\pi \pi \) scattering. Though a bit disappointing, this outcome is not unexpected considering the significant errors connected with the experimental values of these observables and the weak constraints obtained in [24] and [11].
There is one difference, however: we can observe a slight shift of our probability distribution towards a lower ratio of order parameters \(Y=X/Z\), as is confirmed by the mean \(\overline{Y}=1.2\) (\(P(Y\Gamma _{+},\Gamma _{0},a,\pi \pi )\) in Table 7). This could be interpreted as a move of our predictions in the direction of better compatibility with some of the available determinations (Table 2). Here we have to be rather cautious though, because, as we have discussed in Sects. 8 and 9, we can expect some tension between the two sets of data. And indeed, this can be demonstrated when one compares the obtained probability distributions and confidence intervals for Y from \(\eta \rightarrow 3\pi \) (Fig. 3, \(P(Y\Gamma _{+},\Gamma _{0},a)\) in Table 7) with one from \(\pi \pi \) scattering alone (Fig. 5, \(P(Y\alpha _{\pi \pi },\beta _{\pi \pi } )\) in Table 7), which are barely compatible. We can conclude that we need a more precise determination of the value of the \(\pi \pi \rightarrow \pi \pi \) subthreshold parameters, especially for \(\alpha _{\pi \pi }\), to be able to provide a more definite outcome and hopefully solve this puzzle of experimental data pointing in opposite directions.
The results with R left as a free parameter are shown in Figs. 6 and 8. The uncertainties are large and thus it is hard to constrain R without additional information on the chiral order parameters and the remainders. Even in this case a part of the parameter space can be excluded at 2\(\sigma \) C.L. though. The obtained value for R (Fig. 7, Table 8) is compatible with available results (Table 3).
The large uncertainty in the extracted value of R indicates that the dependence of R on the values of the chiral order parameters and the higher order remainders is strong. This could be important information for those determinations of R which use an input from \(\chi \)PT and thus implicitly depend on these uncertainties, such as methods employing a dispersive representation (see Table 3).
We can also evaluate the obtained probability densities for X, Z and Y with R left unconstrained (Table 8, Fig. 7). Note that dismissing the very clear information on R is not a reasonable assumption, we use it only as a test of robustness. And in this case, the results for X and Z seem almost independent on the value of R, which includes the obtained upper bound for the chiral decay constant (10.2).
On the other hand, as Fig. 8 shows, R and the ratio \(Y = X/Z\) seem to be quite strongly correlated. This also provides some additional insight into the large value of Y obtained from \(\eta \rightarrow 3\pi \) data when fixing R = 35.8 ± 2.6. Furthermore, as the \(\pi \pi \) scattering data we use drag Y down to lower values, one can observe a correlated shift in probability densities of R to smaller numbers in Figs. 8, 7 and Table 8.
11 Conclusions
To summarize, we have used statistical methods in the framework of resummed chiral perturbation theory to generate large sets of theoretical predictions for \(\eta \rightarrow 3\pi \) and \(\pi \pi \) scattering observables, dependent on a variety of parameters and assumptions, and confronted them with experimental data.
We have developed a \(\chi ^2\)based analysis, which allowed us to form a basis for preference when choosing between alternative assumptions or models. In particular, it showed us that an approach using different pion masses for the charged and neutral decay channel observables in the isospin limit is significantly more consistent with data, despite violating the isospin relation, than using identical pion mass for both \(\eta \rightarrow 3\pi \) decay modes.
For the main analysis, we have used Bayesian inference to obtain constraints on the values of three flavor chiral order parameters—the chiral decay constant \(F_0\) and the chiral condensate \(\Sigma _0\), which are connected with the spontaneous breaking of chiral symmetry in QCD.
When fixing the light quark difference by input from lattice QCD and using \(\eta \rightarrow 3\pi \) observables only (the decay widths in both channels and the Dalitz plot parameter a), we could exclude a large part of the parameters space at 2\(\sigma \) CL and have observed some correlation between the chiral order parameters. We have obtained an upper bound for the chiral decay constant, \(F_0<81\)MeV at 2\(\sigma \) CL, and have extracted a fairly large value for the ratio of the order parameters \(Y=2\hat{m}B_0/M_{\pi }^2\).
We have found some tension with several of the previous determination of the chiral order parameters, which, however, are neither very consistent with each other. The picture remains unclear, possibly stemming from differences in assumptions as regards the low energy constants, the large \(N_c\) suppressed LEC \(L_4\) being one candidate.
The picture has not changed appreciably when we performed a combined \(\eta \rightarrow 3\pi \) and \(\pi \pi \rightarrow \pi \pi \) analysis. However, we have observed some tension between \(\eta \rightarrow 3\pi \) and \(\pi \pi \) scattering data, which limited our ability to draw more definite conclusions. The possible source is the experimental error of the observables we used, the subthreshold parameters \(\alpha _{\pi \pi }\) and \(\beta _{\pi \pi }\). Specifically, the value of \(\alpha _{\pi \pi }\) proved suspicious in our investigation, as it prefers a very low value for the chiral condensate, which is not very consistent with current expectations.
We have also tried to extract information on the difference of light quark masses, but the uncertainties proved to be very large. This indicates that the dependence on the values of the chiral order parameters and the higher order remainders is strong. The result is consistent with available determinations.
Footnotes
 1.
We will abbreviate these to chiral condensate and chiral decay constant in the following,
 2.
A crash of the chiral condensate at three flavors would also be unexpected in the context of an SU(3) gauge theory with a varying number of light quark flavors; see e.g. [41].
 3.
The choice of such regions could be considered somewhat arbitrary, we constructed them by means of integrating the (discretized) probability densities according to decreasing probabilities starting from the maximal value until the desired confidence level was achieved.
Notes
Acknowledgements
This work was supported by the Czech Science Foundation (Grant No. GACR 1518080S).
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