Patterns of CP violation from mirror symmetry breaking in the $\eta\to\pi^+\pi^-\pi^0$ Dalitz plot

TL;DR

This paper investigates mirror symmetry violation in the eta decay Dalitz plot, revealing that CP-violating amplitudes with isospin 2 are more suppressed than those with isospin 0, providing insights into underlying symmetry breaking.

Contribution

It introduces a method to reconstruct the isospin of CP-violating structures from Dalitz plot data and applies it to recent experiments, uncovering new suppression patterns.

Findings

CP-violating amplitude with isospin I=2 is highly suppressed.

Analysis of experimental data confirms isospin dependence of CP violation.

Provides a new approach to study symmetry breaking in meson decays.

Abstract

A violation of mirror symmetry in the $\eta\to\pi^+\pi^-\pi^0$ Dalitz plot has long been recognized as a signal of C and CP violation. Here we show how the isospin of the underlying C- and CP-violating structures can be reconstructed from their kinematic representation in the Dalitz plot. Our analysis of the most recent experimental data reveals, for the first time, that the C- and CP-violating amplitude with total isospin $I=2$ is much more severely suppressed than that with total isospin $I=0$.

Equations

$$A_{LR}\equiv \frac{N_{+}-N_{-}}{N_{+}+N_{-}}\equiv \frac{1}{N_{\mathrm{tot}}}(N_{+}-N_{-}),$$

$$A(s,t,u)$$

$$M_1^C(s,t,u)$$

$$A(s,t,u)$$

$$N_{\pm }=\frac{1}{256{\pi }^3{M}_{\eta }^3}{\int }_{u\stackrel{>}{{}_{<}}t}dtdu|A(s,t,u){|}^2,$$

$$M_0^{/C}(s,t,u)=(s-t)M_1^{\prime }(u)+(u-s)M_1^{\prime }(t)-(u-t)M_1^{\prime }(s)$$

$$M_2^{/C}(s,t,u)$$

$$X$$

$$Y$$

$$|A(s,t,u){|}^2$$

$$\frac{1}{\xi }|A(s,t,u){|}_{/C}^2=M_1^C[\alpha M_0^{/C}+\beta M_2^{/C}{]}^{\ast }+H.c.$$

$$c$$

$$h$$

$$\text{Re}(\alpha )$$

$$\text{Re}(\beta )$$

$$\text{Im}(\alpha )$$

$$\text{Im}(\beta )$$

$$\text{Re}(\alpha )$$

$$\text{Im}(\alpha )$$

$$\text{Re}(\beta )$$

$$\text{Im}(\beta )$$

$$\text{Re}(\alpha )$$

$$\text{Im}(\alpha )$$

$$\text{Re}(\beta )$$

$$\text{Im}(\beta )$$

$$|{\pi }^{+}{\pi }^{-}{⟩}_{\mathrm{s}}|{\pi }^0⟩+|{\pi }^{+}{\pi }^0{⟩}_{\mathrm{s}}|{\pi }^{-}⟩+|{\pi }^{-}{\pi }^0{⟩}_{\mathrm{s}}|{\pi }^{+}⟩,$$

$$|{\pi }^i{\pi }^j{⟩}_{\mathrm{s}}\equiv \frac{1}{\sqrt{2}}\{|{\pi }^i{\pi }^j⟩+|{\pi }^j{\pi }^i⟩\},$$

$$\underset{L=1}{\underbrace{|{\pi }^i{\pi }^j{⟩}_{I=1,L=1}|{\pi }^k⟩}}$$

$$|{\pi }^i{\pi }^j{⟩}_{\mathrm{a}}\equiv \frac{1}{\sqrt{2}}\{|{\pi }^i{\pi }^j⟩-|{\pi }^j{\pi }^i⟩\},$$

$$|({\pi }^{+}{\pi }^{-}{)}_{I=1}{⟩}_{\mathrm{s}}|{\pi }^0⟩=|{\pi }^{+}{\pi }^{-}{⟩}_a(p_{{\pi }^{+}}-p_{{\pi }^{-}})\cdot p_{{\pi }^0}|{\pi }^0⟩.$$

$$|({\pi }^{+}{\pi }^{-}{)}_{I=0}⟩|{\pi }^0⟩\to M_0(s),$$

$$|({\pi }^{+}{\pi }^{-}{)}_{I=1}⟩|{\pi }^0⟩(p_{{\pi }^{+}}-p_{{\pi }^{-}})\cdot p_{{\pi }^0}\to M_1(s)\frac{u-t}{2},$$

$$|({\pi }^{+}{\pi }^0{)}_{I=1}⟩|{\pi }^{-}⟩(p_{{\pi }^{+}}-p_{{\pi }^0})\cdot p_{{\pi }^{-}}\to M_1(u)\frac{s-t}{2},$$

$$|({\pi }^{-}{\pi }^0{)}_{I=1}⟩|{\pi }^{+}⟩(p_{{\pi }^0}-p_{{\pi }^{-}})\cdot p_{{\pi }^{+}}\to M_1(t)\frac{u-s}{2},$$

$$|({\pi }^{+}{\pi }^{-}{)}_{I=2}⟩|{\pi }^0⟩\to M_2(s),$$

$$|({\pi }^{+}{\pi }^0{)}_{I=2}⟩|{\pi }^{-}⟩\to M_2(u),$$

$$|({\pi }^{-}{\pi }^0{)}_{I=2}⟩|{\pi }^{+}⟩\to M_2(t),$$

$$M_0(s)$$

Full text

I Introduction

The decay $\eta\rightarrow 3\pi$ first came to prominence after the observation of $K_{L}\to\pi^{+}\pi^{-}$ decay and the discovery of $CP$ violation in 1964 Christenson et al. (1964), because it could be used to test whether $K_{L}\to\pi^{+}\pi^{-}$ decay was generated by $CP$ violation in the weak interactions Lee and Wolfenstein (1965); Prentki and Veltman (1965). Rather, $CP$ violation could arise from the interference of the $CP$-conserving weak interaction with a new, “strong” interaction that breaks $C$ and $CP$; this new interaction could be identified through the appearance of a charge asymmetry in the momentum distribution of $\pi^{+}$ and $\pi^{-}$ in $\eta\to{\pi^{+}\pi^{-}\pi^{0}}$ decay Lee and Wolfenstein (1965); Lee (1965); Nauenberg (1965). Since $\eta\to{\pi^{+}\pi^{-}\pi^{0}}$ breaks G parity, isospin $I$ and/or charge-conjugation $C$ must be broken in order for the process to occur. Thus a charge asymmetry could arise from the interference of a $C$-conserving, but isospin-breaking amplitude with a isospin-conserving, but $C$-violating one Lee (1965). Numerical estimates were made by assuming that the isospin-violating contributions were driven by electromagnetism Lee (1965); Nauenberg (1965); Barrett et al. (1966). Since that early work, our understanding of these decays within the Standard Model (SM) has changed completely: the weak interaction does indeed break $CP$ symmetry, through flavor-changing transitions characterized by the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Moreover, isospin breaking in the strong interaction, mediated by the up-down quark mass difference Gross et al. (1979); Langacker and Pagels (1979); Leutwyler (1996), is now known to provide the driving effect in mediating $\eta\to 3\pi$ decay Gasser and Leutwyler (1985a); Anisovich and Leutwyler (1996); Bijnens and Gasser (2002); Bijnens and Ghorbani (2007), with isospin-breaking, electromagnetic effects playing a much smaller role Sutherland (1966); Bell and Sutherland (1968); Baur et al. (1996); Ditsche et al. (2009).

Modern theoretical studies of $\eta\rightarrow 3\pi$ decay focus on a complete description of the final-state interactions within the SM, in order to extract the isospin-breaking, light-quark mass ratio $Q\equiv\sqrt{(m_{s}^{2}-\hat{m}^{2})/(m_{d}^{2}-m_{u}^{2})}$, with $\hat{m}=(m_{d}+m_{u})/2$, precisely Anisovich and Leutwyler (1996); Bijnens and Gasser (2002); Bijnens and Ghorbani (2007); Colangelo et al. (2009); Kampf et al. (2011); Guo et al. (2015); Albaladejo and Moussallam (2017); Colangelo et al. (2017, 2018). There has been no further theoretical study of $CP$ violation in $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ decay since 1966. Since the $\eta$ meson carries neither spin nor flavor, searches for new physics in this system possess special features. For example, $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ decay must be parity $P$ conserving if the $\pi$ and $\eta$ mesons have the same intrinsic parity, so that $C$ violation in this process implies that $CP$ is violated as well. There has been, moreover, much effort invested in the possibility of flavor-diagonal $CP$ violation via a non-zero permanent electric dipole moment (EDM), which is $P$ and time-reversal $T$ violating, or $P$ and $CP$ violating if $CPT$ symmetry is assumed. Studies of flavor-diagonal, $C$ and $CP$ violating processes are largely lacking. We believe that the study of the Dalitz plot distribution in $\eta\rightarrow\pi^{+}(p_{\pi^{+}})\pi^{-}(p_{\pi^{-}})\pi^{0}(p_{\pi^{0}})$ decay is an ideal arena in which to search for $C$ and $CP$ violation beyond the SM. Were we to plot the Dalitz distribution in terms of the Mandelstam variables $t\equiv(p_{\pi^{-}}+p_{\pi^{0}})^{2}$ and $u\equiv(p_{\pi^{+}}+p_{\pi^{0}})^{2}$, the charge asymmetry we have noted corresponds to a failure of mirror symmetry, i.e., of $t\leftrightarrow u$ exchange, in the Dalitz plot. In contrast to that $C$ and $CP$ violating observable, a nucleon EDM could be mediated by a minimal $P$- and $T$-violating interaction, the mass-dimension-four $\bar{\theta}$ term of the SM, and not new weak-scale physics. The $\bar{\theta}$ term can also generate $\eta\to\pi\pi$ and $\eta/\eta^{\prime}\to 4\pi^{0}$ decay, breaking $P$ and $CP$ explicitly, so that limits on the decay rate constrains the square of a $CP$-violating parameter Pich and de Rafael (1991); Gutsche et al. (2017); Zhevlakov et al. (2019); Guo et al. (2012). Since the $\bar{\theta}$ term is $C$ even, it cannot contribute to the charge asymmetry, at least at tree level. Moreover, SM weak interactions do not support flavor-diagonal $C$ and $CP$ violation. Note that the charge asymmetry is linear in $CP$-violating parameters.

The appearance of a charge asymmetry and thus of $C$ (and $CP$) violation in $\eta\rightarrow\pi^{+}(p_{\pi^{+}})\pi^{-}(p_{\pi^{-}})\pi^{0}(p_{\pi^{0}})$ decay can be probed experimentally through the measurement of a left-right asymmetry, $A_{LR}$ Layter et al. (1972):

[TABLE]

where $N_{\pm}$ is the number of events with $u\stackrel{{\scriptstyle>}}{{{}_{<}}}t$, so that the $\pi^{+}$ has more (less) energy than the $\pi^{-}$ if $u>(<)t$ in the $\eta$ rest system. A number of experiments have been conducted over the years to test for a charge asymmetry in $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ decay, with early experiments finding evidence for a nonzero asymmetry Baltay (1966); Gormley et al. (1970, 1968), but with possible systematic problems becoming apparent only later, as, e.g., in Ref. Anastasi et al. (2016). Other experiments find no evidence for a charge asymmetry and $C$ violation Larribe et al. (1966); Layter et al. (1972, 1973); Jane et al. (1974); Ambrosino et al. (2008); Anastasi et al. (2016), and we note that new, high-statistics experiments are planned Gan (2017); Gatto et al. (2016); Beacham et al. (2020). It is also possible to form asymmetries that probe the isospin of the $C$-violating final state: a sextant asymmetry $A_{S}$, sensitive to the $I=0$ state Nauenberg (1965); Lee (1965), and a quadrant asymmetry $A_{Q}$, sensitive to the $I=2$ final state Lee (1965); Layter et al. (1972). These asymmetries are more challenging to measure and are only poorly known Layter et al. (1972). In this paper we develop a method to discriminate between the possible $I=0$ and $I=2$ final states by considering the pattern of mirror-symmetry-breaking events they engender in the Dalitz plot. Mirror-symmetry breaking as a probe of $CP$ violation has also been studied in untagged, heavy-flavor decays Burdman and Donoghue (1992); Gardner (2003); Gardner and Tandean (2004); Petrov (2004), with Ref. Gardner and Tandean (2004) analyzing how different $CP$-violating mechanisms populate the Dalitz plot. We also note Refs. Bediaga et al. (2009, 2012) for Dalitz studies of $CP$ violation in heavy-flavor decays.

II Theoretical Framework

The $\eta\to{3\pi}$ decay amplitude in the SM can be expressed as Gasser and Leutwyler (1985a); Anisovich and Leutwyler (1996)

[TABLE]

where we employ the Mandelstam variables $u$, $t$, and $s=(p_{\pi^{+}}+p_{\pi^{-}})^{2}$ and work to leading order in strong-interaction isospin breaking. Since $C=-(-1)^{I}$ in $\eta\to{3\pi}$ decay Lee (1965), the $C$- and $CP$-even transition amplitude with a $\Delta I=1$ isospin-breaking prefactor must have $I=1$. The amplitude $M(s,t,u)$ thus corresponds to the total isospin $I=1$ component of the $\pi^{+}\pi^{-}\pi^{0}$ state and can be expressed as Anisovich and Leutwyler (1996); Lanz (2011)

[TABLE]

where $M_{I}$(z) is an amplitude with $\pi-\pi$ rescattering in the $z$-channel with isospin $I$. This decomposition can be recovered under isospin symmetry in chiral perturbation theory (ChPT) up to next-to-next-to-leading order (NNLO), ${\cal O}(p^{6})$, because the only absorptive parts that can appear are in the $\pi-\pi$ $S-$ and $P$-wave amplitudes Bijnens and Ghorbani (2007). An analogous relationship exists in $\eta\rightarrow 3\pi^{0}$ decay Anisovich and Leutwyler (1996), though there is no Dalitz plot asymmetry and hence no effect linear in $CP$ violation in that case because the final-state particles are all identical.

Since we are considering $C$ and $CP$ violation, additional amplitudes can appear—namely, total $I=0$ and $I=2$ amplitudes. The complete amplitude is thus

[TABLE]

where $\alpha$ and $\beta$ are unknown, low-energy constants—complex numbers to be determined by fits to the experimental event populations in the Dalitz plot. If they are determined to be nonzero, they signal the appearance of $C$- and $CP$-violatiion. To construct $A_{LR}$ in Eq. (1), we compute

[TABLE]

using Eq. (4) and working to leading order in $CP$ violation. Since the phase space is symmetric and the $CP$-violating terms are antisymmetric under $u\leftrightarrow t$ exchange, we see that the $CP$-violating terms leave the total decay rate unchanged in ${\cal O}(\alpha)$, ${\cal O}(\beta)$.

We now turn to the amplitudes $M^{\not C}_{0,2}(s,t,u)$. Here, too, we introduce functions $M_{I}(z)$ for amplitudes that contain $\pi-\pi$ scattering in the $z$ channel with isospin $I$. After using angular-momentum conservation and the Clebsch-Gordon coefficients for the addition of the possible isospin states, as shown in the appendix, we have

[TABLE]

and

[TABLE]

where the superscripts distinguish the functions that appear in each state of total isospin. In what follows we do not compute $M_{I}^{\prime}(z)$ and $M_{I}^{\prime\prime}(z)$ explicitly, but, rather, estimate them. With this, we can use the experimental studies we consider in this paper to set limits on the possibilities, by constraining $\alpha$ and $\beta$. For context we note that the particular new-physics operators that would give $C$- and $CP$-violation are not well established, though examples have been discussed in the literature Khriplovich (1991); Conti and Khriplovich (1992); Engel et al. (1996); Ramsey-Musolf (1999); Kurylov et al. (2001). From the viewpoint of SM effective field theory (SMEFT) Buchmuller and Wyler (1986); Grzadkowski et al. (2010), we also know that there are many more examples, even in leading-mass dimension, than have been discussed thus far Gardner and Shi (2020). Nevertheless, we can draw conclusions about $M_{I}^{\prime}(z)$ and $M_{I}^{\prime\prime}(z)$ irrespective of the choice of new-physics operator. In particular, since the operators that mediate $I=0$ or $I=2$ amplitudes break $C$, they cannot mediate a $\eta\to\pi^{0}$ transition, as we suppose that the neutral meson states remain of definite $C$-parity in the presence of $C$-violation. Thus if we were to evaluate the decay diagrams in NLO ChPT in these exotic cases, they would have the same decay topology as the diagrams that appear in that order in the SM amplitude for $\eta\to\pi^{+}\pi^{-}\pi^{0}$ decay. Thus there is a one-to-one map of the two-body rescattering terms in the SM to the $C$- and $CP$-violating amplitudes. To proceed, we assume that the phases of the functions $M_{I}(z)$, $M_{I}^{\prime}(z)$ and $M_{I}^{\prime\prime}(z)$ arise from the strong-interaction dynamics of final-state, $\pi-\pi$ scattering of isospin $I$ in channel z, making the phase of each function common to all three isospin amplitudes. Such treatments are familiar from the search for non-SM $CP$ violation, such as in the study of $B\to\pi(\rho\to\pi\pi)$ decays Lipkin et al. (1991); Snyder and Quinn (1993); Gardner and Meissner (2002); Bigi and Sanda (2009). Moreover, at the low energies we consider here, the scattering of the two-pions in the final state is predominantly elastic, as mixing with other final-states can only occur through G-parity breaking. Regardless of the total isospin of the final state pions, the effective Hamiltonian that mediates the decay separates into a $C$- and/or $I$-breaking piece and a $C$- and $I$-conserving piece. Working to leading order in $C$- and/or $I$-breaking, and assuming that the final-state interactions are two-body only, Watson’s theorem Watson (1954), familiar from $K\to\pi\pi$ decays Bigi and Sanda (2009), also applies to this case and makes the phase of the function $M_{I}(z)$ common to the three cases. However, the functions $M_{I}(z)$, $M_{I}^{\prime}(z)$ and $M_{I}^{\prime\prime}(z)$ could differ by polynomial prefactors that depend on $z$. Nevertheless, we believe these effects are relatively negligible, because the energy release in $\eta\to\pi^{+}\pi^{-}\pi^{0}$ decay is small. We illustrate this explicitly later in this section.

We wish to study the possible patterns of $C$- and $CP$-violation across the Dalitz plot, so that we now turn to the explicit evaluation of Eq. (4) and its associated Dalitz distribution. Much effort has been devoted to the evaluation of the SM contribution, with work in ChPT Osborn and Wallace (1970); Gasser and Leutwyler (1985a); Bijnens and Ghorbani (2007), as well as in frameworks tailored to address various final-state-interaction effects Neveu and Scherk (1970); Roiesnel and Truong (1981); Kambor et al. (1996); Anisovich and Leutwyler (1996); Borasoy and Nissler (2005); Colangelo et al. (2009); Schneider et al. (2011); Kampf et al. (2011); Guo et al. (2015); Albaladejo and Moussallam (2017); Colangelo et al. (2017, 2018). In what follows we employ a next-to-leading-order (NLO) ChPT analysis Gasser and Leutwyler (1985a); Bijnens and Ghorbani (2007) because it is the simplest choice in which the $C$- and $CP$-violating coefficients $\alpha$ and $\beta$ can have both real and imaginary parts. A comparison of the NLO and NNLO analyses of Bijnens et al. Bijnens and Ghorbani (2007), noting Table I of Ref. Anastasi et al. (2016), shows that this is an acceptable choice. We thus think it is rich enough to give a basic view as to how our idea works. To compute the $C$-violating amplitudes, we decompose the $I=1$ amplitude into the isospin basis $M_{I}(z)$. As well known Bijnens et al. (2003); Bijnens and Ghorbani (2007); Colangelo et al. (2017); Albaladejo and Moussallam (2017), the isospin decompositions involving the $\pi-\pi$ rescattering functions $J^{r}_{PQ}(s)$ are unique, whereas the polynomial parts of the amplitude are not, due to the relation $s+t+u=3s_{0}$, where $s_{0}=(M_{\eta}^{2}+2M_{\pi^{+}}^{2}+M_{\pi^{0}}^{2})/3$. Thus there are $M_{I}(z)$ redefinitions that leave the $I=1$ amplitude invariant, as discussed in Ref. Bijnens et al. (2003). However, since we assume that strong rescattering effects dominate $M_{I}(x)$, we can demand that the $I=0,2$ amplitudes remain invariant also. As a result, only the redefinition $M_{0}(s)-\frac{4}{3}\delta_{1}$ and $M_{2}(z)+\delta_{1}$, with $\delta_{1}$ an arbitrary constant, survives. In what follows we adopt the NLO analyses of Refs. Gasser and Leutwyler (1985a); Bijnens and Ghorbani (2007), and our isospin decomposition of Ref. Gasser and Leutwyler (1985a) is consistent with that in Bijnens and Ghorbani Bijnens and Ghorbani (2007) — its detailed form can be found in the information in the appendix. Small differences in the numerical predictions exist, however, due to small differences in the inputs used Gasser and Leutwyler (1985a); Bijnens and Ghorbani (2007), and we study their impact explicitly. Returning to the would-be NLO ChPT computation of the total $I=0,2$ amplitudes, we note that $C$- and $CP$-violating four-quark operators are generated by operators in mass dimension 8 in SMEFT Gardner and Shi (2020), so that these amplitudes start beyond ${\cal O}(p^{4})$, though this is not at odds with pulling out a strong rescattering function. The $p^{2}$-dependence of a $C$- and $CP$-violating operator from physics beyond the SM would in part be realized as dimensionless ratios involving the new physics scale and would appear in the prefactors $\alpha$ or $\beta$ as appropriate.

Irrespective of the particular new-physics operator, we note, by analyticity, that the $M_{I}(x)$ for the total $I=0,2$ amplitudes could differ from the SM form, which is driven by the strong $\pi-\pi$ phase shifts, by a multiplicative polynomial factor, nominally of form $1+C_{1}^{I}x/M_{\pi}^{2}+C_{2}^{I}x^{2}/M_{\pi}^{4}+\dots$, where $C_{1}^{I}$ and $C_{2}^{I}$ are constants. (We note polynomials of similar origin appear in the time-like pion form factor Gardner and O’Connell (1998).) We emphasize that in assuming that strong-interaction phases dominate we suppose these corrections to be unimportant. We believe this to be an excellent approximation, which we illustrate through a plot of the functions $M_{I}(s)$, as shown in Fig. 1. The physics of $\pi-\pi$ scattering make the functions $M_{I}(s)$ vary substantially with $s$, whereas $s$ itself only changes by about a factor of 2 in $\eta\to 3\pi$ decay. As a result we expect that the ignored polynomial factors are numerically unimportant, so that their neglect does not impact the conclusions of this paper.

III Results

The Dalitz distribution in $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ is usually described in terms of variables $X$ and $Y$ Weinberg (1960):

[TABLE]

where $Q_{\eta}=T_{\pi^{+}}+T_{\pi^{-}}+T_{\pi^{0}}=M_{\eta}-2M_{\pi^{+}}-M_{\pi^{0}}$, and $T_{\pi^{i}}$ is the $\pi^{i}$ kinetic energy in the $\eta$ rest frame. The decay probability can be parametrized in a polynomial expansion around the center point $(X,Y)=(0,0)$ Anastasi et al. (2016):

[TABLE]

Since the $C$ transformation on the decay amplitude is equivalent to $t\leftrightarrow u$ exchange Gardner and Tandean (2004), we see that the appearance of terms that are odd in $X$ would indicate both $C$ and $CP$ violation. The KLOE-2 collaboration Anastasi et al. (2016) has provided a more precise estimate of the $C$-even parameters in Eq. (10) and bounded the $C$-odd ones. Returning to Eq. (4), we see that the $C$- and $CP$-violating contributions to the decay probability are

[TABLE]

where $\xi\equiv-(M_{K}^{2}/M_{\pi}^{2})(M_{K}^{2}-M_{\pi}^{2})/(3\sqrt{3}F_{\pi}^{2}Q^{2})$, and the existing experimental assessments of $|A(s,t,u)|^{2}_{\not C}$ correspond to the set of odd $X$ polynomials in $|A(s,t,u)|^{2}$. The parameter $N$ drops out in the evaluation of the asymmetries, and the parameters $c,e,h,$ and $l$ are taken from the first line of Table 4.6 in the Ph.D. thesis of Caldeira Balkest$\stackrel{{\scriptstyle\circ}}{{\text{a}}}$hl Caldeira Balkeståhl (2016),

[TABLE]

which fleshes out Ref. Anastasi et al. (2016)—the results emerge from a global fit to the Dalitz distribution. There is a typographical error in the sign of $c$ in Ref. Anastasi et al. (2016). We now turn to the extraction of $\mathrm{Re}(\alpha)$, $\mathrm{Im}(\alpha)$, $\mathrm{Re}(\beta)$, and $\mathrm{Im}(\beta)$ using the experimental data and Eqs. (3,6,7) using the $M_{I}(z)$ amplitudes from ${\cal O}(p^{4})$ ChPT Gasser and Leutwyler (1985a); Bijnens and Ghorbani (2007). We evaluate the denominators of the possible charge asymmetries by computing $\xi^{2}|M_{1}^{C}(z,t,u)|^{2}$ only.

Herewith we collect the parameters needed for our analysis. We compute the phase space with physical masses, so that $s+t+u=3s_{0}$, but the decay amplitudes Gasser and Leutwyler (1985a); Bijnens and Ghorbani (2007) on which we rely, namely, $M(s,t,u)$ in Eq. (2), should be in the isospin limit, implying some adjustment of the input parameters may be needed. We adopt the hadron masses and $\sqrt{2}F_{\pi}=(130.2\pm 1.7)\times 10^{-3}~{}\text{GeV}$ from Ref. Tanabashi and others. (Particle Data Group) for both amplitudes. For the Gasser and Leutwyler (GL) amplitude Gasser and Leutwyler (1985a) we use $M_{\pi}\equiv\sqrt{(2M_{\pi^{\pm}}^{2}+M_{\pi^{0}}^{2})/3}$, $M_{K}\equiv\sqrt{(M_{K^{+}}^{2}+M_{K^{0}}^{2})/2}$, where we discuss our treatment of the two-particle thresholds in the supplement, with $F_{0}=F_{\pi}$, $F_{K}/F_{\pi}=1.1928\pm 0.0026$ Tanabashi and others. (Particle Data Group), and $L_{3}=(-3.82\pm 0.30)\times 10^{-3}$ from the NLO fit with the scale $\mu=0.77~{}\text{GeV}$ Bijnens and Ecker (2014). We use these parameters in the prefactor in Eq. (2) also, as well as $Q=22.0$ Colangelo et al. (2017), to find $\xi=-0.137$. For the Bijnens and Ghorbani (BG) amplitude through ${\cal O}(p^{4})$ Bijnens and Ghorbani (2007), we use $M(s,t,u)=M^{(2)}(s)+M^{(4)}(s,t,u)$ and multiply the prefactor in Eq. (2) by $-(3F^{2}_{\pi})/(M_{\eta}^{2}-M_{\pi}^{2})$ to yield that in Ref. Bijnens and Ghorbani (2007). In the ${\cal O}(p^{2})$ term, which contributes to $M_{0}(s)$, $M^{(2)}(s)=(4M_{\pi}^{2}-s)/F_{\pi}^{2}$, and we use $M_{\pi}$ and $F_{\pi}$ as defined for the GL amplitude Gasser and Leutwyler (1985a). In the ${\cal O}(p^{4})$ term, we use $M_{\pi^{0}}$ and $M_{K^{0}}$ as indicated, as well as $\Delta=M_{\eta}^{2}-M_{\pi^{0}}^{2}$ and $L_{3}$, $L_{5}$, $L_{7}$, $L_{8}$ from fit 10 of Ref. Amoros et al. (2001).

We solve for $\alpha$ and $\beta$ in two different ways for each of the decay amplitudes Gasser and Leutwyler (1985a); Bijnens and Ghorbani (2007). We begin with the GL amplitude Gasser and Leutwyler (1985a), first making a Taylor expansion of Eq. (11) to cubic power in $X$ and $Y$ about $(X,~{}Y)=(0,~{}0)$. We then equate coefficients associated with the $X$, $XY$, $XY^{2}$, and $X^{3}$ terms to $c$, $e$, $h$ and $l$, respectively, and then solve the four equations to obtain $\text{Re}(\alpha)$, $\text{Im}(\alpha)$, $\text{Re}(\beta)$, and $\text{Im}(\beta)$. The resulting values of $\alpha$ and $\beta$ are

[TABLE]

In the first row of Fig. 2 we compare the resulting assessment of Eq. (11) with the KLOE-2 results. Large discrepancies exist, particularly at large values of $X$ and/or $Y$, where the empirical Dalitz plot Anastasi et al. (2016) shows considerable stength. Thus we turn to a second procedure, in which we make a global fit of $\alpha$ and $\beta$ in Eq. (11) to the KLOE-2 results. That is, we assess the Dalitz distribution $N(X,Y)$ and its error by using the Dalitz plot parameters in Eq. (III), discretized onto a $(X,Y)$ mesh with 682 points. To determine $N(X,Y)$ and its error we use the odd-$X$ terms in Eq. (10) with the normalization factor $N=0.0474$ as per the GL amplitude Gasser and Leutwyler (1985a) and compute the covariance matrix using Eq. (12) and the correlation matrix given in Table 4.3 of Ref. Caldeira Balkeståhl (2016). We then fit $|A(s,t,u)|^{2}_{\not C}$ using the GL amplitude Gasser and Leutwyler (1985a) to $N(X,Y)$ using a $\chi^{2}$ optimization to find

[TABLE]

and we show the results of this method in the second row of Fig. 2. Enlarging the $(X,Y)$ mesh to 1218 points incurs changes within $\pm 1$ of the last significant figure. The comparison with experiment shows that the fitting procedure is the right choice. We draw the same conclusion from the use of the BG amplitude Bijnens and Ghorbani (2007), noting that the global fit in that case (with $N=0.0508$) gives

[TABLE]

so that the results are compatible within errors. Using these solutions, we obtain $A_{LR}=(-7.18\pm 4.51)\times 10^{-4}$ using Ref. Gasser and Leutwyler (1985a) and $A_{LR}=(-7.20\pm 4.52)\times 10^{-4}$ using Ref. Bijnens and Ghorbani (2007). These compare favorably with $A_{LR}=(-7.29\pm 4.81)\times 10^{-4}$ that we determine using the complete set of Dalitz plot parameters and the covariance matrix we construct given the information in Ref. Caldeira Balkeståhl (2016). We note that our $A_{LR}$ as evaluated from the Dalitz plot parameters, which are fitted from the binned data, is a little different from the reported value using the unbinned data, i.e., ($-5.0\pm 4.5\stackrel{{\scriptstyle+5.0}}{{{}_{-11}}})\times 10^{-4}$, reported by KLOE-2 Anastasi et al. (2016). The discrepancy is not significant, and we suppose its origin could arise from the slight mismatch between the theoretically accessible phase space and the experimentally probed one, or other experimental issues. Although NLO ChPT does not describe the $CP$-conserving Dalitz distribution well Bijnens and Ghorbani (2007), we find it can confront the existing $CP$-violating observables successfully.

We have shown that the empirical Dalitz plot distribution can be used to determine $\alpha$ and $\beta$. These, in turn, limit the strength of $C$-odd and $CP$-odd operators that can arise from physics beyond the SM Khriplovich (1991); Conti and Khriplovich (1992); Engel et al. (1996); Ramsey-Musolf (1999); Kurylov et al. (2001); Gardner and Shi (2020). That $\beta$ is so much smaller than $\alpha$ can be, in part, understood from the differing behavior of the $M_{I}(z)$, as illustrated in Fig. 1, which follows because the $L=0$, $I=2$ $\pi-\pi$ phase shift is larger than the $L=1$, $I=1$ one for the kinematics of interest Hyams et al. (1973); Gasser and Leutwyler (1985a); Colangelo et al. (2001); Ananthanarayan et al. (2001), making it easier to veto the $I=2$ operators. Crudely, the ratio of $\beta$ to $\alpha$ we have found is that of the SM electromagnetic interactions that would permit a $I=2$ amplitude to appear in addition to a $I=0$ one. The utility of our Dalitz analysis is underscored by our results for the quadrant asymmetry $A_{Q}$ and sextant asymmetry $A_{S}$ defined in Fig. 3. Using Ref. Gasser and Leutwyler (1985a) and Eq. (14), e.g., we find $A_{Q}=(2.85\pm 3.72)\times 10^{-4}$, and $A_{S}=(3.87\pm 4.04)\times 10^{-4}$; the asymmetries by themselves hide the nature of the underlying strong amplitudes. For reference we note the KLOE-2 results using unbinned data Anastasi et al. (2016): $A_{Q}=(1.8\pm 4.5\stackrel{{\scriptstyle+4.8}}{{{}_{-2.3}}})\times 10^{-4}$ and $A_{S}=(-0.4\pm 4.5\stackrel{{\scriptstyle+3.1}}{{{}_{-3.5}}})\times 10^{-4}$, with which our results are compatible within errors.

IV Summary

We propose an innovative way of probing $C$- and $CP$-violation in the $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ Dalitz plot. Working to leading order in charge conjugation $C$ and isospin $I$ breaking, we have shown that the strong amplitudes associated with the appearance of $C$- and $CP$-violation can be estimated from the SM amplitude for $\eta\to{\pi^{+}\pi^{-}\pi^{0}}$ if the decomposition of Eq. (3) holds Anisovich and Leutwyler (1996). We have illustrated this in NLO ChPT, though the use of more sophisticated theoretical analyses would also be possible. New-physics contributions that differ in their isospin can thus be probed through the kinematic pattern they imprint in the Dalitz plot. Our method opens a new window on the study of $C$- and $CP$-violation in $\eta\to{\pi^{+}\pi^{-}\pi^{0}}$ decay, and it holds promise for the high-statistics experiments of the future Gan (2017); Gatto et al. (2016); Beacham et al. (2020).

V Acknowledgments

We thank Jian Liang for his assistance in the numerical analysis, Johan Bijnens for helpful correspondence regarding Ref. Bijnens and Ghorbani (2007), Andrzej Kupsc for a discussion of the KLOE-2 experimental results Anastasi et al. (2016); Caldeira Balkeståhl (2016), and Bastian Kubis for comments that we found helpful to clarifying our presentation. We acknowledge the DOE Office of Nuclear Physics under contract DE-FG02-96ER40989 for partial support.

Appendix A Calculation Details

We begin by showing how Eqs. (6) and (7) emerge from elementary considerations. Working in the isospin limit, a $|\pi^{+}\pi^{-}\pi^{0}\rangle$ state with $J=0$ must obey Bose symmetry, so that it is proportional to

[TABLE]

where “s” denotes a symmetrized combination of distinct pion states. In what follows, as in the $CP$-conserving case, Eq. (3)  Anisovich and Leutwyler (1996), we consider $S$- and $P$-wave $\pi-\pi$ amplitudes only. The symmetrized two-pion states can be written as a $S$-wave $I=0$ or $I=2$ state or as a $P$-wave $I=1$ state. We choose $|\pi^{i}\rangle\equiv|I=1,I_{3}=i\rangle$. For S-waves, we write

[TABLE]

whereas for P-waves we note

[TABLE]

with $i+j+k=0$ contributes to the $J=0$ state. Defining

[TABLE]

we see, e.g.,

[TABLE]

We can also label particular $\eta\to\pi^{+}\pi^{-}\pi^{0}$ decay amplitudes by the isospin of the two-pion state, as used in Eq. (3). Enumerating the possibilities, we find

[TABLE]

which contributes to the total $I=1$ amplitude, $M_{1}^{C}$, only, as well as

[TABLE]

which contribute to the amplitudes with total $I=0,1$, and $2$, and

[TABLE]

which contribute to the total $I=1$ and $2$ amplitudes. Using the Clebsch-Gordan coefficients tabulated in Ref. Tanabashi and others. (Particle Data Group), we find, after redefining $M_{1}/2\sqrt{2}\to M_{1}$ and $M_{2}\sqrt{3/10}\to M_{2}$, that $M_{1}^{C}(s,t,u)$, $M_{0}^{/{\!\!\!\!C}}(s,t,u)$, and $M_{2}^{/{\!\!\!\!C}}(s,t,u)$ are precisely as given in Eqs. (3) Anisovich and Leutwyler (1996), (6), and (7). Note that only the $C$-odd amplitudes are odd under $t\leftrightarrow u$ as needed. Adding the possible total $I$ amplitudes in leading order in $C$, $CP$, and $I$ violation, with their associated coefficients, yields Eq. (4).

We now turn to our isospin decomposition of the $\eta\to\pi^{+}\pi^{-}\pi^{0}$ amplitude of Gasser and Leutwyler through ${\cal O}(p^{4})$ Gasser and Leutwyler (1985a):

[TABLE]

[TABLE]

and

[TABLE]

where $\Delta=M_{\eta}^{2}-M_{\pi}^{2}$, $M_{\pi}^{2}=(2M_{\pi^{+}}^{2}+M_{\pi^{0}}^{2})/3$, and $M_{K}^{2}=(M_{K^{+}}^{2}+M_{\bar{K}^{0}}^{2})/2$. We refer to Ref. Gasser and Leutwyler (1985b) for $J^{r}_{PQ}(z)$, noting Eqs. (8.8-8.10) and (A.11), where P and Q denote the mesons $\pi$, $K$, or $\eta$, and to Ref. Gasser and Leutwyler (1985a) for $a_{i}$ and $d_{i}$. We note that the $J^{r}_{PQ}(z)$ carry renormalization-scale $\mu$ dependence, though cancelling that dependence is beyond the scope of our current approach—we note a similar issue arises in the use of the pion form factor in the analysis of $B\to\pi(\rho\to\pi\pi)$ decay Gardner and Meissner (2002). For this choice of $M_{\pi}$ and the use of physical phase space we need to evaluate the possible two-particle thresholds with care. The rescattering function $J^{r}_{\pi\pi}(z)$ contains $\sigma(s)=\sqrt{1-4m_{\pi}^{2}/z}$. If we use $m^{2}_{\pi}=M_{\pi}^{2}$, then for $M_{I}(z)$ with $z=t$ or $u$ evaluated at its minimum value the argument of the square root is less than zero. To avoid this problem, we use $\sigma(z)=\sqrt{1-(M_{\pi^{\pm}}+M_{\pi^{0}})^{2}/z}$ for $z=t$ or $u$. For $M_{I}(s)$, though, $s_{\rm min}=4M_{\pi^{+}}^{2}$ and this problem does not occur. However, for consistency we use $\sigma(s)=\sqrt{1-4M_{\pi^{+}}^{2}/s}$ for $M_{I}(s)$. Moreover, we note $J^{r}_{\pi\eta}(s)$ contains $\nu(s)=\sqrt{(s-(M_{\eta}-m_{\pi})^{2})(s-(M_{\eta}+m_{\pi})^{2})}$. If we use $m_{\pi}=M_{\pi}$, then for $M_{I}(s)$ at the maximum of $s$, we once again find the argument of the square root to be less than zero. To avoid this, we use $\nu(s)=\sqrt{(s-(M_{\eta}-M_{\pi^{0}})^{2})(s-(M_{\eta}+M_{\pi^{0}})^{2})}$ for $M_{I}(s)$. To be consistent, we use $\nu(z)=\sqrt{(z-(M_{\eta}-M_{\pi^{+}})^{2})(z-(M_{\eta}+M_{\pi^{+}})^{2})}$ for $M_{I}(z)$ with $z=t$ or $u$. As a check of our assessments we have extracted the $C$- and $CP$-conserving Dalitz plot parameters from this amplitude. Describing the $CP$-conserving piece of $|A(s,t,u)|^{2}$ by $N(1+aY+bY^{2}+dX^{2}+fY^{3}+gX^{2}Y)$, recalling Eq. (10), we find using a global fit that $a=-1.326$, $b=0.426$, $d=0.086$, $f=0.017$, and $g=-0.072$. These results compare favorably to the global fit results of Ref. Albaladejo and Moussallam (2017); namely, $a=-1.328$, $b=0.429$, $d=0.090$, $f=0.017$, and $g=-0.081$. That work also uses the decay amplitude of Ref. Gasser and Leutwyler (1985a) through ${\cal O}(p^{4})$ and the same value of $L_{3}$ but includes electromagnetic corrections through ${\cal O}(e^{2}p^{2})$ as well.

In evaluating the BG amplitude Bijnens and Ghorbani (2007) we note that an overall 2 should not appear on the second right-hand side of Eq.(3.23); this is needed for the result to agree with that of Ref. Gasser and Leutwyler (1985a).

Values of the strong functions associated with the $CP$-violating parameters $\text{Re}(\alpha)$, $\text{Im}(\alpha)$, $\text{Re}(\beta)$, $\text{Im}(\beta)$ in Eq. (11) on our analysis grids in $(X,Y)$ are available upon request.