Dispersive analysis of \(\eta \rightarrow 3 \pi \)
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Abstract
The dispersive analysis of the decay \(\eta \rightarrow 3\pi \) is reviewed and thoroughly updated with the aim of determining the quark mass ratio \(Q^2=(m_s^2m_{ud}^2)/(m_d^2m_u^2)\). With the number of subtractions we are using, the effects generated by the final state interaction are dominated by low energy \(\pi \pi \) scattering. Since the corresponding phase shifts are now accurately known, causality and unitarity determine the decay amplitude within small uncertainties – except for the values of the subtraction constants. Our determination of these constants relies on the Dalitz plot distribution of the charged channel, which is now measured with good accuracy. The theoretical constraints that follow from the fact that the particles involved in the transition represent Nambu–Goldstone bosons of a hidden approximate symmetry play an equally important role. The ensuing predictions for the Dalitz plot distribution of the neutral channel and for the branching ratio \(\varGamma _{\eta \rightarrow 3\pi ^0}/ \varGamma _{\eta \rightarrow \pi ^+\pi ^\pi ^0}\) are in very good agreement with experiment. Relying on a known lowenergy theorem that relates the meson masses to the masses of the three lightest quarks, our analysis leads to \(Q=22.1(7)\), where the error covers all of the uncertainties encountered in the course of the calculation: experimental uncertainties in decay rates and Dalitz plot distributions, noise in the input used for the phase shifts, as well as theoretical uncertainties in the constraints imposed by chiral symmetry and in the evaluation of isospin breaking effects. Our result indicates that the current algebra formulae for the meson masses only receive small corrections from higher orders of the chiral expansion, but not all of the recent lattice results are consistent with this conclusion.
1 Introduction
Our world is almost isospin symmetric: the up and the down quarks can be freely interchanged (or replaced by any linear combination of them) inside hadrons almost without any observable consequence. Of course the charge of the two quarks is different, so that after an isospin transformation the charge of the hadronic state might change, but since the electromagnetic interactions are much weaker than the strong ones, we can classify this as a small effect. Besides the charge, the only difference between the two quarks is their mass. In relative terms their mass difference is large, but very small when compared to the mass of a typical hadron: if we interchange the up and down quarks inside a hadron, the mass of the latter barely changes. Observables which are sensitive to isospin violations are therefore particularly interesting, as they offer us rare insights into the sector of the Standard Model Lagrangian which breaks the isospin symmetry. One of them is the decay of the \(\eta \)meson into three pions. This decay would be forbidden by isospin symmetry and moreover it is mainly due to purely strong isospin violations [1, 2]: among the already rare observables sensitive to isospin breaking, this is even more special as it allows to clearly separate the two sources, which are otherwise mostly present at a similar level. To a good approximation the decay rate is proportional to the square of the up and down mass difference. If one were able to accurately calculate the proportionality factor – the modulus squared of the transition amplitude between the \(\eta \) and a threepion state mediated by the third component of the scalar isovector quark bilinear – a measurement of the decay rate would provide a determination of this quark mass difference. This approach has been adopted before, but both, recent improved measurements of the differential decay rates as well as progress on the theory side call for an updated and improved analysis. This is the aim of the present paper, where we give a detailed account of the work reported in Ref. [3].
The calculation of hadronic matrix elements is not an easy task, especially if the aim is high precision. Several methods are available and can be applied with varying degree of success, depending on the circumstances: they range from lattice QCD to chiral perturbation theory (\(\chi \)PT), to dispersive approaches. Decays into three particles are not accessible to lattice calculations yet,^{1} but both the effective field theory approach and dispersion relations can be and have been used to analyze these processes. As it turns out, the main difficulty concerns the evaluation of rescattering effects among the pions in the final state. In particular, the lowest resonance occurring in QCD, the \(f_0(500)\), strongly amplifies the final state interaction in the Swave with \(I=0\). For this reason, the first few terms of the chiral pertubation series do not provide a good description of the momentum dependence of the amplitude, even if the oneloop representation [11] is extended to two loops [12]. We will discuss the limitations of the effective theory in the present case in Sect. 6. Dispersion relations, on the other hand, are perfectly suited to evaluate rescattering effects to all orders [13, 14, 15]. They express the amplitude in terms of a few subtraction constants, which play a role analogous to the lowenergy constants (LEC) of \(\chi \)PT. Those relevant for the momentum dependence of the amplitude can be determined very well on the basis of the experimental information on the Dalitz plot distribution. Theory is needed only for the analogs of those LECs that describe the dependence on the quark masses.
 1.
Until recently, the dispersive analyses relied on a rather crude input for the \(\pi \pi \) phase shifts, which is the essential ingredient in the dispersive calculation. Today a much more accurate representation for this amplitude is available [16, 17].
 2.
Improved calculations of the electromagnetic effects in this decay are available [18] and it is impossible to use these in combination with old dispersive calculations.
 3.
There have been recent, more accurate experimental measurements of the Dalitz plot in the charged channel [19, 20, 21, 22], which challenge the theory to correctly describe this momentum dependence.
 4.
The experimental information concerning the momentum dependence in the neutral channel also improved very significantly [23, 24, 25, 26], but represents a theoretical puzzle, because Chiral Perturbation Theory does not predict the slope correctly, in fact, not even the sign.
 (i)

obtaining numerical solutions of the integral equations which follow from the dispersion relations;
 (ii)

the dispersion relations are analyzed in the isospin limit – isospin breaking effects must be accounted for;
 (iii)

formulate and impose the constraints that follow from the fact that the particles involved in this decay are Nambu–Goldstone bosons of a hidden approximate symmetry.
The plan of the paper is as follows. We set up our dispersive framework in Sect. 2 and review \(\chi \)PT calculations and predictions on this process in Sect. 3. Our dispersive analysis is performed in the isospin limit – the approach used to account for isospin breaking effects is discussed in Sect. 4. In Sect. 5, we describe our fits to the KLOE measurements of the Dalitz plot for \(\eta \rightarrow \pi ^+\pi ^ \pi ^0\) and discuss the importance of the theoretical constraints in this context. The results of the dispersive analysis are compared with the \(\chi \)PT twoloop representation of the decay amplitude in Sect. 6, whereas, in Sect. 7, we analyze the consequences for the decay \(\eta \rightarrow 3\pi ^0\). In Sect. 8, the results are compared with the recent update of the MAMI data on this decay [25]. Sect. 9 discusses our determination of the kaon mass difference in QCD and of the quark mass ratios Q and \(m_u/m_d\). Finally, in Sect. 10, we compare our analysis with related work. Our conclusions in Sect. 11 are followed by a number of appendices containing details of our calculation.
2 Theoretical framework
2.1 Isospin
2.2 Branch cuts, discontinuities
The consequences of causality and unitarity for transitions with three particles in the final state were investigated long ago [30, 31, 32, 33, 34] and many papers concerning the decays \(K\rightarrow 3\pi \) and \(\eta \rightarrow 3\pi \) have appeared since then. In particular, as shown in [13, 14, 15, 35], the final state interaction can reliably be accounted for with dispersion relations. Since the publication of these papers, the \(\pi \pi \) phase shifts have been determined to remarkable precision [16, 17, 36] and the quality of the experimental information about these decays is now also much better. Moreover, the nonrelativistic effective field theory has been set up for these transitions. The application of this method to \(K\rightarrow 3\pi \) turned out to be very successful [37, 38, 39, 40]. These developments have triggered renewed interest in theoretical studies of \(\eta \rightarrow 3\pi \) [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53].
2.3 Dispersion relations, subtractions
We expect that, in the physical region of the decay, the representations (2.17), (2.18) constitute an excellent approximation to the isospin limit of the transition amplitudes. In \(\chi \)PT, the approximation holds up to and including nexttonexttoleading order (NNLO) – in that framework, the decomposition (2.17) is referred to as the ‘reconstruction theorem’ [54].
2.4 Polynomial ambiguities
This demonstrates that the decomposition (2.17) is unique up to a fiveparameter family of polynomials. The transformations specified in (2.20), (2.21) form a Lie group, which we denote by \(G_5\). Under this group, the isospin components \(M_0(s)\), \(M_1(s)\) and \(M_2(s)\) transform in a nontrivial manner, but their sum, \(M_c(s,t,u)\) is invariant.
2.5 Elastic unitarity
We use the standard method proposed in the pioneering papers on the subject and define the angular averages by means of analytic continuation in the square of the mass of the \(\eta \). Reserving the symbol \(M_\eta \) for the physical value of the mass, we denote the corresponding complex variable by M. Starting with a real value of \(M^2\) below \(9M_\pi ^2\), where the \(\eta \) is stable, the physical mass is approached with \(M^2=M_\eta ^2+i\delta \), where \(\delta \) is positive and tends to zero. For \(\text {Re}\,M^2<9M_\pi ^2\), the integral over z in (2.26) runs over values that are in the analyticity domain of the integrand, so that the integral is meaningful as it stands. Since the integrand is an analytic function of z, the path of integration can be deformed without changing the value of the integral, as long as the path stays within the domain of analyticity. Indeed, if \(\text {Re}\,M^2\) is increased above \(9M_\pi ^2\), such a deformation is necessary to avoid the singularities of the integrand. The matter is discussed in some detail in Appendix A.
Gasser and Rusetsky [56] very recently found a more efficient method for the solution of the integral equations. Their approach relies on a formulation of these equations for complex values of the Mandelstam variables and avoids the numerical problems altogether, which are encountered in the method we are using to evaluate the angular averages and are described in Appendix A. They kindly made their numerical results for the fundamental solutions available to us prior to publication – see the ancillary files in [56]. In the vicinity of the critical points, their solutions are significantly more accurate than those obtained with our numerical procedure, while away from these points, their results offer a very welcome check. The numerical results given in the present paper are based on their fundamental solutions – some of our numerical results differ from those quoted in the letter version [3], but in all cases, the difference amounts to a small fraction of the quoted error.
Analytic continuation in the mass of the \(\eta \) fully specifies the elastic unitarity approximation used in the present work. As mentioned in Sect. 2.2, the approximation (2.17), which represents the amplitude in terms of three functions of a single variable, is valid in \(\chi \)PT, up to and including NNLO. This statement holds within the effective theory based on SU(3)\(\times \)SU(3), i.e. includes loops involving kaons or \(\eta \)mesons. Our treatment of elastic unitarity, however, only accounts for the discontinuities generated by elastic collisions among the pions and does not include intermediate states containing heavy members of the Nambu–Goldstone octet.
Albaladejo and Moussallam [48, 49] have set up a dispersive framework for the analysis of the decay \(\eta \rightarrow 3\pi \) which extends elastic unitarity to the quasielastic collisions among the members of the pseudoscalar octet. We compare our approach with theirs in Sect. 10.1. In the range of energies of interest to us and in view of the fact that we use dispersion relations with many subtractions, the polynomial approximation for the contributions from the heavy intermediate states is perfectly adequate. What is important, however, is that the singularities generated by the final state interaction among the pions are properly accounted for and we have checked that this is the case: the elastic unitarity approximation specified above does account for the pionic singularities contained in the chiral representation of the transition amplitude, up to and including two loops.
2.6 Phase shifts
2.7 Integral equations
For our method it is crucial that the dispersion relations used uniquely determine the amplitude in terms of the subtraction constants. With the form (2.16) of these relations, that is not the case, however. There, the subtraction constants are collected in the polynomials \(P_I(s)\). The problem is that the homogeneous equations obtained if these polynomials are set equal to zero admit nontrivial solutions.
Bookkeeping then shows, however, that the dispersion relation (2.16) cannot determine the solution uniquely: the asymptotic behaviour \(M_0(s)\propto s^2\) allows a cubic polynomial for \(m_0(s)\), but only a quadratic one for \(P_0(s)\). Hence the general solution involves four free parameters while the dispersion relation only contains three subtraction constants. Evidently, the phenomenon occurs because the Omnès factor \(\varOmega _0(s)\) tends to zero if s becomes large. This is the case also for \(\varOmega _1(s)\), while the solution of the dispersion relation for \(M_2(s)\) is determined uniquely by the subtraction constants.
2.8 Subtraction constants, fundamental solutions
For the phase shift parametrizations we are using, the integrands vanish above 1.7 GeV. Hence convergence is not an issue – we could use unsubtracted dispersion integrals, i.e. set \(n_I=0\) in (2.32). It is more convenient, however, to instead work with \(n_0=2\), \(n_1=1\), \(n_2=2\), for two reasons: (i) Although the manifold of solutions is exactly the same, for the solutions obtained with \(n_I=0\), the dispersion integrals are quite sensitive to the behaviour of the phase shifts above 0.8 GeV, which is poorly known – the sensitivity is compensated by a corresponding sensitivity of the subtraction constants, but the correlation leads to a clumsy error analysis. (ii) The choice is also more convenient for comparison with earlier work where the dispersion integrals were written in subtracted form.
Figure 2 shows the result for this particular fundamental solution. The comparison of the first and last panels shows that the neutral component of the solution is dominated by the contribution from \(M_0(s)\).
2.9 Taylor invariants
2.10 Nonrelativistic expansion
The LECs \(L_0, \ldots , L_3\) play a role analogous to the subtraction constants \(\alpha _0,\ldots \,\gamma _1\) of the dispersive framework, but there is a qualitative difference: while the LECs are real, the subtraction constants can be complex. Note also that the decomposition of the amplitude into isospin components is unique only up to polynomials. When comparing the components of the NREFT representation with those of dispersion theory, the polynomial ambiguities must be taken into account. This can be done with the method used when matching the dispersive and chiral representations. The polynomial ambiguities only affect the coefficients of the even powers of q. There are analogs of the Taylor invariants – suitable linear combinations of the coefficients \(m_0^{2k}, m_1^{2k}, m_2^{2k}\) – that do not depend on the choice made when decomposing the amplitude into isospin components. Four such invariants are within reach of the twoloop representation. Hence there is a unique dispersive solution with four subtraction constants that matches the generic twoloop representation in the isospin limit. Alternatively, one may compare the dispersive and nonrelativistic amplitudes in the physical region and minimize the difference between the two. We will carry this out for one particular nonrelativistic representation in Sect. 5.9.
3 Chiral perturbation theory
3.1 Current algebra, Adler zero
3.2 \(\chi \)PT to one loop
The chiral perturbation series of the transition amplitude was worked out to NLO in the framework of SU(3)\(\times \)SU(3) in [11]. In this framework, the final state interaction manifests itself through oneloop graphs involving pions as well as kaons or \(\eta \)mesons. The amplitude can be expressed in terms of the meson masses \(M_\pi \), \(M_K\), \(M_\eta \), the decay constants \(F_\pi \), \(F_K\) and the lowenergy constant \(L_3\). We use the numerical values \(F_\pi =92.28(9) \text {MeV}\) [66], \(F_K/F_\pi =1.193(3)\) [27] and rely on the recently improved determination of \(L_3\) from \(K_{\ell 4}\) decay, \(L_3=2.63(46)\cdot 10^{3}\) [67], so that the oneloop representation does not contain any unknowns.
While the dispersive representation yields an accurate description of the momentum dependence in the entire range from \(s=0\) to the physical region and even beyond, the truncated chiral expansion is useful only at small values of s, where it can be characterized by the lowest few coefficients of the Taylor series (2.36). The contributions from the loop graphs are determined by the masses of the NambuGoldstone bosons and the pion decay constant. The tree graphs, on the other hand, yield polynomials of up to \(O(p^4)\) in the momenta. The coefficients of these polynomials are in onetoone correspondence with the Taylor coefficients \(A_0\), \(B_0\), \(C_0\), \(A_1\), \(B_1\), \(A_2\), \(B_2\), \(C_2\). Together with \(F_\pi \), these coefficients thus uniquely determine the oneloop representation.
The constants \(H_0,H_1,H_2,H_3\) contain the essence of the oneloop representation: if they are known, the transition amplitude is uniquely determined by unitarity, to NLO of the chiral expansion (an explicit proof of this statement can be found in Appendix B). In this sense, the momentum dependence of the chiral representation is not of interest – dispersion theory provides better control over that. The general principles that underly dispersion theory, however, do not determine the subtraction constants. That is where \(\chi \)PT can offer useful information.
Remarkably, despite the fact that the \(\eta \) undergoes mixing with the \(\eta ^\prime \), the formula (3.4) only contains \(M_\eta \), while \(M_{\eta '}\) does not occur. The role played by the \(\eta '\) in the lowenergy structure of QCD is well understood. It can be studied in a systematic manner by invoking the large \(N_c\) limit, where the \(\eta ^\prime \) becomes massless and can be treated on the same footing as the Nambu–Goldstone bosons [68]. This framework gives a good understanding of the size of the LEC \(L_7\), which determines the deviation from the Gell–Mann–Okubo formula and enters the lowenergy theorem via the term \(\varDelta _{\mathrm {GMO}}\). Indeed, as shown in Ref. [69], the contribution from this term in the low energy theorem (3.4) fully accounts for the effects generated by \(\eta \)\(\eta '\)mixing at \(O(m_{\mathrm {quark}})\) – it would be wrong to supplement \(\chi \)PT with an extra wheel to account for \(\eta \)\(\eta '\)mixing.
Note that the dependence on the decay constants is suppressed by a factor of \(M_\pi ^2\) – if the two lightest quarks are taken massless, \(H_0\) is fully determined by the masses of the Nambu–Goldstone bosons, up to NNLO contributions. At the physical values of the masses and decay constants, the term proportional to \(\varDelta _{\mathrm {F}}\) amounts to 0.036. The contribution from the chiral logarithms is also small: \(\mathrm {chilogs}=0.037\). The dominating contribution stems from the term \(\varDelta _{\mathrm {GMO}}\) and amounts to 0.103. The net result at one loop reads: \(H_0^{\mathrm {NLO}}=1.176\).
3.3 \(\chi \)PT to two loops
Bijnens and Ghorbani [12] have worked out the chiral perturbation series of the transition amplitude to NNLO. The amplitude retains the form (2.17), but the isospin components \(M_0(s)\), \(M_1(s)\), \(M_2(s)\) pick up additional contributions, which can be expressed in terms of the meson masses and the LECs that occur in the effective Lagrangian. As discussed above, elastic unitarity determines the oneloop representation in terms of the tree graph amplitude up to a polynomial, which can be characterized by the four Taylor invariants \(H_0,\ldots ,H_3\). The situation at NNLO is analogous: elastic unitarity determines the amplitude in terms of the oneloop representation up to a polynomial. Since the amplitude now includes terms of \(O(p^6)\), the polynomial is of higher degree and now contains six independent terms rather than four: \(p_0+p_1\,s+ p_2\, s^2+p_3\,\tau ^2+p_4\,s^3+p_5\, s\,\tau ^2\), with \(\tau \equiv tu\). Hence there are six combinations of Taylor coefficients that are independent of the choice of the decomposition. At two loops, all of the six Taylor invariants \(K_0, \ldots , K_5\) are needed to characterize the representation.
The invariants \(K_0, \ldots , K_5\) can also be used to characterize the solutions of our system of integral equations. The Taylor coefficients of the dispersive representation are given by linear combinations of the six subtraction constants and uniquely determined by these. Knowledge of the subtraction constants thus fixes the Taylor invariants \(K_0, \ldots , K_5\) and vice versa: the degrees of freedom inherent in the twoloop representation are in onetoone correspondence with the degrees of freedom occurring in our integral equations.
3.4 Imaginary parts at two loops
3.5 Matching the dispersive and oneloop representations
At one loop, the Taylor invariants are known within rather small uncertainties. We now work out the dispersive representation that matches the oneloop representation in the sense that the behaviour of the functions \(M_0(s)\), \(M_1(s)\), \(M_2(s)\) at small values of s is the same: the dispersive solution that possesses the same Taylor invariants. More precisely, as we are working with real subtraction constants, we can match only the real parts of the Taylor invariants.
The imaginary parts of the chiral representation vanish for \(s<4M_\pi ^2\). Those of the dispersive representation are different from zero in that region, but are very small there because they exclusively arise from the crossed channels. Above threshold, however, the oneloop representation strongly underestimates the imaginary parts. It is not difficult to see why that is so: the dominating contribution to \(\text {Im}\,M_0\) is the one proportional to \(\sin ^2\!\delta _0\). At oneloop, the representation for the \(\pi \pi \) phase shifts enters at LO, where the scattering length of the \(I=0\) Swave is given by Weinberg’s current algebra result [71]: \(a_0^{\mathrm {LO}}=0.16\) in pion mass units, below the prediction \(a_0=0.220(5)\) [16] by the factor 1.38. The oneloop representation underestimates the imaginary part of \(M_0\) roughly by the square of this factor.
3.6 Adler zero at one loop
Figure 4 shows that the final state interaction generates curvature, but does not significantly affect the position of the Adler zero: at LO, it occurs at \(s_A=\frac{4}{3}M_\pi ^2\), while at one loop, the real part along the line \(s=u\) vanishes at \(s_A=1.40 M_\pi ^2\). Note that the behaviour of the amplitude in the vicinity of the zero involves large values of t: for \(s=u\simeq \frac{4}{3}M_\pi ^2\), we get \(t_A\simeq 15.7\, M_\pi ^2\), i.e. \(\sqrt{t_A}\simeq 550\,\text {MeV}\). As far as the isospin components \(M_0(s)\) and \(M_1(s)\) are concerned, only their behaviour at small arguments of order \(s\simeq s_A\) matters, but \(M_2(s)\) is needed for \(s\simeq t_A\) as well as for \(s\simeq s_A\). Adler’s lowenergy theorem thus concerns the behaviour of the amplitude not only at small values of s and u, but also in the vicinity of \(t=t_A\). In particular, the contributions from kaon loops to \(M_2(t_A)\) are relevant. The fact that these do not move the position of the zero far away from the place where it occurs in current algebra shows that they do obey the constraints imposed by chiral symmetry.
3.7 Neutral decay mode
As noted above, in connection with the imaginary parts, the chiral representation only offers a crude, semiquantitative description of the final state interaction. The comparison of the LO and NLO representations for \(M_n(s)\) shows that, at the center of the Dalitz plot, the effects generated by this interaction are large: the oneloop contributions modify the tree level amplitude by more than 50%. We conclude that the truncated chiral series does not have the accuracy required to make a meaningful statement about the slope.
4 Isospin breaking corrections
The decay \(\eta \rightarrow 3\pi \) violates isospin conservation. As discussed in Sect. 2.1, the dominating contribution to the transition amplitude can be represented in the form (2.4), as a product of the factor \((M_{K^0}^2M_{K^+}^2)_{\mathrm {QCD}}\) which breaks isospin symmetry and the factor \(M_c(s,t,u)\) which is invariant under isospin rotations. The basic properties of the amplitude \(M_c(s,t,u)\) were discussed in the preceding sections – we now turn to the remainder, which is of order \(O[e^2,(m_um_d)^2]\). While the effects due to \((m_u m_d)^2\) are tiny, those from the electromagnetic interaction must properly be taken into account when comparing theory with experiment. In particular, the e.m. selfenergy of the charged pion generates a mass difference to the neutral pion which affects the phase space integrals quite significantly.
In the literature, the corrections of order \(O[e^2,(m_um_d)^2]\) have been calculated by several groups, to different levels of accuracy – i.e. to different orders of the expansion in the isospin breaking parameters. In the present paper we will rely on the work of Ditsche, Kubis and Meißner (DKM) [18], who evaluated the transition amplitude within the effective theory relevant for QCD+QED, to first nonleading order of the chiral expansion and to order \(e^2\) in the electromagnetic interaction, with unequal up and down quark masses and in the presence of real as well as virtual photons. An earlier calculation by Baur, Kambor and Wyler [72], performed in the same framework, did not include effects of order \(e^2(m_um_d)\). These are of second order in isospin breaking and were deemed to be negligible. Ditsche, Kubis and Meißner, however, correctly observe that while terms of order \((m_um_d)^2\) are indeed negligible, there are a number of effects which scale as \(e^2(m_um_d)\) and should be taken into account, like real and virtual photon corrections to the purely strong amplitude, and also, and most importantly, effects related to the pion mass difference, which are in particular responsible for the presence of cusps in the Dalitz plot of \(\eta \rightarrow 3 \pi ^0\).
Isospin breaking also affects the phase shifts of \(\pi \pi \) scattering. We take these from the solution of the Roy equations reported in [16], which is done in the isospin limit. Our dispersive analysis is also carried out in that limit. In order to correct our results for isospin breaking effects, we make use of Chiral Perturbation Theory. We first study the effects of isospin breaking in this framework, comparing the representation of Ditsche, Kubis and Meißner [18], which does account for isospin breaking, with the one of Gasser and Leutwyler [11], which concerns the isospin limit. Our estimates for the size of the isospin breaking effects in the physical amplitudes rely on the assumption that these effects factorize, at least approximately. The branching ratio \(B=\varGamma _{\eta \rightarrow 3\pi ^0}/\varGamma _{\eta \rightarrow \pi ^+\pi ^\pi ^0}\) provides a strong test of the assumptions that underly our analysis.
4.1 Kinematics
The Adler zero discussed in Sect. 3.6 occurs along the line \(s_c=u_c\), which is indicated as a dashed line, but the relevant value of \(s_c\) is around \(\frac{4}{3}M_\pi ^2\), which is outside the range shown in this figure. The symmetry with respect to \(t\leftrightarrow u\) implies that an Adler zero also occurs along the line \(s_c=t_c\), at the same value of \(s_c\).
The amplitude relevant for the decay into \(3\pi ^0\) is invariant under the exchange of the three Mandelstam variables also in the presence of isospin breaking. Each of the three channels contains a pair of branch points at \(4M_{\pi ^0}^2\) and \(4M_{\pi ^+}^2\). The right panel of Fig. 6 shows that the three straight lines with \(s_n\), \(t_n\) or \(u_n\) equal to \(4M_{\pi ^0}^2\) touch the boundary of the physical region, while the other three branch cuts run across this region and manifest themselves as cusps in the Dalitz plot distribution. The relations between \(s_n\), \(\tau _n\) and the variables \(X_n,Y_n\) used in the figure are obtained from (4.2) by replacing \(M_{\pi ^+}\) with \(M_{\pi ^0}\), while those among the variables s, \(\tau \) and X, Y of the isospin symmetric world are reached with the substitutions \(M_{\pi ^+}\rightarrow M_\pi \), \(M_{\pi ^0}\rightarrow M_\pi \).
4.2 Isospin breaking at one loop
Remarkably, despite these additional singularities, the oneloop representation obeys elastic unitarity also in the presence of photons: the amplitude \(M^{\mathrm {DKM}}_c(s_c,t_c,u_c)\) can be expressed in terms of three functions of a single variable according to (2.17) and \(M^{\mathrm {DKM}}_n(s_n,t_n,u_n)\) retains the form (2.18). Only the explicit expressions for the components are modified and the relation (2.19) between the components relevant for the charged and neutral decay modes is lost. As it is the case without isospin breaking, for the charged decay mode one function of a single variable is needed for the schannel (Swave) and two functions (Swave and Pwave) for the tand uchannels. For the neutral decay mode, a single function \(M_n^{\mathrm {DKM}}(s)\) again suffices (Swave), but it now differs from the combination \(M_0^{\mathrm {DKM}}(s)+\frac{4}{3}M_2^{\mathrm {DKM}}(s)\) of amplitudes relevant for the charged mode.
The decay is necessarily accompanied by the emission of real photons and the comparison with the data must properly account for that. The main features of the phenomenon are universal and are thoroughly discussed in the literature [73]. Up to and including \(O(e^2)\), the rate of the decay \(\eta \rightarrow \pi ^+\pi ^\pi ^0\) contains two contributions, one from the square of the amplitude relevant for the decay without real photons in the final state, the other from the square of the amplitude for the emission of one real photon. It is wellknown that both of these contributions are infrared divergent and that, in the sum of the two, the infinities cancel. The only physical remnant of the infrared divergences is that the probability for generating a real photon depends logarithmically on the upper limit set for the energy of the emitted photon. In the comparison with the data, the maximal photon energy in the rest frame of the \(\eta \), which is denoted by \(E_{\mathrm {max}}\), is determined by the experimental resolution.
4.3 Selfenergy effects
In the decay \(\eta \rightarrow \pi ^+\pi ^\pi ^0\), the selfenergy of the charged pion directly affects the kinematics, as it is relevant for the size of the physical region and for the value of \(s_c+t_c+u_c\). The selfenergy of the charged pion increases its mass and hence reduces the phase space available in the charged decay mode – since phase space is small, this makes a significant difference, which must be accounted for. In early work on \(\eta \)decay, this was done only very crudely: in the calculation of the decay rate, the square of the isospin symmetric amplitude was simply integrated over the physical phase space rather than the isospin symmetric one.
The oneloop representation allows us to separate the selfenergy effects from the remaining contributions generated by the electromagnetic interaction: the amplitude can be evaluated at the physical masses of the mesons even if e is set equal to zero. The left panel of Fig. 7 depicts the square of the ratio \(K_c^e=M_c^{\mathrm {DKM}}(s,t,u)/M_c^{\mathrm {DKM}}(s,t,u)_{e=0}\), along the lines \(s=u\) and \(t=u\). It shows that the remaining electromagnetic contributions vary in the narrow range \(0.997< K_c^e^2< 1.022\). As seen in the right panel, the square of the correction factor \(K_n^e=M_n^{\mathrm {DKM}}(s,t,u)/M_n^{\mathrm {DKM}}(s,t,u)_{e=0}\) relevant for the neutral channel is also of the order of 1%, but nearly constant over the entire physical region: \(0.98757< K_n^e^2< 0.98765\). This implies that in the Dalitz plot distribution of the decay \(\eta \rightarrow 3\pi ^0\), the corrections generated by the electromagnetic interaction are totally dominated by the selfenergy effects.
4.4 Kinematic map for \(\eta \rightarrow \pi ^+\pi ^\pi ^0\)
The extension to the decay \(\eta \rightarrow 3\pi ^0\) meets with a technical problem: the map obtained by applying the above construction to the corresponding transition amplitude does take the physical region of the neutral Dalitz plot onto the isospin symmetric one, but does not respect Bose statistics, because it does not treat s on equal footing with t and u. As shown in Appendix C, this shortcoming is easily cured – the kinematic map specified in (C.1)–(C.5) does preserve the symmetry under exchange of s, tand u as well as the boundary and the center of the physical region. In the following, we use this map to analyze isospin breaking effects in the neutral channel.
4.5 Applying the kinematic map to the oneloop representation
4.6 Correcting the dispersive solutions for isospin breaking effects
 (i)
We first apply the kinematic map, replacing the solutions \(M_c\), \(M_n\) of our integral equations by the amplitudes \(\tilde{M}_c\), \(\tilde{M}_n\). In the charged channel, the explicit expression reads \(\tilde{M}_c(s_c,t_c,u_s)\equiv M_c(\tilde{s}_c,\tilde{t}_c,\tilde{u}_c)\), where \(\tilde{s}_c\), \(\tilde{t}_c\), \(\tilde{u}_c\) are specified in (4.13). Since this operation takes the constraint \(s_c+t_c+u_c=M_\eta ^2+2M_{\pi ^+}^2+M_{\pi ^0}^2\) into \(\tilde{s}_c+\tilde{t}_c+\tilde{u}_c=M_\eta ^2+3M_\pi ^2\), it ensures that the solutions \(M_c(s,t,u)\) are used only for values of the Mandelstam variables that obey \(s+t+u=M_\eta ^2+3M_\pi ^2\) – this is where they are uniquely defined. Moreover, the map takes center and boundary of the physical Dalitz plot into center and boundary of the isospin symmetric phase space. Analogous statements hold for the neutral channel – the kinematic map relevant in that case is specified in Appendix C.
 (ii)We assume that the remaining isospin breaking effects can be estimated with the oneloop representation and approximate the physical amplitude with$$\begin{aligned} M^{\mathrm {phys}}_c(s,t,u)= & {} K_c(s,t,u)\tilde{M}_c(s,t,u),\nonumber \\ M^{\mathrm {phys}}_n(s,t,u)= & {} K_n(s,t,u)\tilde{M}_n(s,t,u). \end{aligned}$$(4.18)
Experimental values of the Dalitz plot parameters of \(\eta \rightarrow \pi ^+ \pi ^ \pi ^0\). The two entries for KLOE(2016) correspond to their fits with 4 and 5 free coefficients, respectively (fit#3 and fit#4)
Experiment  \(a\)  \(b\cdot 10\)  \(d \cdot 10^2\)  \(f \cdot 10\)  \(g \cdot 10^2\) 

Gormley (1970) [74]  1.17(2)  2.1(3)  6(4)  –  – 
Layter (1973) [75]  1.080(14)  0.34(27)  4.6(3.1)  –  – 
CBarrel (1998) [76]  1.22(7)  2.2(1.1)  6(fixed)  –  – 
KLOE (2008) [19]  \(1.090(5)(^{+8}_{19})\)  1.24(6)(10)  \(5.7(6)(^{+7}_{16})\)  1.4(1)(2)  – 
WASA (2014) [20]  1.144(18)  2.19(19)(37)  8.6(1.8)(1.8)  1.15(37)  – 
BESIII (2015) [21]  1.128(15)(8)  1.53(17)(4)  8.5(1.6)(9)  1.73(28)(21)  – 
\(\hbox {KLOE}^a\) (2016) [22]  1.104(3)  1.420(29)  7.26(27)  1.54(6)  0 
\(\hbox {KLOE}^b\) (2016) [22]  1.095(3)  1.454(30)  8.11(33)  1.41(7)  \(4.4(9)\) 
While in the neutral channel, the residual corrections affect the Dalitz plot distribution only very little, the momentum dependence of the amplitude relevant for the charged decay mode is not properly accounted for by the kinematic map. The contribution from the triangle graph is singular at \(s=4M_{\pi ^+}^2\), but we have removed that singularity by subtracting the Coulomb pole specified in (4.5). As shown in Appendix B.4, the spike occurring there does not arise from the triangle graph, but from the interference between the contributions generated by the branch cuts in the schannel (final state interaction among the pairs \(\pi ^+\pi ^\) and \(\pi ^0\pi ^0\)) with those in the t and uchannels due to \(\pi ^\pm \pi ^0\) pairs. We assume that the oneloop approximation does provide a decent estimate for the distortion of the discontinuities generated by the electromagnetic interaction and expect that multiplying the amplitudes of the charged and neutral decay modes with the ratios \(K_c=M_c^{\mathrm {DKM}}/M_c^{\mathrm {GL}}\) and \(K_n=M_n^{\mathrm {DKM}}/M_n^{\mathrm {GL}}\) yields a good approximation of the physical distribution. This implies, in particular, that we are accounting for the cusps that run through the physical region of the decay \(\eta \rightarrow 3\pi ^0\) only in oneloop approximation. We will compare the resulting parameter free prediction for the Dalitz plot distribution of the decay \(\eta \rightarrow 3\pi ^0\) with experiment in Sect. 7 – this comparison offers another good check on the internal consistency of our framework.
5 Dalitz plot distribution for \(\eta \rightarrow \pi ^+\pi ^\pi ^0\)
5.1 Experiment

In view of the much larger statistics, KLOE data dominate any common fit; the inclusion of the WASA data barely shifts the parameters and any outcome of the fit.

The compatibility among the two data sets is marginal: a common fit (with six subtraction constants, i.e. five fit parameters) gives \(\chi ^2_{\mathrm {K}}=371\) for 371 data points and \(\chi ^2_{\mathrm {W}}=84\) for 59 data points.

Fitting WASA data by themselves gives a much better \(\chi ^2\): \(\chi ^2_{\mathrm {W}}=49\), but this would be totally incompatible with KLOE, as the corresponding \(\chi ^2\) is huge.
5.2 Fitting the KLOE distribution for \(\mathbf \eta \rightarrow \pi ^+\pi ^\pi ^0\)
Since the normalization of the amplitude drops out in the Dalitz plot distribution, the value of \(H_0\) is irrelevant – the discrepancy function is independent thereof. We fix it at the central value obtained at one loop, \(H_0=1.176\). The relation (2.38) between \(H_0\equiv K_0\) and the subtraction constants thus ties \(\alpha _0\) to \(\beta _0\) according to \(\alpha _0=0.85940.08736\,\beta _0\), so that \(\chi _{\mathrm {K}}^2\) contains six independent real parameters: \(\beta _0\), \(\gamma _0\), \(\delta _0\), \( \beta _1\), \(\gamma _1\), \(\varLambda _{\mathrm {K}}\).
5.3 Dispersive fits to the KLOE data without theoretical constraints
The left panel of Fig. 9 corresponds to the one on the left of Fig. 8: \(X_c=0\) implies \(t_c=u_c\). While Fig. 8 concerns the correction factor \(K_c^2\) used to account for some of the isospin breaking effects, we are now considering the Dalitz plot distribution of the full amplitude. The comparison shows that the spike occurring in \(K_c^2\) near \(s_c=4M_{\pi ^+}^2\) also manifests itself in the Dalitz plot distribution near \(Y_c=0.895\), but in rather modest form. For the reasons given in Sect. 4.3, the spikes in \(K_c^2\) and in \(D_c\) are of opposite sign. A dedicated experimental study is required to resolve the structure in the vicinity of \(s_c=4 M_{\pi ^+}^2\).
5.4 Theoretical constraints
Comparison of the matching solution \(\mathrm {fit\chi _4}\) with fits to the KLOE Dalitz plot distribution for \(\eta \rightarrow \pi ^+\pi ^\pi ^0\). The presence or absence of the label \(\chi \) indicates whether or not the theoretical discrepancy (5.6) is included in the minimization procedure and the index specifies whether four, five, or six subtraction constants are taken different from zero (in the chosen normalization, \(\alpha _0\) is tied to \(\beta _0\) according to \(\alpha _0=0.85940.08736\,\beta _0\)). For fits obtained by dropping either the experimental or the theoretical part of the discrepancy function, the values of \(\chi ^2_{\mathrm {K}}\) or \(\chi ^2_{\mathrm {\chi }}\) are put in brackets
\(\beta _0\)  \(\gamma _0\)  \(\delta _0\)  \(\beta _1\)  \(\gamma _1\)  \(\chi ^2_{\mathrm {K}}\)  \(\chi ^2_{\mathrm {th}}\)  

\(\mathrm {fit}\chi _4\)  16.9(1.7)  \(29.5(10.6)\)  –  6.6(2.3)  –  (801)  0 
\(\mathrm {fitK}_4\)  17.6(7)  \(35.2(7.2)\)  –  5.9(8)  –  390  (0.67) 
\(\mathrm {fitK}\chi _4\)  17.5(6)  \(35.0(7.2)\)  –  6.0(7)  –  390  0.59 
\(\mathrm {fitK}_5\)  13.3(2.2)  23.8(26.9)  \(147(66)\)  13.4(3.6)  –  379  (46) 
\(\mathrm {fitK}\chi _5\)  16.6(8)  \(20.1(9.1)\)  \(38(17)\)  7.8(1.1)  –  384  1.43 
\(\mathrm {fitK}_6\)  \(20.0(10.2)\)  \(35.6(89.3)\)  \(75(91)\)  77(19)  \(308(88)\)  371  (1005) 
\(\mathrm {fitK}\chi _6\)  16.2(1.2)  \(20.8(10.1)\)  \(38(17)\)  8.5(2.2)  \(3.8(10.7\))  384  1.47 
Figure 10 displays the behaviour of the real parts belonging to the various dispersive solutions all the way down to \(s=0\) (while the curves for the Dalitz plot distribution shown in Fig. 9 account for the corrections due to isospin breaking, those for ReM represent the isospin symmetric solutions as they are). Remarkably, in the entire range shown, \(\mathrm {fitK}\chi _6\) runs close to fit\(\chi _4\), the matching solution specified in Sect. 3.5.
In addition to the representations fit\(\chi _4\), \(\mathrm {fitK}_4\) and \(\mathrm {fitK}\chi _6\) we discussed above, Fig. 10 shows a fourth solution, \(\mathrm {fitK}_6\). The only difference between this solution and \(\mathrm {fitK}_4\) is that \(\delta _0\) and \(\gamma _1\) are not set equal to 0, but are treated as free parameters. Accordingly, this fit follows the data even more closely: \(\chi _{\mathrm {K}}^2=371\) for 371 data points and 6 free parameters. Figure 9 shows that, in the physical region, the Dalitz plot distributions belonging to \(\mathrm {fitK}_4\) and \(\mathrm {fitK}_6\) are nearly the same. Outside the physical region, however, \(\mathrm {fitK}_6\) goes astray: this solution of our system of integral equations is not acceptable, because it does not have an Adler zero at all. The clash with chiral symmetry also manifests itself in the Taylor invariants: \(\mathrm {fitK}_6\) yields \(\text {Re}\,h_3^{\mathrm {K_5}}=59.8\), for instance, which differs from the theoretical estimate \(h_3=6.3(2.0)\) in (3.7) by 28 \(\sigma \). This indicates that – with six subtraction constants – there is too much freedom in the space of solutions for the experimental information about the Dalitz plot distribution to control the behaviour of the transition amplitude outside the physical region.
The fact that \(\mathrm {fitK}\chi _6\) does have an Adler zero at \(s_A=1.39\,M_\pi ^2\) shows that the theoretical constraints do provide the missing information: the only difference between \(\mathrm {fitK}_6\) and \(\mathrm {fitK}\chi _6\) is that the latter accounts for these while the former does not. The theoretical constraints barely matter in the physical region, but play an important role in the extrapolation to small values of s. The properties of the amplitude at small values of s are essential, because theory is needed to determine the normalization of the amplitude. Since the relevant Taylor invariant, \(H_0\), represents a linear combination of the subtraction constants \(\alpha _0\) and \(\beta _0\), it concerns the value and the first derivative of the component \(M_0(s)\) at \(s=0\).
5.5 Error analysis
The uncertainties in our results are dominated by the statistical errors. These are determined by the behaviour of the discrepancy function in the vicinity of the minimum. In connection with the fits to the measured Dalitz plot distribution of the charged decay mode, the normalization constant \(H_0\) is irrelevant – we keep it fixed at the value found at one loop. Also, since none of the observables of interest in the present context depends on \(\varLambda _{\mathrm {K}}\), we fix this parameter at the minimum, which is nearly the same for all fits: \(\varLambda _{\mathrm {K}}\simeq 0.938\). The discrepancy function \(\chi ^2_{\mathrm {tot}}\) then depends on five independent real variables, which can, for instance, be identified with \(\beta _0\), \(\gamma _0\), \(\delta _0\), \(\beta _1\), \(\gamma _1\). We rely on the Gaussian approximation, which exploits the fact that, in the vicinity of the minimum, the discrepancy function can be approximated by the truncated Taylor series in all five variables. The calculation is described in detail in Appendix D.
The uncertainties inherent in the input used for the \(\pi \pi \) phase shifts must also be accounted for. These were discussed in Sect. 2.6. We have worked out the response of the dispersive representation to variations in the Roy solutions of [16], not only below 800 MeV where the uncertainties are small, but also at higher energies where dispersion theory does not provide strong constraints – for details see Appendix E. The resulting uncertainties in the subtraction constants are small compared to the Gaussian errors discussed above, except for \(\gamma _0\): this term is relatively sensitive to the high energy tail of the dispersion integrals – the corresponding uncertainty is comparable to the Gaussian error.
The kinematic map we are using to embed the isospin symmetric dispersive representation in the physical world accounts for the effects due to the mass difference between the charged and neutral pions only rather crudely. We rely on the oneloop approximation of Ditsche, Kubis and Meißner [18] to correct for all other effects that (i) are generated by the e.m. interaction and (ii) are not taken care of when applying radiative corrections to the data. We consider the difference between our results and those obtained by neglecting the isospin breaking effects altogether and estimate the uncertainty of our treatment of these effects at 30% of that difference.
The errors listed in Table 2 are obtained by adding the Gaussian errors, those from the \(\pi \pi \) phase shifts and those related to isospin breaking in quadrature,
5.6 Number of subtraction constants, significance of theoretical constraints
The number of subtraction constants occurring in the dispersive form of the chiral representation increases with the order: four subtraction constants at NLO, six at NNLO, etc. We impose theoretical constraints based on the NLO representation of \(\chi \)PT – four subtraction constants are a suitable choice in this context, but our framework does leave room for two further subtractions. In the present section, we compare the solutions of our integral equations obtained with four, five or six subtraction constants and discuss the role of the theoretical constraints.
The approach in [43] differs from ours as it relies on the NNLO representation of \(\chi \)PT [12]. Six subtraction constants are used ab initio to impose the theoretical constraints. In particular, the representation obtained in this way invokes the estimates for the LECs obtained from resonance saturation in the scalar channel – our analysis avoids the use of such estimates. For a comparison of their results with ours, we refer to Sect. 10.
The first two lines in Table 2 represent two extremes: while fit\(\chi _4\) only relies on theory, \(\hbox {fitK}_4\) only relies on experiment. For a detailed comparison of these two solutions, we refer to the end of Sect. 5.3. Table 2 shows that the central values of all of the subtraction constants of \(\hbox {fitK}_4\) are within the uncertainty range of fit\(\chi _4\) and vice versa. In other words, the fit to the data automatically satisfies the theoretical constraints. This can also be seen in the value \(\chi ^2_{\mathrm {th}}=0.67\) obtained with \(\hbox {fitK}_4\): the central values of \(h_1\), \(h_2\), \(h_3\) obtained from the KLOE data are all in the predicted range.
The entries for \(\chi _{\mathrm {K}}^2\), on the other hand, show that fit\(\chi _4\) differs strongly from \(\hbox {fitK}_4\): while the latter represents an excellent fit of the 371 data points with \(\chi _{\mathrm {K}}^2=390\), the former yields a value of \(\chi ^2_{\mathrm {K}}\) that is more than twice as large. Superficially, this may give the impression that the matching solution is ruled out by experiment, but this is by no means the case. In view of the uncertainties attached to the predictions for \(h_1\), \(h_2\), \(h_3\), the matching procedure leads to an entire family of solutions – fit\(\chi _4\) merely represents the central one of these. The very fact that \(\hbox {fitK}_4\) is a member of this family shows that the KLOE data on the Dalitz plot distribution of \(\eta \rightarrow \pi ^+\pi ^\pi ^0\) confirm the theoretical estimates based on the assumption that the strong interaction possesses a hidden approximate symmetry.
In the derivation of fitK\(\chi _4\), both the KLOE data and the theoretical constraints are made use of. The comparison with \(\hbox {fitK}_4\) shows, however, that this barely makes any difference. In particular, the values of \(\chi _{\mathrm {th}}^2\) and \(\chi _{\mathrm {K}}^2\) obtained with these two fits are nearly the same.
The solution \(\hbox {fitK}_5\) differs from \(\hbox {fitK}_4\) in that the subtraction constant \(\delta _0\) is not set equal to zero, but is treated as a free parameter. Table 2 shows that the solution then changes quite drastically: (1) the minimum occurs at a value of \(\delta _0\) that differs from zero by about two standard deviations, (2) the quantities \(\beta _0\), \(\gamma _0\) and \(\beta _1\) are also pushed outside the range found with \(\hbox {fitK}_4\) or fitK\(\chi _4\) and (3) the value of \(\chi _{\mathrm {th}}^2\) becomes very large. This shows that \(\hbox {fitK}_5\) very strongly violates the theoretical constraints. The situation is similar to the one encountered with \(\hbox {fitK}_6\) in Sect. 5.4: the data are not accurate enough to pin down more than four parameters. Both \(\hbox {fitK}_5\) and \(\hbox {fitK}_6\) must be discarded – they represent unphysical solutions of our integral equations.
The theoretical constraints domesticate the manifold of solutions if more than four subtraction constants are treated as free parameters. In fact, it does then not make much of a difference whether five or six subtraction constants are treated as free parameters. In either case, the solution is consistent with the theoretical constraints and the common subtraction constants agree within errors. Moreover, fitK\(\chi _6\), which treats \(\gamma _1\) as a free parameter, yields a result with a broad uncertainty range – the value \(\gamma _1=0\) that corresponds to fitK\(\chi _5\) is within that range. The discrepancy function \(\chi _{\mathrm {th}}^2\) punishes strong deviations from the values of the Taylor invariants obtained at one loop. The fit yields \( \text {Re}\,h_1^{\mathrm {K\chi _6}}=4.52(14) \), \(\text {Re}\,h_2^{\mathrm {K\chi _6}}=21.7(4.3) \), \(\text {Re}\,h_3^{\mathrm {K\chi _6}}= 7.3(1.7)\). The comparison with (3.7) shows that, within errors, these numbers are consistent with the estimates based on \(\chi \)PT.
The shape of the Dalitz plot distribution is tightly constrained by experiment. Indeed, Fig. 9 shows that for the behaviour in the physical region, it barely makes a difference whether four or six subtraction constants are treated as free parameters. The numbers for \(\chi _{\mathrm {K}}^2\) in Table 2 confirm this: the fits fitK\(\chi _4\), fitK\(\chi _5\) and fitK\(\chi _6\) all describe the data very well. We conclude that, as far as the momentum dependence in the physical region is concerned, the description of the observed behaviour does not require more than four subtraction constants.
Value of the amplitude at the center of the Dalitz plot: sensitivity to the number of subtraction constants
\(\hbox {fitK}_4\)  fitK\(\chi _4\)  fitK\(\chi _5\)  fitK\(\chi _6\)  

\(N_1\)  1.371(22)  1.372(22)  1.499(64)  1.494(66) 
To discuss the implications of this result, we consider the correlation between \(N_1\) and the slope a of the Dalitz plot distribution at the center, that is, the term linear in \(Y_c\) in (5.1). Figure 11 shows that it makes a significant difference whether the subtraction constant \(\delta _0\) is set equal to zero (\(\hbox {fitK}_4\), fitK\(\chi _4\)) or treated as a free parameter (fitK\(\chi _5\), fitK\(\chi _6\)). If \(\delta _0\) is set equal to zero then \(N_1\) is determined very sharply. In fact, the solution then becomes so stiff that the result for \(N_1\) is outside the range obtained if \(\delta _0\) is allowed to float. In somewhat milder form, the problem also manifests itself in Table 2: the value \(\delta _0=0\) is about two standard deviations away from the results obtained with fitK\(\chi _5\) or fitK\(\chi _6\). This shows that setting \(\delta _0=0\) amounts to introducing a systematic theoretical error, which pulls the amplitude down by about 9%.
Four subtraction constants do suffice to properly describe the momentum dependence in the physical region of the decay, but to cope with the theoretical constraints that follow from the fact that the particles involved in this decay are Nambu–Goldstone bosons of a hidden approximate symmetry, an extrapolation from the physical region all the way down to the Adler zero is required. We conclude that with only four subtractions, the dispersive representation does not provide a controlled extrapolation: \(\delta _0\) cannot simply be set equal to zero, but needs to be determined by experiment.
For \(\gamma _1\), the situation is different: since the value \(\gamma _1=0\) is close to the center of the range obtained if this parameter is allowed to float, it does not make much of a difference whether or not we keep it fixed at zero. The advantage of using six subtractions rather than five is that the uncertainties associated with the contributions from the high energy tails of the dispersion integrals are then reduced. For this reason, we identify our central solution with fitK\(\chi _6\).
5.7 Imaginary parts of the subtraction constants
Central values and errors for two versions of the central solution: while for \(\mathrm {fitK}\chi _6\), the subtraction constants are taken real, in the case of \(\mathrm {FitK\chi _6}\), they are instead calculated from the twoloop prediction for the imaginary parts of the Taylor coefficients
\(\text {Re}\,\beta _0\)  \(\text {Re}\,\gamma _0\)  \(\text {Re}\,\delta _0\)  \(\text {Re}\,\beta _1\)  \(\text {Re}\,\gamma _1\)  \(\chi ^2_{\mathrm {K}}\)  \(\chi ^2_{\mathrm {th}}\)  

\(\mathrm {fitK}\chi _6\)  16.2(1.2)  \(20.8(10.1)\)  \(38(17)\)  8.5(2.2)  \(3.8(10.7)\)  384.1  1.47 
\(\mathrm {FitK\chi _6}\)  16.2(1.2)  \(21.0(10.0)\)  \(38(17)\)  8.6(2.2)  \(4.8(10.7)\)  384.8  1.58 
Taylor invariants and position of the Adler zero for the two variants of the central solution
\(\text {Re}\,K_1\)  \(\text {Re}\,K_2\)  \(\text {Re}\,K_3\)  \(\text {Re}\,K_4\)  \(\text {Re}\,K_5\)  \(s_A\)  

\(\mathrm {fitK}\chi _6\)  4.51(25)  25.6(5.4)  \(3.7(1.8)\)  90.2(5.0)  52.9(7.0)  1.39(11)\(M_\pi ^2\) 
\(\mathrm {FitK\chi _6}\)  4.55(24)  25.8(5.2)  \(3.6(1.9)\)  90.8(5.2)  52.3(6.9)  1.38(11)\(M_\pi ^2\) 
Table 4 compares the real parts of the subtraction constants belonging to FitK\(\chi _6\) with those of fitK\(\chi _6\), which are real by construction. It shows that the differences between the two versions of our central solution are negligibly small compared to the uncertainties therein.
Table 5 shows that the same conclusion is reached if instead of the real parts of the subtraction constants we compare the real parts of the Taylor invariants \(\text {Re}\,K_1, \ldots , \text {Re}\,K_5\) or the position of the Adler zero for the two variants of our central solution. The Adler zero is determined to an accuracy of about 8% and occurs in the immediate vicinity of the current algebra prediction, \(s_A=4/3\,M_\pi ^2\).
Since the difference between the two versions of the central solution is in the noise of our calculation, we do not pursue it further. In Sect. 6, where we discuss the difference between the twoloop representation of \(\chi \)PT and the dispersive representation that matches it at low energies, we consider the version FitK\(\chi _6\), because it matches the imaginary parts as well as the real parts. Throughout the remainder of the paper, however, where we draw the conclusions from our analysis, we stick to real subtraction constants and work with the version fitK\(\chi _6\) of the central solution.
5.8 Dalitz plot coefficients of our central solution
The parametrization (5.1) amounts to a polynomial in the Mandelstam variables s, t, u. Unitarity generates branch points at the boundary of the physical region (the corresponding cusps in the real part of the amplitude can be seen e.g. in Fig. 10). Outside the physical region, a polynomial parametrization of the Dalitz plot distribution cannot provide a reliable improvement of the current algebra formula, \(D_c^{\mathrm {LO}}=(3\,s4M_\pi ^2)^2/(M_\eta ^2M_\pi ^2)^2\). The dispersive framework we are using does account for the singularities required by unitarity, but as discussed in Sect. 5.6, a fit to the KLOE distribution that simply treats the subtraction constants as free parameters leads to solutions that violate chiral symmetry. We are exploiting the fact that this symmetry imposes strong conditions on the amplitude at small values of s, in particular also near the Adler zero. Although these conditions do not significantly constrain the amplitude in the physical region, they are essential for the interpretation of the experimental results in the framework of the Standard Model.
5.9 Comparison with the nonrelativistic effective theory
We conclude that the twoloop representation of NREFT yields a decent approximation of the momentum dependence also for \(\eta \)decay. In the case of kaondecay, the contributions due to the electromagnetic interaction were worked out in the framework of NREFT and the cusps generated by the transition \(\pi ^0\pi ^0\rightarrow \pi ^+\pi ^\rightarrow \pi ^0\pi ^0\) were studied in detail. The twoloop representation of Ref. [38] does properly account for the mass difference between the charged and neutral pions – an evident advantage compared to our analysis, which takes care of the mass difference only in a purely kinematic way. For those electromagnetic effects that do not show up in the selfenergies of the pions, we are relying on the relativistic oneloop representation [18]. The work done in the framework of NREFT [39, 40] would provide the basis for a more thorough analysis of the contributions generated by the electromagnetic interaction, but we must leave this for future work.
The numerical values found for the subtraction constants of \(\hbox {fitNRK}_4\) are very different from those of the dispersive solutions listed in Table 2. One of the reasons is that the normalization differs: while the nonrelativistic twoloop representation is normalized by setting \(L_0=1\), the solutions in Table 2 are normalized by fixing the Taylor invariant \(H_0\) at the value found at one loop. The Taylor invariants are outside the reach of the nonrelativistic effective theory. We can instead fix the normalization such that the magnitude of the amplitude at the center of the Dalitz plot is the same as for our central solution, fitK\(\chi _6\). This is achieved by simply stretching all of the LECs: \(L_n\rightarrow \lambda L_n\), with \(\lambda =2.353\). The subtraction constants of \(\hbox {fitNRK}_4\) must be stretched by the same factor.
There is a further difference: for the dispersive solution to match the NR representation, the subtraction constants must be allowed to have an imaginary part – those of the solutions listed in Table 2 are real. We investigated the sensitivity of our results to the imaginary parts of the subtraction constants in Sect. 5.7. There, we observed that, in the chiral expansion, the Taylor invariants become complex at NNLO. We worked out the dispersive solution obtained if the imaginary part of the Taylor invariants are taken from the twoloop representation of the relativistic effective theory and found that the imaginary parts do not significantly affect our results. Matching with the NR effective theory at two loops confirms this experience: although the subtraction constants of \(\hbox {fitNRK}_4\) have sizeable imaginary parts while those of the solutions listed in Table 2 are real, the results obtained for quantities of physical interest are in the same ballpark. As we are not in a position to properly account for isospin breaking effects, we do not continue the comparison with the nonrelativistic framework further, but will briefly return to related work in Sect. 10.2.
Figure 12 shows that the Dalitz plot distributions of the two representations can barely be distinguished, in the entire physical region and for \(\eta \rightarrow \pi ^+\pi ^\pi ^0\) as well as for \(\eta \rightarrow 3\pi ^0\). Note the difference in the scale used in the two panels. In the left panel, the difference between the nonrelativistic fit to KLOE and our central solution can barely be seen, but it does show up in the right panel: the cusps generated by the final state interaction represent an isospin breaking effect, which is clearly seen in the band belonging to fitK\(\chi _6\), but is absent in the other Dalitz plot distributions, because these are shown in the isospin limit. Visibly, \(D_n=1+2\alpha (X_n^2+Y_n^2)+\cdots \) stays close to 1, with a negative value of the slope parameter \(\alpha \).
6 Anatomy of the twoloop representation
As discussed in Sect. 3.3, elastic unitarity determines the NNLO representation of \(\chi \)PT in terms of the one valid at NLO, up to a polynomial. The nonpolynomial part does not contain any unknowns, but the polynomial does, in the form of the lowenergy constants that occur in the effective Lagrangian at \(O(p^6)\) – for some of these, only crude theoretical estimates are available. Note that the twoloop representation is unique up to a real polynomial. To consistently compare the dispersive and chiral representations at \(O(p^6)\) of the chiral expansion, the subtraction constants must be given the proper imaginary part. In particular, for the central solution, we need to consider the version \(\mathrm {FitK}\chi _6\), so that the imaginary parts of the Taylor invariants do agree with those of the twoloop representation.
6.1 Final state interaction at two loops
In Sect. 3.5, we determined the solution of our integral equations which matches the oneloop representation of \(\chi \)PT at low energies: fit\(\chi _4\). We now extend this to the twoloop level, exploiting the fact that the contributions from the loop graphs are determined by the oneloop representation and do not involve any unknowns. For the explicit numerical evaluation of these contributions, we rely on the work of Bijnens and Ghorbani, more precisely on the code provided by these authors [70]. Concerning the tree graph contributions, we make use of the fact that these are polynomials in the momenta. Instead of calculating the coefficients of the polynomials with the effective Lagrangian and then inserting the available estimates for the LECs contained therein, we determine the polynomial part in such a way that the amplitude matches our central solution at low energies. In the sum over the isospin components, the polynomial part contains six independent coefficients, which are in onetoone correspondence with the Taylor invariants \(K_0, \ldots , K_5\). In order to construct the twoloop representation that matches FitK\(\chi _6\), we simply need to match these invariants.
Figure 13 compares the isospin components of the twoloop representation with those of \(\mathrm {FitK\chi _6}\). Below threshold, the two representations can barely be distinguished from one another. The components with \(I=1\) and \(I=2\) of the twoloop representation closely follow those of the central solution even for \(s>4M_\pi ^2\) (note that the range shown for \(M_2(s)\) is substantially wider than for the other components, because this is of interest in connection with the position of the Adler zero – see below). In \(M_0(s)\), however, a significant difference can be seen in the physical region. It implies that the real part of the isospin combination relevant for the transition \(\eta \rightarrow 3\pi ^0\), \(M_n^{\mathrm {NNLO}}(s)=M_0^{\mathrm {NNLO}}(s)+\frac{4}{3}M_2^{\mathrm {NNLO}}(s)\) nearly follows a straight line. This answers the question raised above: the twoloop representation accounts sufficiently well for the final state interaction only for \(s\lesssim 5 M_\pi ^2\). Above that energy, the lowest resonance of QCD, the \(f_0(500)\), manifests itself. The corresponding pole occurs on the second sheet, in the vicinity of \(s_{\mathrm {pole}}\simeq (441i \,272\, \text {MeV})^2\simeq 6.2i \,12.3 \,M_\pi ^2\) [64, 78] (the arrows in Fig. 13 indicate the real part of the pole position). Although the resonance is very broad – the pole is far away from the real axis – the truncated expansion in powers of momentum cannot properly cope with it above \(5M_\pi ^2\), not even at NNLO.
As discussed in Sect. 3.7, the curvature of the function \(M_n(s)\) determines the slope parameter \(\alpha \) of the neutral decay mode. Since the curvature of \(M^{\mathrm {NNLO}}_n(s)\) nearly vanishes, the slope of this representation is very small – numerically, we obtain \(\alpha ^{\mathrm {NNLO}}= + 0.002\). In the neutral channel, the NNLO representation of the Dalitz plot distribution can thus barely be distinguished from the horizontal line in Fig. 5, which indicates the tree level result. This is lower than the value \(\alpha = +0.011\) that belongs to the NLO curve, which is also shown in Fig. 5, or the twoloop estimate \(\alpha =+0.013(32)\) given in [12], but the discrepancy with the experimental value \(\alpha =0.0318(15)\) [66] is not removed. We conclude that a substantial part of the discrepancy is due to the fact that the twoloop result does not fully account for the enhancement of the final state interaction generated by the resonance \(f_0(500)\). Closely related aspects of the same problem were discussed already earlier, by Schneider, Kubis and Ditsche (see in particular Sect. 4.3 of Ref. [42]).
The Adler zero of \(\text {Re}\,M_n^{\mathrm {NNLO}}(s,t,u)\) occurs at \(s_A=1.35(11)\,M_\pi ^2\), remarkably close to the value \(s_A=1.37(11)\) where the real part of FitK\(\chi _6\) has its zero. By construction, the isospin components belonging to the twoloop approximation \(M^{\mathrm {NNLO}}(s,t,u)\) agree with those of the dispersive representation at small values of \(s=u\), but as discussed in Sect. 3.6, the behaviour of the sum over the isospin components at small values of \(s=u\) is not controlled exclusively by their behaviour in that region, but also depends on the properties of the comparatively small component \(\text {Re}\,M_2(s)\) in the vicinity of \(s = 16 M_\pi ^2\). Figure 13 shows that even there, the twoloop approximation follows the dispersive representation for \(M_2(s)\) rather well. This explains why that approximation is rather accurate also in the vicinity of the Adler zero.
\(\text {Re}\,K_0\)  \(\text {Re}\,K_1\)  \(\text {Re}\,K_2\)  \(\text {Re}\,K_3\)  \(\text {Re}\,K_4\)  \(\text {Re}\,K_5\)  \(s_A\)  

NNLO  1.176(53)  4.55(24)  25.8(5.2)  \( 3.6(1.9)\)  90.8(5.2)  52.3(6.9)  1.33(14)\(M_\pi ^2\) 
BG  1.27  3.88  37.2  \(6.2\)  113(34)  73  1.17\(M_\pi ^2\) 
6.2 Contribution from the lowenergy constants at NNLO
Finally, we compare the polynomial part of the amplitude of Bijnens and Ghorbani [12] with the twoloop representation constructed in the preceding section. The numbers in the row NNLO of Table 6 represent central values and uncertainties of the Taylor invariants belonging to that representation – by construction, these coincide with the invariants of the dispersive solution FitK\(\chi _6\). The values in the row BG are obtained with the code [70] mentioned earlier.
We recall that the experimental information about the Dalitz plot distribution exclusively concerns the relative size of the invariants, not the invariants themselves. The value quoted for \(\text {Re}\,K_0\) relies on theory, more precisely on the expansion of \(K_0\) in powers of the masses of the three lightest quarks. This expansion starts with \(K_0=1+O(m_{\mathrm {quark}})\). As discussed in Sect. 3.2, the coefficient of the nexttoleading term of the expansion can be worked out from the oneloop representation of the transition amplitude, which does not involve any unknowns. Numerically, the correction is of typical size: \(K_0=1+0.176+O(m_{\mathrm {quark}}^2)\). The error quoted in Table 6 is based on the estimate of the higher order contributions described in Sect. 3.2. The table shows that the value obtained for \(\text {Re}\,K_0\) from the estimates used for the LECs in [12] is outside our range (disregarding the uncertainty in the number 1.27, the difference amounts to \(1.7\sigma \)). Since \(K_0\) is not plagued by infrared singularities – in particular, this invariant remains finite in the limit \(M_\pi \rightarrow 0\) – we see no reason why it should pick up unusually large corrections from higher orders and stick to the value quoted in the table.
The value of \(K_0\) is important for the determination of the kaon mass difference and of the quark mass ratio Q, to be discussed in Sect. 9, but in the present section, we compare the chiral and dispersive representations for the Dalitz plot distribution of the charged channel, the slope \(\alpha \) of the Zdistribution in the neutral channel and the position of the Adler zero with our central solution – these quantities only involve the ratios \(K_1/K_0,\ldots ,K_5/K_0\). We set \(\text {Re}\,K_0=1.176\) and fix the imaginary parts with the twoloop representation of Bijnens and Ghorbani [12].
As pointed out in Sect. 3.3, the Taylor invariant \(K_4\) does not get any contribution from the LECs of \(O(p^6)\). The corresponding entry for \(\text {Re}\,K_4\) in the table includes our uncertainty estimate from Eq. (3.9). The value obtained with our central solution is indeed within the range of this prediction (the imaginary parts are identical by construction). \(\text {Re}\,K_3\) also agrees within the uncertainties attached to our central solution, but for \(\text {Re}\,K_1\), \(\text {Re}\,K_2\) and \(\text {Re}\,K_5\), the two results differ by up to \(2\sigma \). We conclude that the values of some of the LECs used in [12] are not consistent with the experimental information on \(\eta \rightarrow 3\pi \) available today.
As discussed in Sect. 6.1, a direct comparison of the twoloop representation with the data in the physical region is not meaningful – the \(f_0(500)\) is the stumbling block. Dispersion theory is needed to establish a controlled connection between the region that is accessible to experiment and the domain \(s\lesssim 5M_\pi ^2\), where the twoloop approximation for \(M_0(s)\) is sufficiently accurate.
The Taylor invariants provide the bridge. The dispersive representation reliably determines the behaviour of the amplitude in the physical region in terms of these. Their imaginary parts are known to NNLO of the chiral expansion. Using this, and keeping \(\text {Re}\,K_0\) fixed at the central value, the KLOE data on the Dalitz plot distribution of \(\eta \rightarrow \pi ^+\pi ^\pi ^0\) imply that the real parts of the remaining five invariants are in the range indicated in the row NNLO of Table 6.
As already mentioned, unitarity fixes the twoloop representation for \(M_c(s,t,u)\) in terms of known quantities up to a real polynomial. The polynomial contains six independent coefficients that are in onetoone correspondence with the real parts of the Taylor invariants \(K_0,\ldots ,K_5\). In the representation of the amplitude obtained with \(\chi \)PT, the Taylor invariants represent linear combinations of some of the LECs of \(O(p^6)\). In particular, those relevant for the scalar channel with \(I=0\) contribute, which are notoriously difficult to estimate because the contribution from the \(f_0(500)\) to the corresponding spectral functions is not easily accounted for. The experimental information about the Taylor invariants and their correlations obtained from our analysis should make it possible to reliably determine these particular couplings, which also enter in many other applications of \(\chi \)PT. An update of the LECs of \(\chi \)PT (for a recent review, see [79]) that accounts for this information would be of considerable interest, but is beyond the scope of the present work.
7 Consequences for \(\mathbf \eta \rightarrow 3\pi ^0\)
7.1 Branching ratio
7.2 Dispersive representation of the Dalitz plot distribution
7.3 Slope
Since \(\alpha \) is very small, details of the evaluation matter. In particular, as demonstrated in Sect. 3.7, \(\alpha \) is very sensitive to the final state interaction. As an example, consider isospin breaking. Although the isospin breaking effects in the decay \(\eta \rightarrow 3\pi ^0\) are small, dropping them in the calculation of the slope changes the central value of the prediction from \(0.0303\) to \(0.0327\). Details of the evaluation also matter in the analysis of the data: the number quoted in (7.8) is the derivative of the Zdistribution at \(Z=0\). In the past, the experimental determination of the slope was instead determined by fitting the data with the linear formula \(1+2\alpha Z\) on a finite range of Z values. The sensitivity of the result to this range and to the fact that – at the accuracy reached – the curvature of the distribution cannot be neglected will be discussed in Sect. 7.7.
7.4 Experiment
Various experimental and theoretical results for the slope parameter \(\alpha \). We have added systematic and statistical uncertainties in quadrature. The PDG average is based on the experimental results listed here. For comparison, the above numbers are visualized in Fig. 15
\(\alpha \)  

GAMS2000 (1984)  − 0.022(23)  [80] 
Crystal Barrel@LEAR (1998)  − 0.052(20)  [81] 
Crystal Ball@BNL (2001)  − 0.031(4)  [82] 
SND (2001)  −0.010(23)  [83] 
WASA@CELSIUS (2007)  − 0.026(14)  [84] 
WASA@COSY (2008)  − 0.027(9)  [85] 
Crystal Ball@MAMIB (2009)  − 0.032(3)  [23] 
Crystal Ball@MAMIC (2009)  − 0.032(3)  [24] 
KLOE (2010)  − 0.0301(\(_{49}^{+41}\))  [26] 
BESIII (2015)  − 0.055(15)  [21] 
PDG average  − 0.0318(15)  [66] 
Crystal Ball@MAMIA2 (2018)  − 0.0265(10)(9)  [25] 
Kambor et al. (1996)  \(0.007\)  [14] 
Bijnens and Gasser (2002)  − 0.007  [86] 
Bijnens and Ghorbani (2007)  0.013(32)  [12] 
Schneider et al. (2011)  − 0.025(5)  [42] 
Kampf et al. (2011)  − 0.044(4)  [43] 
JPAC (2016)  − 0.025(4)  [47] 
Albaladejo and Moussallam (2017)  − 0.0337(12)  [49] 
This work  − 0.0303(12) 
The most precise determination of the Dalitz plot distribution and its slope parameter \(\alpha \) is based on the data collected at the Mainz Microtron: 1.8 million events were analyzed at MAMIB [23], another three million \(\eta \rightarrow 3\pi ^0\) decays were collected at MAMIC [24] and, very recently, the A2 Collaboration came up with an update based on altogether 7 million events [25]. KLOE has performed such a measurement too [26], on the basis of about half a million events. The PDG average \(\alpha =0.0318(15)\) [66] is largely dominated by the MAMI measurements. As discussed in the preceding section, the result for \(\alpha \) is sensitive to the range over which the data are approximated with the linear formula \(1+2\alpha Z\). A more controlled determination that does not rely on this approximation became possible only very recently [25]. We will discuss it in detail in Sect. 8.
7.5 Zdistribution
The band in Fig. 16 shows the result obtained for the Zdistribution from our central solution, \(\mathrm {fitK}\chi _6\). The width of the band represents the uncertainties in \(d_n^Z\), which are worked out as described in Sect. 5.5. The data points represent the Zdistribution obtained by the A2 collaboration at MAMI [25]. In earlier accounts of the data collected at MAMI, the normalization of the Zdistribution was fixed by fitting the data with the linear approximation, \(d_n^Z=1+2\alpha Z\), but at the accuracy reached, this is not legitimate any more, because the curvature cannot be neglected. In Ref. [25], the normalization of the Zdistribution is left open. When comparing these data with our prediction, we multiply the observed distribution by the factor \(\varLambda \), which is treated as a free parameter. Visibly, the resulting normalized distribution, \(\varLambda \,d_n^{\mathrm {Z\; exp}}\), is in excellent agreement with the prediction. Quantitatively, we obtain \(\varLambda =0.974\), \(\chi ^2=24.9\) for 30 data points and one free parameter.
7.6 Mdistribution
7.7 Polynomial approximation
7.8 Strength of the cusps
The polynomial approximation (7.13) is adequate only in the singularityfree part of the physical region. We now turn to the remainder, \(Z>Z^{\mathrm {cusp}}\), where the cusps do manifest themselves. The pioneering work of Budini, Fonda and Cabibbo [87, 88] on the physics of the cusps occurring in the decays \(K^+\rightarrow \pi ^+\pi ^0\pi ^0\) and \(K_L\rightarrow 3\pi ^0\) and the subsequent thorough analysis in [37, 38, 39, 40, 89, 90] led to a very satisfactory understanding of the phenomenon. As shown in [37, 38, 39, 40], it can be analyzed by means of nonrelativistic effective theory. Indeed, the precision of the data on kaon decays even allows a determination of \(\pi \pi \) scattering lengths [37, 38, 39, 40, 59, 88, 89]. The situation for \(\eta \rightarrow 3\pi ^0\) is essentially the same as for \(K_L\rightarrow 3 \pi ^0\), but the knowledge is much more limited, both experimentally and theoretically. The work reported in two theoretical investigations [41, 42] will briefly be discussed in Sect. 10.2.
Polynomial representations for the decay \(\eta \rightarrow 3\pi ^0\). The parametrization is specified in Eq. (7.18). The first two lines represent fits to the MAMI data for the Zdistribution. The next three lines show polynomial fits to the MAMI data on the Dalitz plot distribution – two of these stem from Table I of Ref. [25]. The lower half of the table contains polynomial approximations to various dispersive representations obtained within our framework. The coefficients \(\alpha \), \(\beta \) and \(\gamma \) are determined with a fit in the region \(Z<Z\mathrm {cusp}\approx 0.597\), where \(\delta \) does not contribute (18 bins of the Zdistribution and 266 bins of the Dalitz plot distribution are in this region – the values quoted for \(\chi ^2_{\mathrm {M}}\) give the contributions to the discrepancy function from these bins). The values of \(\delta \) are obtained by fitting the remaining 140 bins of the Dalitz plot distribution, varying \(\alpha \), \(\beta \), \(\gamma \) in the range found in the first step. The asterisks mark values used as input
\(\alpha \)  \(\beta \)  \( \gamma \)  \(\delta \)  \(\chi ^2_{\mathrm {M}}\)  \(\chi ^2_{\mathrm {K}}\)  \(\chi ^2_{\mathrm {th}} \)  

fitMZ  − 0.0265(59)  \(+\) 0.0017(96)  10.2  
\(\hbox {fitMZ}_1\)  − 0.0267(15)  \(+\) 0.0019\(^*\)  10.2  
fitMD  − 0.0301(64)  − 0.0069(18)  \(+\) 0.0087(110)  − 0.027(14)  343  
fit#9 [25]  − 0.0265(10)  − 0.0073(10)  \(0^\star \)  − 0.017(7)  408  
fit#10 [25]  − 0.0247(30)  − 0.0070(12)  − 0.0023(40)  − 0.015(7)  363  
fit\(\chi _4\)  − 0.0222(117)  − 0.0039(7)  \(+\) 0.0015(8)  − 0.0169(4)  352  0  
\(\hbox {fitK}_4\)  − 0.0310(17)  − 0.0043(3)  \(+\) 0.0021(3)  − 0.017(4)  354  390  0.67 
\(\hbox {fitKM}_4\)  − 0.0303(13)  − 0.0042(4)  \(+\) 0.0020(2)  − 0.017(4)  352  391  0.46 
fitK\(\chi _6\)  − 0.0307(17)  − 0.0052(5)  \(+\) 0.0019(3)  − 0.017(4)  352  384  1.47 
fitKM\(\chi _6\)  − 0.0296(12)  − 0.0055(4)  \(+\) 0.0018(3)  − 0.017(4)  387  383  5.12 
With the values of the coefficients in (7.16), (7.19), the parametrization (7.18) reproduces our dispersive representation of the Dalitz plot distribution within 0.6 permille, throughout the physical region. It does not quite reach the remarkable precision of the polynomial representation on the disk \(Z<Z^{\mathrm {cusp}}\), presumably because the extrapolation of the first few terms of the Taylor series does not describe the background underneath the cusps very accurately – the presence of the resonance \(\hbox {f}_0\)(500) may accurately be accounted for only in the dispersive representation.
The error in the result for \(\delta \) reflects the uncertainties of the dispersive representation. These subject the coefficients \(\alpha \), \(\beta \), \(\gamma \) to the errors listed in (7.16) and also lead to correlations among them. When minimizing the discrepancy in the region \(Z>Z^{\mathrm {cusp}}\), the errors then propagate into \(\delta \). The evaluation shows that the strength of the cusps is rather sensitive to the uncertainties in the isospin breaking corrections – the corresponding contribution to the error budget is even slightly larger than the Gaussian error, while the one from the noise in the phase shifts is negligible.
The prediction for the slope mainly relies on the experimental information concerning the Dalitz plot distribution of \(\eta \rightarrow \pi ^+\pi ^\pi ^0\) – the theoretical constraints are not important in this connection. This can be seen by comparing the polynomial approximations for the two dispersive solutions obtained if either the data on this decay or the theoretical constraints are ignored: fit\(\chi _4\) versus \(\hbox {fitK}_4\) – the first represents the matching solution, which exclusively relies on theory, while the second is instead based on the KLOE data alone. The coefficients of the corresponding polynomial approximations are listed in Table 8. The comparison shows that the two representations of the Dalitz plot distribution in the neutral channel are consistent with one another. Concerning \(\delta \), the results are even the same and for \(\beta \), there is not much of a difference, either. For fit\(\chi _4\), however, the uncertainties in \(\alpha \) and \(\gamma \) are much larger than for \(\hbox {fitK}_4\): in this regard, the theoretical constraints are much weaker than the experimental ones.
8 Fits to the MAMI data
8.1 Zdistribution
8.2 Dalitz plot distribution on the disk \(Z<Z^{\mathrm {cusp}}\)
Next, we consider the MAMI data on the Dalitz plot distribution. As noted above, each event is represented by 6 different points in the physical region. The binning in the variables \(X_n\), \(Y_n\) does preserve the symmetry under \(X_n\rightarrow X_n\), but not the one under reflections at the lines \(\varphi =\pm \, 30^\circ \). Accordingly, a subset of bins that contains each event exactly once does not exist.
A polynomial fit to the MAMI data on the Dalitz plot distribution that does not invoke dispersion theory at all is listed in the entry fitMD of Table 8: the coefficients \(\alpha \), \(\beta \), \(\gamma \) are determined with a fit to the data in those bins that are contained in the disk \(Z<Z^{\mathrm {cusp}}\), where the Taylor series converges and where \(\delta \) does not contribute. Treating the overall normalization of the experimental distribution as a free parameter, the fit returns the central values for \(\alpha \), \(\beta \), \(\gamma \) listed in the table, together with \(\varLambda _{\mathrm {M}}=0.976\) and \(\chi ^2=343.3\) for 266 data points and 4 parameters. The errors are obtained in the same way as for the subtraction constants of the dispersive representation, except that the discrepancy function now contains an additional parameter, \(\varLambda _{\mathrm {M}}\). The result for \(\alpha \) and \(\gamma \) confirms what we found when fitting the Zdistribution: fitMD and fitMZ agree within errors. The uncertainties are large, but the values are strongly correlated. In contrast to fitMZ, however, the likelihood of fitMD is not satisfactory: \(\chi ^2/\mathrm {dof}=1.31\). Since the polynomial approximation of the dispersive representation is very accurate in the disk \(Z<Z^{\mathrm {cusp}}\), we consider it very unlikely that the problem originates in the lack of flexibility of the parametrization.
8.3 Cusps
To evaluate the strength of the cusps for fitMD, we use the same procedure as in the construction of an approximate representation for our central dispersive solution: keep the values of \(\alpha \), \(\beta \) and \(\gamma \) fixed at fitMD, vary \(\delta \) and minimize the difference between the parametrization (7.18) and the data in the region \(Z>Z^{\mathrm {cusp}}\). The quality of the fit is worse than for the bins contained in the disk \(Z<Z^{\mathrm {cusp}}\): \(\chi ^2=233\) for 140 data points and 1 free parameter, \(\chi ^2/\mathrm {dof}=1.68\). The error calculation follows the same steps: first determine \(\delta \) for prescribed values of \(\alpha \), \(\beta \), \(\gamma \), \(\varLambda _{\mathrm {M}}\), then vary these within the range obtained when minimizing the discrepancy in the disk \(Z<Z^{\mathrm {cusp}}\), accounting for the correlations among them. Finally, the additional uncertainty arising from the statistical fluctuations in the region \(Z>Z^{\mathrm {cusp}}\) is added in quadrature. For \(\delta \), the error is dominated by the contribution from the uncertainties and correlations encountered in the first step. Table 8 shows that the result for fitMD is consistent with our prediction, also concerning \(\delta \). Although the cusps do not stick out from the fluctuations visible in Fig. 17, the quantitative analysis on the basis of formula (7.18) does confirm their presence.
For the dispersive representation of the amplitude, it does not make much of a difference whether the slope is determined with a fit in the disk \(Z<Z^{\mathrm {cusp}}\) or in the entire physical region. Fitting the parametrization (7.18) to our central solution fitK\(\chi _6\) in the entire physical region, we obtain \(\alpha =0.0307(18)\), \(\beta =0.0049(5)\), \(\gamma =0.0018(3)\), \(\delta =0.016(4)\). These numbers barely differ from those quoted in Table 8 for the polynomial approximation to fitK\(\chi _6\). This shows that the dispersive representation provides a stable extrapolation from the region below \(Z^{\mathrm {cusp}}\) to the region where the cusps occur.
When fitting data with the polynomial approximation, the situation is very different, because the correlation between the behaviour at small values of Z and in the region where the cusps manifest themselves is then absent. This is illustrated with two fits taken from Table I of Ref. [25], which are also based on the combined data of Runs I and II, but use all three sextants with \(X_n>0\). Apart from that, the analysis differs from ours only in one respect: while we determine the coefficients \(\alpha \), \(\beta \), \(\gamma \) with a fit to the data in the disk \(Z<Z^{\mathrm {cusp}}\) and make use of those in the remaining bins exclusively to estimate the strength of the cusps, fit#9 and fit#10 treat all coefficients on the same footing (except that in the case of fit#9 \(\gamma \) is set to zero). The comparison of the two illustrates the strong correlation between \(\alpha \) and \(\gamma \): the uncertainty in the result for the slope becomes much smaller if \(\gamma \) can be taken as known. Note that for all of the entries in Table 8, the values quoted for \(\chi ^2_{\mathrm {M}}\) refer to the 266 independent bins in the disk \(Z<Z^{\mathrm {cusp}}\).
The three polynomial representations fitMD, fit#9 and fit#10 agree within uncertainties, but the latter two have substantially smaller errors. The left panel of Fig. 19 illustrates the difference, which arises because the polynomial terms grow with Z; extending the region over which the approximation is fit to the data leads to smaller errors in the coefficients. While fitMD is consistent with our prediction (7.16), (7.19), the values obtained for \(\alpha \) and\(\beta \) with fits #9 and #10 are not. In fact, the entries for \(\chi ^2_{\mathrm {M}}\) show that, in the region \(Z<Z^{\mathrm {cusp}}\), the polynomial approximation to our prediction follows the data more closely than these two fits. Concerning the parameter \(\delta \), which measures the strength of the cusps, however, they are in very good agreement with our prediction.
The main problem we are facing here is that one is dealing with small effects. In current algebra approximation, the Dalitz plot distribution is flat, \(D_n^{\mathrm {LO}}(X_n,Y_n)=1\). The MAMI data do allow an accurate measurement of the slope \(\alpha \) of the distribution, but what remains is tiny: for our prediction, the difference \(D_n^{\mathrm {phys}}(X_n,Y_n)1 2\,\alpha \, Z\) stays below 7 permille, throughout the region \(Z<Z^{\mathrm {cusp}}\), where the Taylor series converges. Although the set we are analyzing is based on more than 7 million events, the statistical errors in the mean value of the Dalitz plot distribution for a given bin are of order 8 permille and the systematic ones must be small compared to this for the measurement to be sound. Isospin breaking effects are by no means negligible at this level of accuracy. In the approximation we are using, they yield a positive contribution to the slope: \(\delta \alpha = + 0.0024(7)\). At \(Z=Z^{\mathrm {cusp}}\), it affects the value of the Dalitz plot distribution by about 3 permille. Note also that the cusps are visible in the physical region only because the physical masses of the charged and neutral pions differ – isospin breaking is crucial for an accurate analysis of the Dalitz plot distribution in the region \(Z > Z^{\mathrm {cusp}}\). The fact that the result obtained for the branching ratio agrees with experiment gives us confidence that our estimates for the effects due to isospin breaking in the integrals over the square of the amplitude are adequate, but resolving the Dalitz plot distribution at the level of accuracy needed to reliably determine small quantities like \(\beta \) and \(\gamma \) and to measure the strength of the cusps is a different matter.
8.4 Dispersive analysis of the MAMI data
We first allow for only four subtraction constants, set \(\delta _0=\gamma _1=0\) and denote the simultaneous fit to the KLOE and MAMI data by fitKM4. Table 8 shows that the inclusion of the MAMI data lowers the value of the slope \(\alpha \) from \(0.0310(17)\) (\(\hbox {fitK}_4\)) to \(0.0303(13)\) (\(\hbox {fitKM}_4\)), while the coefficients \(\beta \), \(\gamma \), \(\delta \) nearly stay put. The ratio \(\chi ^2_{\mathrm {M}}/\mathrm {dof}= 1.34\) shows that the quality of the fit is not satisfactory, even slightly worse than for the polynomial representation fitMD, where \(\chi ^2_{\mathrm {M}}/\mathrm {dof}= 1.31\). On the other hand, the value \(\chi ^2_{\mathrm {th}}=0.46\) indicates that, although the theoretical constraints that follow from the presence of a hidden approximate symmetry are not made use of in the derivation of \(\hbox {fitKM}_4\), the MAMI data for \(\eta \rightarrow 3\pi ^0\) are consistent with these, as well as with the KLOE data for \(\eta \rightarrow \pi ^+\pi ^\pi ^0\).
If more than four subtraction constants are treated as free parameters, the minimization again goes astray. When analyzing the KLOE data we found that simply adding the term \(\chi _{\mathrm {th}}^2\) to the discrepancy function suffices to ensure that the theoretical constraints are respected. In the present case, this is not the case, however: the contributions from the 371 and 406 data points of KLOE and MAMI, respectively, overwhelm the one from the theoretical part of the discrepancy function. The minimum occurs at \(\chi _{\mathrm {th}}^2=5.12\), indicating that the constraints are still violated – fitKM\(\chi _6\) does not represent a physically acceptable solution of our integral equations. For the determination of Q, the extrapolation below threshold is needed and the theoretical constraints do play an essential role in this connection.
As far as the behaviour in the physical region is concerned, however, fitKM\(\chi _6\) does represent an acceptable parametrization of the amplitude. The violation of the theoretical constraints can be cured without significantly changing the behaviour of the amplitude there. It suffices, for instance, to give the theoretical discrepancy in \(\chi _{\mathrm {tot}}^2=\chi _{\mathrm {K}}^2+\chi _{\mathrm {M}}^2+\chi _{\mathrm {th}}^2\) more weight. If we multiply that term by 3, the value of \(\chi _{\mathrm {th}}^2\) falls to 1.20 while \(\alpha \), \(\beta \), \(\gamma \), \(\delta \) nearly stay put at the values obtained for fitKM\(\chi _6\) listed in Table 8. The white ellipse in the right panel of Fig. 19 illustrates the result. The comparison shows that fitKM\(\chi _6\) is close to \(\hbox {fitKM}_4\), consistent with fitMZ and fitMD (MAMI data alone) as well as with our prediction, fitK\(\chi _6\) (KLOE data plus theoretical constraints). The result for \(\beta \), \(\gamma \) and \(\delta \) can barely be distinguished from the prediction. The inclusion of the MAMI data reduces the value of the slope, irrespective of whether four or six subtraction constants are allowed. As emphasized in Ref. [25], these data imply a smaller value than the average \(\alpha =0.0318(15)\) quoted by the Particle Data Group [66].
9 Kaon mass difference and quark mass ratios
9.1 Mass difference between charged and neutral kaons
9.2 Electromagnetic contributions to the meson masses, Dashen theorem
9.3 Determination of the quark mass ratio Q
Theoretical results for the quark mass ratio Q (statistical and systematic uncertainties added in quadrature)
Q  

Gasser and Leutwyler (1975)  30.2  [106] 
Weinberg (1977)  24.1  [107] 
Gasser and Leutwyler (1985)  23.2(1.8)  [11] 
Donoghue et al. (1993)  21.8  [99] 
Kambor et al. (1996)  22.4(9)  [14] 
Anisovich and Leutwyler (1996)  22.7(8)  [15] 
Walker (1998)  22.8(8)  [108] 
Amoros et al. (2001)  21.3  [109] 
Martemyanov and Sopov (2005)  22.8(4)  [110] 
Bijnens and Ghorbani(2007)  23.2  [12] 
Kastner and Neufeld (2008)  20.7(1.2)  [111] 
Kampf et al. (2011)  23.1(7)  [43] 
Lanz (2011)  21.31(\(^{+59}_{50}\))  [118] 
FLAG (\(N_f = 2 + 1\)) (2016)  22.5(8)  [27] 
FLAG (\(N_f = 2 + 1 + 1\)) (2016)  22.2(1.6)  [27] 
BMW (\(N_f = 2 + 1\)) (2016)  23.4(6)  [92] 
JPAC (2017)  21.6(1.1)  [47] 
Albaladejo and Moussallam (2017)  21.5(1.0)  [49] 
RM123 (\(N_f = 2 + 1 + 1\)) (2017)  23.8(1.1)  [93] 
This work  22.1(7) 
Table 9 compares our value of Q with results found in the literature. The numbers listed are either given in the quoted papers or are calculated from the estimates for the quark masses or mass ratios given therein. The first crude estimate for the masses of the three lightest quarks within QCD, \(m_u\simeq 4\) MeV, \(m_d\simeq 6\) MeV, \(m_s\simeq 135\) MeV [106] appeared in 1975 – the entry in the first line is calculated from these numbers. The value given in the second line is obtained from the current algebra formulae for \(M_{\pi ^+}^2\), \(M_{K^+}^2\) and \(M_{K^0}^2\), corrected for electromagnetic selfenergies with Dashen’s theorem [107] (tree approximation of \(\chi \)PT). The significance of the quark mass ratio Q for the chiral expansion of the meson masses was noticed only in 1985 [68]. The third line represents the result of a \(\chi \)PT calculation to one loop [11], where the quantity \(\kappa \equiv 1/Q^2\) was determined from the experimental decay rate. Note that, at that time, the rate was still subject to substantial uncertainties – since then, the value of \(\varGamma _{\eta \rightarrow \pi ^+\pi ^\pi ^0}\) quoted by the Particle Data Group increased by more than three standard deviations: from 197(29) eV to 299(11) eV. As the result for Q is inversely proportional to the fourth root of the rate, the oneloop result 23.3(1.8) quoted in Ref. [11] drops to \(Q=20.9(1.6)\) if the erroneous input used for the width is corrected.
9.4 Chiral expansion of the meson masses
The upshot of the above discussion is that, in QCD, the chiral expansion of the squares of the Nambu–Goldstone masses is dominated by the leading terms. At the physical values of \(m_u\), \(m_d\), \(m_s\), the corrections \(\varDelta _S\), \(\varDelta _R\), \(\varDelta _{M_K}\) from the higher order terms were found to be remarkably small and the lowenergy theorem (9.14) suggests that \(\varDelta _Q\) is even smaller. We emphasize that these statements concern the dependence of the meson masses on the masses of the quarks and do not apply to the expansion in powers of the momenta. The example of \(\pi \pi \) scattering shows that even within SU(2)\(\times \)SU(2), the expansion in powers of the momenta picks up sizeable contributions from the final state interaction already at threshold. It is essential that our analysis relies on dispersion theory for the momentum dependence – as discussed in detail in Sect. 6, \(\chi \)PT does not describe the momentum dependence of the transition amplitude sufficiently well in the physical region of the decay, even if the contributions arising at NNLO of the chiral perturbation series are taken into account.
9.5 Comparison with the lattice results for Q
Finally, we compare our results for Q with the most recent determinations on the lattice. Table 9 shows that, while the results reviewed in the FLAG report [27] for simulations with 3 or 4 flavours are quite consistent with ours, the most recent determinations, BMW (\(N_f=2+1\)) [92] and RM123 (\(N_f=2+1+1)\) [93] are higher than our value (9.12) by 1.5 and 1.4 standard deviations, respectively. As mentioned in Sect. 9.1, the results obtained in these references for the kaon mass difference are consistent with ours. Also, the uncertainties in the values of the isospin limits \(M_\pi \) and \(M_K\) are much too small to explain the discrepancy. Hence the difference must arise from the correction term \(\varDelta _Q\) in the lowenergy theorem (9.13), which is beyond the accuracy of our calculation.
We only list the central values – since the quantities \(\hat{M}_{K^0}^2\hat{M}_{K^+}^2\), R and Q are strongly correlated, a meaningful error estimate requires knowledge of the correlations and is thus beyond our reach. The outcome for \(\varDelta _S\) and \(\varDelta _R\) confirms that the first order corrections are small, but \(\varDelta _R\) is of the same sign as \(\varDelta _S\): on the right hand side of (9.30), the two contributions cannot possibly cancel. Hence the result for \(\varDelta _Q\) is in conflict with the expectation that effects of second order are smaller than those of first order.
While completing the present work, the Fermilab Lattice, MILC & TUMQCD collaborations came up with a new lattice determination of the quark masses [115]. Unfortunately, the paper does not contain a result for the ratio Q, but neglecting correlations and adding errors in quadrature, the mass ratios which are given therein, \(S= 27.182(46)(56)(1)\) and \(m_u/m_d=0.4517(55)(101)\), imply \(Q=22.1(3)\) and \(R=34.7(1.0)\). The central values are very close to our numbers in Eqs. (9.12) and (9.28). Accordingly, the outcome of this calculation appears to be consistent with a coherent chiral expansion of the meson masses and to confirm that the corrections to the current algebra formulae are small. Although the paper focuses on the determination of the masses of the heavy quarks, the ratios \(m_u/m_d\) and \(m_s/m_{ud}\) are given to remarkable accuracy. In particular, the precision claimed for S is breathtaking – the quoted uncertainty is about four times smaller than for the FLAG value (9.21) we are relying on and the uncertainty in the outcome for Q is smaller than ours by more than a factor of two. Concerning the comparison with [92, 93], the main difference is that the calculation is done within QCD rather than QCD + QED. The outcome for the masses \(m_u\), \(m_d\) and \(m_s\) is corrected for e.m. effects, but for details of the procedure used, the reader is referred to a forthcoming paper by the MILC collaboration.
10 Comparison with other work
10.1 Dispersive approaches

The phase shifts adopted in [14, 15] were taken from [116], whereas we are now able to use solutions of Roy equations matched to \(\chi \)PT [16, 57].

At that time, accurate data on the Dalitz plot in the charged channel were not available yet, so that the best one could do to fix the subtraction constants was to match them to \(\chi \)PT.

The available \(\chi \)PT calculation was at one loop, and therefore there was no possibility to go beyond four subtraction constants.

The treatment of isospin breaking corrections available at that time [72] was not yet as complete as the one provided in [18].
Figure 20 amounts to an update of a picture drawn by Anisovich and Leutwyler, more than twenty years ago, in order to illustrate the effects generated by the final state interaction [15]. The framework underlying that paper is essentially the same as the one used in the construction of the matching solution fit\(\chi _4\) in Sect. 3.5: a dispersive analysis with four subtraction constants, which are determined by imposing theoretical constraints derived from \(\chi \)PT. The figure concerns the behaviour of the real part of the amplitude \(M_c(s,t,u)\) along the line \(s=u\), in the isospin limit.
In the present work, the convention used for the value of the pion mass in the isospin limit is irrelevant, because we account for isospin breaking when comparing our calculation with experiment. In Fig. 20, however, it does matter: the straight line that shows the behaviour at leading order (LO), for instance, depends on it. We identify the isospin limit of the pion mass with the mass of the charged pion, while in [15], the mass of the neutral pion was used. If isospin breaking corrections are not applied, that choice is preferable because isospin breaking in the masses of the pions is dominated by electromagnetism, which barely affects the mass of the neutral pion. We correct for the difference in the same way as for the isospin breaking corrections, using \(\chi \)PT. At LO, the transformation of the amplitude from one convention to the other amounts to a mere rescaling of the vertical axis, by the factor \(M_{\pi ^+}^2/M_{\pi ^0}^2\,(M_\eta ^2M_{\pi ^0}^2)/(M_\eta ^2M_{\pi ^+}^2)\simeq 1.074\). At oneloop, the isospin limit of the chiral representation is given by \(M_c^{\mathrm {GL}}(s,t,u)\) and the real parts are readily worked out for \(M_\pi =M_{\pi ^0}\) as well as for \(M_\pi =M_{\pi ^+}\). The ratio of the real parts remains roughly constant, but at a slightly larger value. We expect this to be the case for the dispersive representation as well – the red curve in Fig. 20 is obtained from the one shown in the old figure by stretching the values with the oneloop result for the ratio of the real parts.
For comparison, the open circles in Fig. 20 show the real part of the amplitude belonging to the matching solution, fit\(\chi _4\). The main difference between this representation and the one obtained in Ref. [15] is that the \(\pi \pi \) phase shifts are now known much more precisely. The figure shows that the old calculation underestimates the amplification of the amplitude by the final state interaction at threshold, but overestimates its growth with the energy.
The figure also shows the outcome of two more recent calculations [43, 47]. Kampf, Knecht, Novotný and Zdrádhal [43] have adopted a dispersive approach as well, but instead of solving the dispersion relations numerically, they have solved them analytically by iterations, stopping at the second iteration. This corresponds to a twoloop \(\chi \)PT representation from the analytic point of view, but the subtraction constants are not exactly related to the LEC of \(\chi \)PT, as the authors explain in their paper. In this connection, we refer to the detailed comparison of the dispersive approach with the twoloop representation of \(\chi \)PT given above (Sect. 6). Their approach also differs from ours in the way the normalization of the amplitude is fixed from theory: while we use the value of the Taylor invariant \(K_0\), they use the imaginary part of the amplitude along the line \(t=u\).
Figure 20 compares their result for the real part of the amplitude along the line \(s=u\) with the outcome of the present work. By construction, both representations reproduce the Dalitz plot distribution of KLOE – in the physical region of the decay, they are nearly the same up to normalization. Below threshold, however, the difference is very clearly visible: at small values of s, where current algebra predicts the occurrence of an Adler zero at \(s=\frac{4}{3}M_\pi ^2\), the amplitude of Kampf et al. goes astray. We encountered a similar phenomenon in Sect. 5.4: Fig. 10 shows that our calculation also goes astray if we allow for 6 subtraction constants and fit the data on the Dalitz plot distribution by treating these as free parameters. According to Martin Zdráhal [119], this deficiency can be repaired without affecting significantly the rest of the calculation and in particular the fit to data, but detailed results for this improved analysis within their approach have not been published. Note also that their work does not account for isospin breaking corrections. The published value \(Q=23.3(8)\) is significantly higher than ours, but in view of the shortcomings of the underlying analysis, this does not come as a surprise.
More recently, the JPAC collaboration [45, 46, 47] has also analyzed \(\eta \rightarrow 3 \pi \) decays, and in particular KLOE data, with a dispersive approach and the aim to determine the value of Q. The spirit is similar to the one adopted here, but the way in which the dispersion relations for this process are solved differs significantly from ours and isospin breaking corrections are not applied. The authors make an approximate treatment of the lefthand cut for the partial wave amplitudes, and assume that it can be well described by a polynomial. As we have demonstrated here (following [15]), the iterative procedure for deriving solutions of the dispersion relation converges fast and takes into account crossed channels (responsible for the lefthand cut) exactly. It is possible that the polynomial approximation adopted in [46, 47] works reasonably well, but having the exact solution available, this becomes an academic question. We are indebted to Igor Danilkin for providing us with the numerical values shown in Fig. 20. In the physical region of the decay, their results are consistent with ours and the same holds for the value obtained for the quark mass ratio, \(Q=21.6(1.1)\). Unfortunately, the method used does not work below the physical region, so that the behaviour in the vicinity of the Adler zero cannot be compared.
In Refs. [50, 51, 52, 53] Kolesár and Novotný take a very different point of view from the one adopted here – namely that the reason for the bad convergence of \(\chi \)PT for this decay is understood and has to do with large finalstate rescattering effects – and try to identify the reasons for the bad convergence within the framework of the socalled resummed Chiral Perturbation Theory (rChPT) [120, 121]. In this approach, vacuum fluctuations of \(\bar{s} s\) pairs are treated in a special way and their effect resummed. Their size is left unconstrained, which implies that both the SU(3) condensate and decay constant are treated as free parameters, having possibly a very different value than their SU(2) counterparts. The idea is very intriguing and if one could find a way to rigorously determine the size of these SU(3) parameters, this would be a very interesting result.
The present work shows that rescattering effects can be accounted for in a systematic, nonperturbative manner. Causality and unitarity determine the momentum dependence of the transition amplitude up to a set of subtraction constants – \(\chi \)PT is used exclusively to work out the constraints on these constants arising from chiral symmetry. Our analysis, in particular, does not rely on the chiral expansion for quantities that contain strong infrared singularities and are notoriously difficult to deal with in \(\chi \)PT.
Very recently, Albaladejo and Moussallam [48] have shown how to extend the dispersive formalism we have used in the present work to include the effect of inelastic twobody effects, like \(\bar{K}K\) and \(\eta \pi \). This remarkable and very useful technical advance allowed them to explicitly take into account effects related to narrow resonances in the oneGeV region, like the \(a_0(980)\) and the \(f_0(980)\). From their numerical analysis, they conclude that the effect on the determination of Q are of the order of 0.2 units, and therefore much smaller than the error. They also invoke the KLOE data on the Dalitz plot distribution in the charged channel to constrain their representation and to predict the coefficients of the distribution in the neutral channel. Setting \(\gamma =0\), they obtain \(\alpha =0.0337(12)\), \(\beta =00054(1)\), to be compared with our result (7.16). While our value for \(\alpha \) is smaller than theirs by about 2 \(\sigma \), we do confirm their value of \(\beta \). The difference may in part arise because their analysis does not account for isospin breaking corrections, in part because the terms proportional to \(\alpha \) and \(\beta \) in the Taylor series (7.13) provide a decent approximation only in the immediate vicinity of \(Z=0\). As discussed in Sect. 7, the curvature term \(\gamma \) affects the behaviour away from the center of the physical region – setting it to zero distorts the result for \(\alpha \). At any rate, we consider it very unlikely that the difference has to do with the presence of inelastic channels. The plots shown in [48] indicate that – in the physical region of the decay – the effects generated by these are well described by a polynomial. In our calculation, such contributions are absorbed in the subtraction constants. We do therefore not expect that explicitly accounting for inelastic channels would lead to a significant change in our results.
10.2 Nonrelativistic effective field theory
A different approach which has been applied to \(\eta \rightarrow 3 \pi \) decays is the one relying on a nonrelativistic Lagrangian. This has been very successful in describing \(K \rightarrow 3 \pi \) decays and in particular the cusp structure at the opening of the \( \pi ^+ \pi ^\) channel in the \(2 \pi ^0\) spectrum of the \(K^\pm \rightarrow \pi ^\pm 2 \pi ^0\) decay [37, 38, 39, 40]. In this framework one makes a nonrelativistic expansion both at the level of the Lagrangian as well as in the calculation of rescattering effects. The importance of the latter is controlled by the scattering lengths, which happen to be small (as a consequence of the Nambu–Goldstoneboson nature of the pions): technically, the NREFT also relies on an expansion in the scattering lengths. From the calculation point of view, rescattering effects are taken care of automatically by the loop expansion of quantum field theory. A significant advantage of this approach is that one does not rely on an expansion in the quark masses: the treelevel decay amplitude near to threshold is expanded in the spatial momentum squared, and the coefficients of this expansion are treated as free parameters. Which means that in this approach one does not have to worry about the slow convergence of \(\chi \)PT for the scattering lengths, for example, because these are by definition the physical values. The only question that matters in this case is whether one is close enough to threshold that the nonrelativistic expansion works.
The nonrelativistic approach is applied to the decay \(\eta \rightarrow 3\pi \) in Refs. [41, 42]. The mass difference between the charged and neutral pions is accounted for and the cusp due to the opening of the \(\pi ^+ \pi ^\) channel in the \(\pi ^0 \pi ^0\) spectrum of the decay \(\eta \rightarrow 3 \pi ^0\) is analyzed in detail. Moreover, fitting the free parameters in the nonrelativistic representation of the transition amplitude to the KLOE data available at the time, the authors of Ref. [41] did obtain a negative value for the slope \(\alpha \) in the neutral channel, as observed. A comparison of the predicted Dalitz plot in the neutral channel with the data by MAMIC shows that the calculation is in reasonable agreement with the data: in particular that, as one moves from treelevel to one and then to two loops (in the NR expansion), the curves obtained move towards the data and show a good convergent behaviour.
It is worth emphasizing here the difference between our approach and the NR expansion: while in a dispersive treatment rescattering effects (in the S and P waves) are treated exactly, the NR expansion applies a perturbative scheme to account for these. However, the treatment of isospin breaking effects can be done in a theoretically much cleaner way within the NR approach. We have relied on oneloop \(\chi \)PT and a factorization hypothesis, which can only be approximately correct. To exemplify the difference between the two approaches it is useful to compare the Dalitz plot in the neutral channel: in the NR approach the strength of the cusp effect is exactly described in terms of the Swave scattering lengths, according to a venerable lowenergy theorem [87]. If these were taken from experiment, then the strength of the cusp would be correct by definition.
In Ref. [42] this approach has been further refined and extended to include isospin breaking corrections beyond the \(\pi ^+\pi ^0\) mass difference, and a complete set of formulae describing these decays in the NR expansion have been provided. In this paper the question whether fitting the Dalitz plot data in the charged channel correctly reproduces the Dalitz plot in the neutral channel has been addressed thoroughly. The conclusion is similar to the one obtained by Gullström et al. [41], namely that the agreement with the data in the neutral channel is marginal. In particular, only at the two loop level does the value of \(\alpha \) become negative, and only after a partial resummation of rescattering effects does it get close to the measured value. For the coefficients of the Dalitz plot distribution in the neutral channel, Schneider et al. [42] obtain \(\alpha =0.0246(49)\), \(\beta = 0.0042(7)\), \( \gamma =0.0013(4)\), based on matching to \(\chi \)PT and resummation of bubble graphs. Although the ingredients of this calculation are quite different from ours, the comparison with the numbers in (7.13) shows that the qualitative properties of the prediction for the Dalitz plot distribution in the neutral channel are the same.
Reference [42] also proposes a different approach to the determination of \(\alpha \) within the NREFT formalism: the authors derive an exact relation (in the isospin limit) between the Dalitz plot parameters in the charged channel and the slope \(\alpha \) in the neutral channel and show that if one inputs the parameters measured by KLOE and estimates the imaginary part of a combination of Dalitz plot parameters (defined as \(\text {Im}\,\,\bar{a}\)) within the NR expansion, one obtains a value for \(\alpha \) which is only in marginal agreement with the measured value. This remains true even after calculating isospin breaking corrections. We have analyzed this apparent clash in some detail and came to the conclusion that the estimate of the parameter \(\text {Im}\,\,\bar{a}\) within the NR expansion does not seem to be reliable. The reasoning is as follows: if we fit the KLOE data and calculate the slope at \(Z=0\) with our dispersive representation we get \(\alpha =0.0302(13)\), in agreement with the PDG value. This evaluation accounts for isospin breaking effects. As discussed in Sect. 5.8, the polynomial approximation to our central solution agrees well with the experimental determination by KLOE. If we now insert these numbers in Eq. (6.9) of Ref. [42] and rely on their estimate of \(\mathrm {Im}\,\bar{a}\) we get \(\alpha = 0.0474\), in substantial disagreement with our own direct determination. Since Eq. (6.2) of Ref. [42] is algebraically exact, and the estimate of the isospin breaking effects (leading to Eq. (6.9)) only gives a small correction, the problematic step must be in the estimate of \(\text {Im}\,\,\bar{a}\).
An even better test of the NREFT approach would be to analyze the data along the lines of Sect. 5.9
11 Summary and conclusions
 1.The essential properties of the framework we are using to analyze the transition amplitude of the decay \(\eta \rightarrow 3\pi \) were derived long ago [30, 31, 32]. The decay violates the conservation of isospin. Since chiral symmetry suppresses the electromagnetic interaction in this transition [2], the dominating contribution arises from QCD and is proportional to the difference \(m_dm_u\) of quark masses. It is convenient to normalize the amplitude withwhere \(\hat{M}_{K^0}\) and \(\hat{M}_{K^+}\) denote the kaon masses in QCD.$$\begin{aligned} A_{\eta \rightarrow \pi ^+\pi ^\pi ^0}=\frac{\hat{M}_{K^0}^2\hat{M}_{K_+}^2}{3\sqrt{3} F_\pi ^2}\,M_c(s,t,u) \end{aligned}$$(11.1)
 2.
The first part of the present paper reviews the dispersion theory of the amplitude \(M_c(s,t,u)\) in the isospin limit (\(e\rightarrow 0\), \(m_u\rightarrow m_d\)), where this function also determines the amplitude relevant for the transition \(\eta \rightarrow 3\pi ^0\). We follow the dispersive analysis set up in [15], which exploits the fact that, at low energies, the angular momentum barrier suppresses the imaginary parts of the D and higher partial waves. Neglecting these, the amplitude can be decomposed into three isospin components, which only depend on a single variable: \(M_0(s)\), \(M_1(s)\), \(M_2(s)\) – see Eq. (2.17).
 3.
Elastic unitarity determines the discontinuities of the isospin components across the branch cuts associated with collisions among pairs of pions, in terms of the S and Pwave \(\pi \pi \) phase shifts. We write the corresponding dispersion relations in the form (2.33), allowing for six subtraction constants: \(\alpha _0\), \(\beta _0\), \(\gamma _0\), \(\delta _0\), \(\beta _1\), \(\gamma _1\). These relations represent a set of integral equations that uniquely determine the amplitude in terms of the subtraction constants. Moreover, since the equations are linear in the subtraction constants, the general solution is given by a linear combination of six fundamental solutions that can be determined once and for all.
 4.
At the experimental accuracy reached, the electromagnetic interaction cannot be ignored. In particular, the e.m. selfenergy of the charged pion modifies the amplitude obtained from QCD quite significantly. We rely on the representation of Ditsche, Kubis and Meißner [18], who evaluated the transition amplitude within the effective theory of QCD+QED, to first nonleading order of the chiral expansion and to order \(e^2\) in the electromagnetic interaction. Their analysis in particular also accounts for the emission of the soft photons that necessarily accompany the decay as well as for the Coulomb pole generated by the attraction among the charged pions in the final state. We assume that the data are radiatively corrected in accordance with their analysis.
 5.
A substantial part of the e.m. interaction can be accounted for with a purely kinematic map that takes the physical phase space of the decay \(\eta \rightarrow \pi ^+\pi ^\pi ^0\) onto the phase space of the isospin symmetric world. Applying this map and removing the Coulomb pole, the isospin breaking corrections reduce to an approximately constant numerical factor, except near \(s=4M_{\pi ^+}^2\), where a visible structure due to the interference of the branch cuts from \(\pi ^+\pi ^\) and \(\pi ^0\pi ^0\) intermediate states remains (left panel of Fig. 8). Isospin breaking in the decay \(\eta \rightarrow 3\pi ^0\) can be treated analogously. In that case, a Coulomb pole does not occur. Instead there is a small cusp due to the virtual transition \(\pi ^0\pi ^0\rightarrow \pi ^+\pi ^\rightarrow \pi ^0\pi ^0\) (right panel of Fig. 8). Those isospin breaking effects that are not taken care of by the kinematic map are accounted for only in oneloop approximation.
 6.
The theoretical constraints that follow from the fact that the pions are Nambu–Goldstone bosons of a hidden approximate symmetry can be worked out by means of Chiral Perturbation Theory. The representation of the amplitude obtained on this basis does have the structure of Eq. (2.17), up to and including NNLO. The only qualitative difference compared to the dispersive framework we are using is that the chiral representation corresponds to an extended version of elastic unitarity, which also accounts for the discontinuities generated by \(K\bar{K}\), \(\eta \eta \) and \(\pi \eta \) intermediate states. In the region relevant for \(\eta \) decay, the contributions generated by these singularities are very small and well described by their Taylor expansion in powers of s. As we are working with sufficiently many subtractions, they can be absorbed in the subtraction constants.
 7.
At leading order of the chiral expansion (current algebra), the transition amplitude of the decay \(\eta \rightarrow \pi ^+\pi ^\pi ^0\) is independent of t and u, grows linearly with s and has an Adler zero at \(s=\frac{4}{3}M_\pi ^2\): \(M_c(s,t,u)=(3s4M_\pi ^2)/(M_\eta ^2M_\pi ^2)\). Although the zero occurs outside the physical region, the data on the Dalitz plot distribution beautifully confirm its presence: ignoring the theoretical constraints altogether and allowing only four subtraction constants, the dispersive representation yields a very good fit of the data (Sect. 5.3, \(\hbox {fitK}_4\)). Along the line \(s=u\), the real part of this representation indeed passes through zero at \(s=1.43M_\pi ^2\), close to the place where current algebra predicts this to happen.
 8.
The information provided by \(\chi \)PT is essential, because the Dalitz plot distribution leaves the normalization of the amplitude open. To establish contact between the dispersive and chiral representations, we consider the region where the uncertainties in the latter are smallest, i.e. focus on small values of s in \(M_0(s)\), \(M_1(s)\), \(M_2(s)\) and compare Taylor coefficients. The requirement that the oneloop representation, which does not involve any unknowns, yields an acceptable approximation at low energies allows us to consistently combine the two. In particular, we normalize the dispersive representation with the oneloop value of the coefficient \(H_0\), accounting for the higher order contributions merely by attaching an uncertainty estimate to this value.
 9.
There is an alternative to \(\hbox {fitK}_4\), which we denote by fit\(\chi _4\): a dispersive representation that also uses only four subtraction constants, but incorporates the theoretical information instead of the one obtained at KLOE. It is uniquely determined by the requirement that the isospin components of the dispersive representation match those of the oneloop representation at small values of s. Figure 3 shows that the oneloop approximation accurately follows the dispersive representation only below threshold – in the physical region, it underestimates the strength of the final state interaction. This manifests itself particularly clearly in the Dalitz plot distribution of the neutral decay mode: Fig. 5 shows that the curvature of the two representations differs even in sign.
 10.
The same deficiency also shows up at two loops: the lowest resonance of QCD, the \(f_0(500)\), is not described well enough even at NNLO of the chiral expansion. This implies that the twoloop representation does not have the necessary accuracy in the physical region – a meaningful comparison of theory and experiment is possible only in the framework of dispersion theory. The problem is illustrated in Fig. 13, which compares our central solution with the twoloop representation that matches it at low energies.
 11.
We emphasize that the analysis reported here became possible only very recently, with the accurate measurement of the Dalitz plot distribution for the decay \(\eta \rightarrow \pi ^+\pi ^\pi ^0\) at KLOE [22]. For the central solution of our system of equations, the errors arising from the experimental and theoretical uncertainties are of comparable size – \(\eta \)decay is a showcase for a fruitful interplay between theory and experiment.
 12.
As discussed in detail in Sect. 5.5, the simpler framework obtained by dropping the subtraction constants \(\delta _0\) and \(\gamma _1\) is too stiff – doing this amounts to imposing constraints that distort the transition amplitude. The need for the term \(\delta _0s^3\) in the subtraction polynomial of \(M_0(s)\) also shows up in connection with the polynomial approximation of the kaon loops: the contributions from the \(K\bar{K}\) cuts to \(M_0(s)\) are not accounted for sufficiently well by a quadratic polynomial, but a cubic one does suffice. Moreover, working with six subtraction constants has the advantage that – in the region of interest – the solutions are then not sensitive to the high energy tails of the dispersion integrals, where elastic unitarity does not represent a good approximation. In the error analysis, the uncertainties associated with the high energy tails are booked together with those in the phase shifts at low energies, where the Roy equations provide very good control – with six subtraction constants, the net uncertainty from these sources is very small.
 13.
The decomposition of the amplitude \(M_c(s,t,u)\) into its isospin components \(M_0(s)\), \(M_1(s)\), \(M_2(s)\) is unique only up to polynomials [see Eqs. (2.20), (2.21)]. For the dispersive representation, the ambiguity is disposed of when bringing the dispersion relations to the form (2.33). Alternatively, the solutions can be characterized by invariant combinations of Taylor coefficients: two solutions yield the same representation \(M_c(s,t,u)\) if and only if these invariants are the same. This allows us to unambiguously characterize the twoloop representation that matches our central solution at low energies (see Sect. 6.2). A corresponding update of the lowenergy constants occurring in the effective Lagrangian at \(O(p^6)\) would be of considerable interest but is beyond the scope of the present work.
 14.
Isospin symmetry leads to a prediction for the branching ratio of the neutral and charged decay modes, \(B=\varGamma _{\eta \rightarrow 3\pi ^0}/\varGamma _{\eta \rightarrow \pi ^+\pi ^\pi ^0}\). The result of our calculation, \(B=1.44(4)\) is in good agreement with the values \(B=1.426(26)\) and \(B=1.48(5)\) quoted by the Particle Data Group [66].
 15.
The Dalitz plot distribution of the decay \(\eta \rightarrow 3\pi ^0\) can be expanded in powers of the variables \(X_n\), \(Y_n\). In the region where the series converges, \(X_n^2+Y_n^2<0.6\), our prediction is remarkably well approximated by the polynomial (7.13) – the coefficients are specified in (7.16). In the remainder of the physical region, the singularities generated by the final state interaction manifest themselves as cusps. The dominating contribution from these is described by the formula (7.17). Although they are too weak to stick out from the fluctuations in the data, the quantitative analysis does confirm their presence at the strength required by dispersion theory.
 16.
The MAMI data on the decay \(\eta \rightarrow 3\pi ^0\) [23, 24, 25] allow a strong test of our calculation. Isospin symmetry implies that the amplitude of this transition is described by the combination \(M_n(s)\equiv M_0(s)+\frac{4}{3}M_2(s)\) of the isospin components relevant for the charged channel – the KLOE data thus lead to a parameter free prediction for this decay. Figure 16 shows that the calculated distribution is in excellent agreement with the MAMI results.
 17.
The recent update provided by the A2 collaboration [25] now allows an analysis of the Dalitz plot distribution that goes beyond the linear approximation. The data in the neutral channel do not by themselves determine the slope very accurately, but impose a strong correlation between the slope \(\alpha \) and the curvature \(\gamma \). Dispersion theory provides the missing element as it determines the curvature within narrow limits. Our analysis, which relies on the KLOE data for \(\eta \rightarrow \pi ^+\pi ^\pi ^0\) and on the theoretical constraints that follow from the presence of a hidden approximate symmetry, predicts both the slope and the curvature rather precisely: \(\alpha = 0.0303(12)\), \(\gamma =0.0019(3)\). The slope is somewhat smaller than the average \(\alpha =0.0318(15)\) quoted by the Particle Data Group [66]. Including the MAMI data [25] in the dispersive analysis, we obtain a result that is even a little smaller: \(\alpha = 0.0294(10)\). Unfortunately, the likelihood of the fits to the MAMI results is not satisfactory: \(\chi ^2_{\mathrm {M}}/\mathrm {dof}=1.25\) for the polynomial fit to these data alone and \(\chi ^2_{\mathrm {M}}/\mathrm {dof}=1.27\) for the dispersive fit, which combines them with the data from KLOE.
 18.
Our result \(\hat{M}_{K^0}^2\hat{M}_{K^+}^2 = 6.3(4) 10^{3}\,\text{ GeV }^2\) for the kaon mass difference in QCD agrees with recent determinations of the electromagnetic selfenergies on the lattice [92, 93]. We thus confirm that the strong infrared singularities occurring in the chiral expansion of the kaon selfenergies subject the Dashen theorem to a large correction from higher orders. For the parameter which measures the size of this correction, we find \(\epsilon =0.9(3)\).
 19.
Finally, we invoke the lowenergy theorem which relates the kaon mass difference to the ratio \(Q^2\equiv (m_s^2m_{ud}^2)/(m_d^2m_u^2)\) of quark masses [68]. The theorem can be compared with the Gell–Mann–Okubo formula, but there is an important difference: while that formula only holds at leading order of the chiral expansion and picks up corrections of first nonleading order, the relation relevant for Q receives corrections only at nexttonexttoleading order. This implies that, instead of expressing the decay rate in terms of the kaon mass difference, we can just as well express it in terms of the quark mass ratio Q. Conversely, the measured decay rates in the charged and neutral channels yield two independent determinations of this mass ratio. The two results agree very well with one another – combining them, we obtain \(Q =22.1(7)\), where the error includes all sources of uncertainty encountered in the calculation, including an estimate for the neglected higher order contributions in the chiral series.
 20.
The ratio \(S\equiv m_s/m_{ud}\) is now known remarkably well from lattice calculations. With the value \(S=27.30(34)\) quoted by FLAG for simulations with four quark flavours [27], our result for Q leads to \(R\equiv (m_sm_{ud})/(m_dm_u)= 34.2(2.2)\) and \(m_u/m_d=0.44(3)\). These numbers indicate that, within QCD, the chiral expansion of the square of the Nambu–Goldstone masses is dominated by the leading terms, i.e. by the linear formulae of current algebra. At the physical values of \(m_u\), \(m_d\), \(m_s\), the higher order contributions amount to remarkably small corrections.
 21.
While the outcome of our calculation for the kaon mass difference in QCD agrees with the lattice results within errors, the values obtained for the isospin breaking quantities Q, R and \(m_u/m_d\) in two of the three most recent lattice calculations [92, 93] do not. We point out that the discrepancy concerns the size of the corrections arising in the lowenergy theorems for the corresponding ratios of meson masses. While the pattern obtained with our result for Q leads to a coherent picture, these lattice results imply that the corrections in R and S, which are of first order in chiral symmetry breaking are smaller than those in Q, despite the fact that the latter represent contributions of second order. In Sect. 9.5, we indicate a way to resolve this conundrum by means of a lattice simulation within QCD.
 22.
In the plane of the quark mass ratios \(m_u/m_d\) and \(m_s/m_d\), a given value of Q corresponds to an ellipse, while a given value of S corresponds to a straight line. The yellow band in the left panel of Fig. 21 represents the region allowed by our result for Q, while the grey band represents the region allowed by the lattice result for S quoted by FLAG. For comparison, the figure also indicates the first estimates of the three lightest quark masses [106, 107], which appeared shortly after the discovery of QCD. The hexagon represents the rough estimates for the range in the variables S, R and \(m_u/m_d\) where the chiral expansion yields a coherent picture, obtained many years ago [122].
The right panel focuses on the region of physical interest and includes recent results obtained on the lattice. In particular, it compares the outcome of our work with the region allowed by the lattice results according to FLAG [27] and to the Particle Data Group [66]. The outcome of the three most recent lattice calculations (BMW [92], RM123 [93], Bazavov et al. [115]) is also indicated – the regions shown are obtained by treating the values obtained for S and \(m_u/m_d\) as statistically independent.^{15}
 23.
In Sect. 10, our analysis is compared with related work. There are two significant improvements compared to the early dispersive analyses in Refs. [14, 15]: the experimental information about \(\eta \)decay improved very substantially and the phase shifts of \(\pi \pi \) scattering are now under much better control. Concerning the properties of the Dalitz plot distribution, the various investigations are now in reasonable agreement. In order to establish contact with QCD and to extract information about the quark masses from \(\eta \)decay, however, the theoretical constraints that follow from the fact that the pions and the \(\eta \)meson are NambuGoldstone bosons of a hidden approximate symmetry play a crucial role. These constraints can be analyzed in a controlled manner in the framework of \(\chi \)PT, but care must be taken not to leave the region where the first few terms of the chiral perturbation series provide a decent approximation. Some of the analyses found in the literature, for instance, rely on matching the dispersive and chiral representations directly in the physical region of the decay. Since the first few terms of the chiral perturbation series do not represent a good approximation there, this leads to incorrect conclusions.
 24.
The nonrelativistic effective theory provides a representation of the transition amplitude for the decay \(K\rightarrow 3\pi \) that works very well [37, 38, 39, 40]. The method even leads to a coherent analysis of the contributions from the electromagnetic interaction. Since \(M_\eta \) is not much larger than \(M_K\), this approach can be expected to work for \(\eta \rightarrow 3\pi \) as well. We have verified that the amplitude of Ref. [38] indeed fits the KLOE data perfectly well. Moreover, in the isospin limit and in the physical region, the NR framework yields an excellent approximation of our solutions. The subtraction constants of the dispersive solutions that match the NR amplitude have a sizeable imaginary part, but, throughout the physical region, the difference between the two representations is very small, for the imaginary part as well as for the real part. This demonstrates that the NR effective theory provides a suitable framework for the analysis of \(\eta \)decay.
 25.
It is not a straightforward matter to establish contact between the nonrelativistic effective theory and the quark masses which occur in the QCD Lagrangian. Our approach relies on the assumption that, in the vicinity of the Adler zero, the oneloop representation of \(\chi \)PT provides a good approximation. The Adler zero is outside the region where the truncated expansion of the nonrelativistic effective theory represents a good approximation, but the link can be established by matching the dispersive and nonrelativistic representations in the isospin limit: (i) Determine the Dalitz plot distributions in the charged and neutral channels within the nonrelativistic framework. (ii) Take the isospin limit of the transition amplitude and expand it in powers of the spatial momenta of the three pions in the rest frame of the \(\eta \). (iii) Match the coefficients of this expansion – the analogues of the scattering lengths – to those of the generic dispersive representation. It would be most interesting to carry this out, but we leave this for the future.
Footnotes
 1.
 2.
The relative phase of the amplitudes for the charged and neutral channels depends on the convention used to specify the phase of the oneparticle states. We are working with \(\pi ^{\pm }\rangle =(\pi ^1\rangle \pm i \pi ^2\rangle ) /\sqrt{2}\), \(\pi ^0\rangle =\pi ^3\rangle \).
 3.
The mass of the \(\eta \) is protected from isospin breaking: the e.m. selfenergy vanishes at leading order of the chiral expansion and the expansion of \(M_\eta \) in powers of the difference \(m_dm_u\) only starts at \(O(m_dm_u)^2\). The difference between the physical mass of the \(\eta \) and its value in the isospin limit is beyond the accuracy of our calculation.
 4.
In the letter version of the present paper, we shortened the presentation by working with a single set of invariants, completing the set {\(H_0\), \(H_1\), \(H_2\), \(H_3\)} with \(H_4\equiv K_4\) and \(H_5\equiv K_5\).
 5.
Throughout, numerical values of dimensionful quantities are given in GeV units.
 6.
An analogous phenomenon occurs at one loop, where the invariant \(H_3\) does not pick up any contribution from the effective Lagrangian of \(O(p^4)\).
 7.
Value obtained for the convention we are using, where \(M_\pi =M_{\pi ^+}\).
 8.
Since the symmetry with respect to \(\tau \leftrightarrow \tau \) also holds in the presence of isospin breaking, the first term in the Taylor series of \(\tau [s_c,\tau _c]\) with respect to \(\tau _c\) vanishes.
 9.
The original notation allowed for additional terms (c, e) with odd powers of \(X_c\). Since crossing symmetry implies that the amplitude is even under \(X_c\rightarrow X_c\), we are omitting these.
 10.
We stick to the notation introduced by Schneider et al. [42]
 11.
The numerical values differ slightly from those given in Ref. [3], partly because the experimental results for the decay rates quoted by the Particle Data Group have changed in the meantime, partly because we improved the accuracy of the numerical representation of the fundamental solutions. As the shift in the central values amounts to less than a tenth of the quoted uncertainties, it is without significance.
 12.
Whenever three errors are given they are in the order: statistical, systematic, and systematic related to QED (quenching and finite volume).
 13.
In the notation used in that reference, \(\varDelta _S\) stands for \(\varDelta _M\).
 14.
In the notation of Ref. [68], \(\varDelta _{M_K}\) stands for \((M_\eta ^2+M_\pi ^2)/(3M_\eta ^2+M_\pi ^2)\varDelta _{\mathrm {GMO}}\) and involves the LECs \(L_5\), \(L_6\) and \(L_7\).
 15.
 16.
Appendices A and B concern the isospin limit. To simplify the notation we again drop the bar and use the symbols \(M_\pi \), \(M_\eta \) for the masses in the isospin limit.
 17.
Notes
Acknowledgements
We are indebted to Jürg Gasser and Akaki Rusetsky for letting us use the solutions of the integral equations, which they obtained with an entirely new method that yields significantly more accurate results than the one we used ourselves. Moreover, we warmly thank them as well as Bastian Kubis for carefully reading the manuscript, in particular for comments on the NREFT approach and on the analytic properties of the amplitude. Very useful information about the MAMI results on \(\eta \rightarrow 3\pi ^0\) – in particular also a sampling of the data that accounts for the indistinguishability of the three pions in the final state – was provided by Sergey Prakhov and is gratefully acknowledged. We also thank P. Adlarson, J. Bijnens, L. Caldeira Balkeståhl, I. Danilkin, A. Fuhrer, K. Kampf, A. Kupść, B. Moussallam, S. Simula and P. Stoffer for useful information. This work is supported in part by Schweizerischer Nationalfonds and the U.S. Department of Energy (contract DEAC0506OR23177) and National Science Foundation (PHY1714253).
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