Patterns of CP violation from mirror symmetry breaking in the $η\toπ^+π^-π^0$ Dalitz plot
Susan Gardner Jun Shi







[]gardner@pa.uky.edu
Department of Physics and Astronomy, University of Kentucky,
Lexington, Kentucky 40506, USA
 
[]jun.shi@uky.edu
Department of Physics and Astronomy, University of Kentucky,
Lexington, Kentucky 40506, USA




 
A violation of mirror symmetry in the η→π^+π^-π^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.
 

Patterns of CP violation from mirror symmetry breaking in the η→π^+π^-π^0 Dalitz plot
    Jun Shi
    December 30, 2023
=====================================================================================








§ INTRODUCTION

The decay η→ 3π first came to prominence after the 
observation of 
K_L →π^+ π^- decay and the discovery 
of CP violation in 1964 <cit.>, because 
it could be used to test whether K_L →π^+ π^- decay was generated by 
CP violation 
in the weak interactions <cit.>. 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 π^+ and π^- in 
 decay <cit.>. 
Since  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 <cit.>. 
Numerical estimates were made by assuming that 
the isospin-violating contributions 
were driven by electromagnetism <cit.>. 
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 <cit.>, is now known to 
provide the driving effect in mediating η→ 3 π 
decay <cit.>, with
isospin-breaking, electromagnetic effects 
playing a much smaller role <cit.>. 

Modern theoretical studies of η→ 3π 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 ≡√((m_s^2 - m̂^2)/(m_d^2 - m_u^2)), with m̂=(m_d + m_u)/2, 
precisely <cit.>.  There has been 
no further theoretical study of CP violation 
in η→π^+π^-π^0 decay since 1966. Since the η meson carries
neither spin nor flavor, searches for new physics in this system possess special features. 
For example, η→π^+π^-π^0 
decay must be parity P conserving if the π and η 
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 
η→π^+ (p_π^+) π^- (p_π^-) π^0 (p_π^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 ≡ (p_π^- + p_π^0)^2 and u ≡ (p_π^+ + p_π^0)^2, 
the charge asymmetry we have noted corresponds to a failure of mirror symmetry, i.e., 
of t↔ 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 θ̅ term of the SM, and not new weak-scale physics. 

The θ̅ term can also generate η→ππ and η/η^'→ 4π^0 decay,
breaking P and CP explicitly, so that limits on the decay rate constrains
the square of a CP-violating 
parameter <cit.>. Since the
θ̅ 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 
η→π^+ (p_π^+) π^- (p_π^-) π^0 (p_π^0) decay
can be probed experimentally through the measurement of a 
left-right asymmetry, A_LR <cit.>: 

    A_L R≡N_+ - N_-/N_+ + N_-≡1/N_ tot (N_+ - N_-) ,

where N_± is the number of events with u>_<t, 
so that the π^+ has 
more (less) energy than the π^- if u > (<) t in the η rest system. 
A number of experiments have been conducted over the years to test for a charge asymmetry 
in η→π^+π^-π^0 decay, with early experiments
finding evidence for a nonzero asymmetry <cit.>, 
but with possible systematic problems becoming apparent only later, as, e.g., in 
Ref. <cit.>. 
Other experiments find no evidence for a charge asymmetry and 
C violation <cit.>, and we note that new, high-statistics experiments are planned <cit.>.
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 <cit.>, and 
a quadrant asymmetry A_Q, sensitive to the I=2 final state <cit.>. 
These asymmetries are more challenging to measure and are only poorly known <cit.>. 
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 <cit.>, with 
Ref. <cit.> analyzing how different CP-violating mechanisms populate the 
Dalitz plot. We also note Refs. <cit.> for Dalitz studies
of CP violation in heavy-flavor decays. 




§ THEORETICAL FRAMEWORK

The  decay amplitude in the SM 
can be expressed as <cit.>

    A (s, t, u)    =    - 1/Q^2M_K^2/M_π^2M_K^2 -
       M_π^2/3 √(3) F_π^2 M(s,t,u),

where we employ the Mandelstam variables u, t, and s= (p_π^+ + p_π^-)^2
and work to leading order in strong-interaction isospin breaking.





Since C=-(-1)^I in  decay <cit.>, the C- and CP-even 
transition amplitude with a Δ 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 π^+π^-π^0 state and 
can be expressed as <cit.>
  
    M^C_1(s, t, u)    =    M_0 (s) + (s - u) M_1 (t) + (s - t) M_1 (u) 
       +    M_2 (t) + M_2(u) - 2/3 M_2 (s)  ,

  where M_I(z) is
  
  an amplitude with π-π 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), 
O(p^6),
because the only absorptive parts that can 
appear are in the π-π S- and P-wave amplitudes <cit.>. 
An analogous relationship 
exists in η→ 3π^0 decay <cit.>, 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 

    A(s,t,u) 
       =    - 1/Q^2M_K^2/M_π^2M_K^2 -
       M_π^2/3 √(3) F_π^2 M^C_1(s,t,u) 
           + α M^C̸_0(s,t,u) 
    + β M^C̸_2(s,t,u)  ,

where α and β 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_L R in Eq. (<ref>), we compute 

    N_± = 1/256 π^3 M_η^3∫_u >_<t dt du   | A(s, t, u)|^2   ,

using Eq. (<ref>) and working to leading order in CP violation. 

Since the phase space is symmetric and the CP-violating terms are antisymmetric 
under u ↔ t exchange, we see that 
the CP-violating terms leave the total 
decay rate unchanged in O(α), O( β). 

We now turn to the amplitudes M ^C̸_0,2(s,t,u). 
Here, too, we introduce functions M_I (z) for amplitudes that contain 
 π-π 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 

    M^C̸_0(s,t,u) = (s - t) M'_1(u) + (u - s) M'_1(t)
     - (u-t) M'_1(s)

and

    M^C̸_2(s,t,u)    =    (s - t) M”_1 (u) + (u - s) M”_1 (t) 
    
          + 2(u-t)M”_1(s) + √(5)[M”_2 (u) - M”_2 (t)]  ,
    
where the superscripts distinguish the functions that appear in each
state of total isospin. 
In what follows we do not compute 

M_I^' (z) and M_I^'' (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

α and β. 
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 <cit.>. 
From the viewpoint of SM effective field theory (SMEFT) <cit.>, we also know that
there are many more examples, even in leading-mass dimension, than have been
discussed thus far <cit.>.  Nevertheless, 
we can draw conclusions  
about M_I^' (z) and M_I^'' (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 η→π^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
η→π^+π^- π^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'(z) and M_I”(z) arise
from the strong-interaction dynamics of final-state, π-π 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 →π (ρ→ππ)
decays <cit.>.
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 <cit.>,
familiar from K →ππ decays <cit.>, 
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'(z) and M_I”(z) could differ by polynomial prefactors
that depend on z. Nevertheless, we believe these effects are relatively negligible, because the
energy release in η→π^+ π^- π^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. (<ref>) and its associated Dalitz distribution. 
Much effort has been devoted to the evaluation of the SM contribution, 
with work in ChPT <cit.>, 
as well as in frameworks tailored to address various final-state-interaction 
effects <cit.>.  
In what follows we employ a next-to-leading-order (NLO) ChPT 
analysis <cit.>
because it is the simplest choice in which 
the C- and CP-violating coefficients α and β can have both real and imaginary parts. 
A comparison of the NLO and NNLO analyses of 
Bijnens et al. <cit.>, noting Table I of Ref. <cit.>,  
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 <cit.>, 
the isospin decompositions involving 
the π-π rescattering functions J^r_P Q (s) are unique, whereas 
the polynomial parts of the amplitude are not, 
due to the relation 
s + t + u = 3 s_0, where s_0=(M_η^2 + 2M_π^+^2 + M_π^0^2)/3. 
Thus there are M_I(z) redefinitions that leave the I=1 
amplitude invariant, as discussed in Ref. <cit.>. 
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) - 4/3δ_1 and M_2 (z) + δ_1, 
with δ_1 an arbitrary constant, survives.
In what follows we adopt the NLO analyses of Refs. <cit.>, 
and our isospin decomposition of Ref. <cit.> is 
consistent with that in Bijnens and Ghorbani <cit.> — 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 <cit.>, 
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 <cit.>, 
so that these amplitudes start beyond 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 α
or β 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 π-π phase shifts, 
by a multiplicative polynomial factor, nominally of form 1 + C_1^I x/M_π^2 
+ C_2^I x^2/M_π^4 + …, where C_1^I and C_2^I are constants.

(We note polynomials of similar origin appear in the time-like 
pion form factor <cit.>.)

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. <ref>. The physics of π-π scattering make the 
functions M_I(s) vary substantially with s, whereas s itself only 
changes by about a factor of 2 in η→ 3π 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. 










§ RESULTS

The Dalitz distribution in η→π^+ π^- π^0 
is usually described in terms of variables 
X and Y <cit.>: 

    X    ≡   √(3)T_π^+ - T_π^-/Q_η =√(3)/2 M_η Q_η (u - t),
      Y    ≡   3 T_π^0/Q_η - 1 = 3/2 M_η Q_η [(M_η - M_π^0)^2 - s] - 1 ,

where 
Q_η = T_π^+ + T_π^- + T_π^0 = M_η - 2 M_π^+ -
   M_π^0, and T_π^i is the π^i kinetic energy in the η rest frame. 
The decay probability 
can be parametrized in a polynomial
expansion around the center point (X,Y) = (0 ,0) <cit.>: 

    |A (s, t, u)|^2    =    N (1 + a Y + b Y^2 + c X + d X^2 + e X Y  
        +    f Y^3 + g X^2 Y + h X Y^2 + l X^3 + …)  .
  
Since the C transformation on the decay amplitude
is equivalent to t ↔ u exchange <cit.>, we see that 
 the appearance of terms that are 
 odd in X 
would indicate both C and CP violation. 
The KLOE-2 collaboration <cit.> has provided a more precise estimate of the 
C-even parameters in Eq. (<ref>) and bounded the C-odd ones. 
Returning to Eq. (<ref>), we see that the C- and CP-violating 
contributions to the decay probability 
are
 
    1/ξ|A(s,t,u)|^2_C̸= M^C_1[α M_0^C̸ + β M_2^C̸]^∗ 
    + H.c. 
       =   2Re(α)[Re(M_1^C)Re(M_0^C̸) 
    +Im(M^C_1)Im(M_0^C̸) ]
       -     2Im(α)[Re(M^C_1)Im(M_0^C̸) 
    -Im(M_1^C)Re(M_0^C̸) ]
       +    2Re(β)[Re(M_1^C)Re(M_2^C̸) 
    + Im(M_1^C)Im(M_2^C̸) ]
       -     2Im(β)[Re(M_1^C)Im(M_2^C̸) 
    - Im(M_1^C)Re(M_2^C̸)]
     ,

where ξ≡ - (M_K^2/M_π^2) (M_K^2 - M_π^2)/(3 √(3) F_π^2 Q^2), 
and the existing experimental assessments of |A(s,t,u)|^2_C̸ 
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∘ahl <cit.>, 

    c    =   (-4.34± 3.39)× 10^-3,   e = (2.52± 3.20)× 10^-3,
    h    =   (1.07 ± 0.90)× 10^-2,   l = (1.08± 6.54)× 10^-3 ,

which fleshes out Ref. <cit.>—the results emerge from a 
global fit to the Dalitz distribution. 
There is a typographical error in the sign of c in Ref. <cit.>. 

We now turn to the extraction of 
Re(α), Im(α), Re(β), and Im(β) 
using the experimental data and 
Eqs. (<ref>,<ref>,<ref>) using the M_I(z) amplitudes from 
O(p^4) ChPT <cit.>. We evaluate the 
denominators of the possible charge asymmetries by computing ξ^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=3 s_0, 
but the decay amplitudes <cit.> on which we rely, namely, 
M(s,t,u) in Eq. (<ref>), should be in the isospin limit, implying 
some adjustment of the input parameters may be needed. We adopt 
the hadron masses and √(2)F_π=(130.2± 1.7)× 10^-3 GeV 
from Ref. <cit.> for both amplitudes. 
For the Gasser and Leutwyler (GL) amplitude <cit.>  we use 
M_π≡√((2M_π^±^2 + M_π^0^2)/3), 
M_K ≡√((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_π, F_K/F_π = 1.1928±0.0026 <cit.>, 
and L_3= (-3.82 ± 0.30) × 10^-3 from the NLO fit 
with the scale μ=0.77 GeV <cit.>. 
We use these parameters in the prefactor 
in Eq. (<ref>) also, as well as Q=22.0 <cit.>, 
to find ξ=-0.137. 
For the Bijnens and Ghorbani (BG) amplitude through 
O(p^4) <cit.>, we use 
M(s,t,u) = M^(2)(s) + M^(4)(s,t,u) and multiply the prefactor in Eq. (<ref>) 
by -(3F^2_π)/(M_η^2 - M_π^2)
to yield that in Ref. <cit.>. 
In the O(p^2) term, which contributes to M_0(s), 
M^(2)(s)= (4M_π^2 - s)/F_π^2, and 
we use M_π and F_π as defined for the GL amplitude <cit.>. 
In the O(p^4) term, we use M_π^0 and M_K^0 as indicated, 
as well as 
Δ=M_η^2 - M_π^0^2 and L_3, L_5, L_7, L_8 from fit 10 of 
Ref. <cit.>. 

We solve for α and β in two different ways for each of the decay 
amplitudes <cit.>. We begin with the GL 
amplitude <cit.>, 
first making a Taylor expansion of Eq. (<ref>)
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 
Re(α),  Im(α), Re(β), and Im(β). 
The resulting values of α and β are 

    Re(α)    =     16± 24  ,
    Re(β)    =    (-1.5± 2.7)× 10^-3 , 
    Im(α)    =    -20± 29 , 
    Im(β)    =    (-1.3± 4.7)× 10^-3 .




In the first row of Fig. <ref> we compare the 
resulting assessment of Eq. (<ref>) with the KLOE-2 results. 
Large discrepancies exist, particularly at large values of X and/or Y, 
where the empirical Dalitz plot <cit.> shows considerable stength. 
Thus we turn to a second procedure, in which we make a global fit of 
α and β in Eq. (<ref>) 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. (<ref>), 
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. (<ref>) with the normalization factor N=0.0474
as per the GL amplitude <cit.> and compute the 
covariance matrix using Eq. (<ref>) and 
the correlation matrix given in Table 4.3 of Ref. <cit.>. 
We then fit |A(s,t,u)|^2_C̸ using the GL amplitude <cit.> 
to N(X,Y) using a χ^2 optimization to find 

    Re(α)    =     -0.65± 0.80 ,
    Im(α)    =    0.44± 0.74 , 
    Re(β)    =    (-6.3± 14.7)× 10^-4 ,
    Im(β)    =    (2.2± 2.0)× 10^-3 ,

and we show the results of this method in the second row 
of Fig. <ref>. 
Enlarging the (X,Y) mesh
to 1218 points incurs changes within ±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 <cit.>, noting that the global
fit in that case (with N=0.0508) gives 

    Re(α)    =     -0.79± 0.91 ,
    Im(α)    =    0.61± 0.93 , 
    Re(β)    =    (-1.4± 2.3)× 10^-3 ,
    Im(β)    =    (2.3± 1.4)× 10^-3 ,

so that the results are compatible within errors. 
Using these solutions, we obtain A_LR=(-7.18± 4.51)× 10^-4 
using Ref. <cit.> and A_LR=(-7.20± 4.52)× 10^-4 using
Ref. <cit.>. These compare favorably with 
A_LR=(-7.29± 4.81)× 10^-4 that we determine 
using the complete set of Dalitz plot parameters and the covariance matrix we
construct given the information in Ref. <cit.>. 
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± 4.5 +5.0_-11)× 10^-4, reported by 
KLOE-2 <cit.>. 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 <cit.>, 
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 α and β. 
These, in turn, limit the strength of 
C-odd and CP-odd operators that can arise from physics beyond the 
SM <cit.>. 
That 
β is so much smaller than α can be, in part, understood from the 



differing behavior of
the M_I(z), as illustrated in Fig. <ref>, which



follows



because the
L=0, I=2 π-π phase shift is larger than the L=1, I=1 one
for the kinematics of
interest <cit.>,
making it easier to veto the I=2 operators.
Crudely, the ratio of β to α 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. <ref>. 
Using Ref. <cit.> and Eq. (<ref>), e.g., 
we find 
 A_Q=(2.85± 3.72)× 10^-4, and A_S = (3.87± 4.04)× 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 <cit.>: 
 A_Q=(1.8± 4.5 +4.8_-2.3)× 10^-4 
and A_S = (-0.4 ± 4.5 +3.1_-3.5)× 10^-4, with 
which our results are compatible within errors. 






§ SUMMARY

We propose an innovative way of probing 
C- and CP-violation 
in the η→π^+π^-π^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  
if the decomposition of Eq. (<ref>) holds <cit.>. 
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 
 decay, and it 
holds promise for the high-statistics
experiments of the future <cit.>.




§ ACKNOWLEDGMENTS

We thank Jian Liang for his assistance in the numerical analysis, 
Johan Bijnens for helpful correspondence regarding Ref. <cit.>, 
Andrzej Kupsc for a discussion of the KLOE-2 experimental 
results <cit.>, 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.


*


§ CALCULATION DETAILS
  



We begin by showing how Eqs. (<ref>) and (<ref>) emerge 

from elementary considerations. Working in the isospin limit, 
a |π^+ π^- π^0⟩ state with J=0 must obey Bose symmetry, so that 
it is proportional to 


    |π^+ π^- ⟩_ s  |π^0 ⟩ + 
    |π^+ π^0 ⟩_ s |π^- ⟩ + |π^- π^0 ⟩_ s | π^+ ⟩ ,

where “s” denotes a symmetrized combination of distinct pion states.
In what follows, as in the CP-conserving case, Eq. (<ref>)  <cit.>, we consider 
S- and P-wave π-π 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 
|π^i ⟩≡ | I=1, I_3=i⟩. 
For S-waves, we write 

    |π^i π^j ⟩_ s≡1/√(2){ | π^i π^j ⟩ + | π^j π^i ⟩} ,




whereas for P-waves we note 

    | π^i π^j ⟩_I=1, L=1 | π^k ⟩_L=1

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

    |π^i π^j ⟩_ a≡1/√(2){ |π^i π^j ⟩ - |π^j π^i ⟩} ,
 
we see, e.g., 

    |(π^+ π^-)_I=1⟩_ s |π^0⟩
    = |π^+ π^-⟩_a  
    (p_π^+ - p_π^-) · p_π^0 |π^0⟩ .


We can also label particular 
η→π^+ π^-  π^0 

decay amplitudes by the isospin of the two-pion state, 
as used in Eq. (<ref>). Enumerating the possibilities, we find 




    |(π^+π^-)_I=0⟩ | π^0 ⟩→ M_0(s)  ,

which contributes to the total I=1 amplitude, M_1^C, only, 
as well as 

    |(π^+π^-)_I=1⟩ | π^0 ⟩ (p_π^+ - p_π^-) · p_π^0→ M_1(s) u-t/2 , 
           |(π^+π^0)_I=1⟩  | π^- ⟩ (p_π^+ - p_π^0) · p_π^-→ M_1(u) s-t/2 , 
           |(π^-π^0)_I=1⟩  | π^+ ⟩ (p_π^0 - p_π^-) · p_π^+→ M_1(t) u-s/2 ,

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

    |(π^+π^-)_I=2⟩ | π^0 ⟩→ M_2(s)   , 
           |(π^+π^0)_I=2⟩  | π^- ⟩→ M_2(u)   , 
           |(π^-π^0)_I=2⟩  | π^+ ⟩→ M_2(t)   ,

which contribute to the total I=1 and 2 amplitudes. 
Using the Clebsch-Gordan coefficients tabulated in Ref. <cit.>, we find, after 
redefining M_1/2√(2)→ M_1 and M_2 √(3/10)→ 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. (<ref>) <cit.>, (<ref>), and (<ref>).  
Note that only the C-odd amplitudes are odd under t ↔ u as needed. 
Adding the possible total I amplitudes in leading order in C, CP, and I violation, 
with their associated coefficients, yields Eq. (<ref>). 


We now turn to 

our isospin decomposition of the 
η→π^+ π^- π^0 amplitude of Gasser and Leutwyler
through O(p^4) <cit.>:


    M_0 (s)
          =   [ 2 (s - s_0)/Δ + 5/3]
      1/2 F_π^2 (2 s - M_π^2) J^r_ππ (s) 
       +   1/6 F_π^2 Δ (4 M_K^2 - 3 M_η^2 - M_π^2) (s
      - 2 M_π^2) J_ππ^r (s)
        +    1/4 F_π^2 Δ[ - 6 s^2 + s (5 M_π^2 + 4
      M_K^2 + 3 M_η^2) 
                    - 4 M_K^2 ( M_η^2 + 1/3 M_π^2
      ) ] J_K K^r(s)
        +    M_π^2/3 F_π^2 Δ( 2 s - 11/3
      M_π^2 + M_η^2 ) J_πη^r(s)  - M_π^2/2 F_π^2 J_ηη^r(s)
       -     3 s/8 F_π^2(3 s - 4 M_K^2)/(s - 4 M_K^2)(J^r_KK(s) - J^r_KK(0)-1/8π^2)
        +    [ 1 + a_1 + 3 a_2Δ + a_3 (9 M_η^2 - M_π^2) + 2/3 d_1.
            .   + 8
      M_π^2/3 Δ d_2 ]( 1 + 3 s - s_0/Δ) + a_4 - 8/3M_π^2/Δ d_1 
       -   3/Δ( 2 μ_π +  μ_K )(s-s_0)
       +   (4L_3/F_0^2Δ - 1/64π^2 F_π^2 Δ)
    (4/3s^2 - 9s_0s + 9s_0^2 )
        -   1/64π^2 F_π^2 Δ3(s-s_0)(4M_π^2 + 2M_K^2)  ,


    M_1 (z)      =    1/4 Δ F_π^2[ (z - 4 M_π^2) 
    J^r_ππ   (z) 
           + ( 1/2 z - 2 M_K^2 ) J_K K^r (z) ] ,

and 

    M_2 (z) 
          =    ( 1 - 3/2z - s_0/Δ) [ -
      1/2 F_π^2 (z - 2 M_π^2) J^r_ππ (z) .
          .+ 1/4F_π^2 (3 z- 4 M_K^2) J_K K^r (z) 
    + M_π^2/3 F_π^2 J_πη^r (z) ] 
          + (1/64π^2 F_π^2 Δ - 4L_3/F_0^2Δ)z^2  ,

where 
Δ=M_η^2 - M_π^2, 
M_π^2=(2M_π^+^2 + M_π^0^2)/3, and M_K^2 = (M_K^+^2 + M_K̅^0^2)/2. 
We refer to Ref. <cit.> 

for 
J^r_P Q (z), noting Eqs. (8.8-8.10) and (A.11), 

where P and Q denote the mesons 
π, K, or η, and to Ref. <cit.>

for a_i and d_i. We note that the J^r_P Q (z) carry renormalization-scale μ
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→π (ρ→ππ)
decay <cit.>. 

For this choice of M_π and the use of physical phase space we need to evaluate
the possible two-particle thresholds with care. The rescattering function 
J^r_ππ(z) contains σ(s) = √(1-4m_π^2/z). 
If we use m^2_π = M_π^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 σ(z) = √(1-(M_π^± + M_π^0)^2/z) 
for z=t or u. 
For M_I(s), though, s_ min=4M_π^+^2 and this problem does not occur. 
However, for consistency we use σ(s) = √(1-4M_π^+^2/s) 
for M_I(s). 
Moreover, we note
J^r_πη(s) contains ν(s)=√((s-(M_η-m_π)^2)(s-(M_η+m_π)^2)). 
If we use m_π = M_π, 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  ν(s)=√((s-(M_η-M_π^0)^2)(s-(M_η+M_π^0)^2)) 
for M_I(s). To be consistent, we 
use ν(z)=√((z-(M_η-M_π^+)^2)(z-(M_η+M_π^+)^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. (<ref>), 

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. <cit.>; 

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. <cit.>

through O(p^4) and the same value 
of L_3 but includes electromagnetic corrections through O(e^2p^2) as well. 

In evaluating the BG amplitude <cit.>

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. <cit.>. 


Values of the strong functions associated with the CP-violating parameters
Re(α), Im(α), Re(β), Im(β)
in Eq. (<ref>)

on our analysis grids in (X,Y) are available upon request. 







