Correlations of $C$ and $CP$ violation in $\eta\to \pi^0\ell^+\ell^-$ and $\eta'\to \eta\ell^+\ell^-$

Based on recent progress in the systematic analysis of $C$ and $CP$ violation in the light-meson sector, we calculate the $C$-odd transition amplitudes $\eta\to\pi^0\ell^+\ell^-$ and $\eta'\to\eta\ell^+\ell^-$. Focusing on long-distance contributions driven by the lowest-lying hadronic intermediate states, we work out the correlations between these beyond-the-Standard-Model signals and the Dalitz-plot asymmetries in $\eta \rightarrow \pi^0 \pi^+ \pi^-$ and $\eta' \rightarrow \eta \pi^+ \pi^- $, using dispersion theory.

One may use the formalisms addressed in the previous paragraph to investigate the correlation of T -odd and P -even forces between different decays.In this sense we consider C-and CP -violating radiative decays of η and η ′ .In the following, we denote both mesons with η (′) .Given that η (′) as well as π 0 have the eigenvalue C = +1 but photons have C = −1, C is violated in general if η (′) decays into an arbitrary number of uncharged pions and an odd number of photons.This consideration also holds for radiative decays of the η ′ into an η.In the following we will focus on the radiative C-odd decays η → π 0 γ ( * ) and η ′ → ηγ ( * ) .To shorten the notation we will refer to both processes collectively by X → Y γ ( * ) .Angular momentum conservation demands the final state to be in a relative P -wave.Consequently, parity is conserved and the decays at hand additionally violate CP , thus offering an opportunity to investigate ToPe forces.The decay into a real, transverse photon violates both gauge invariance and the conservation of angular momentum [7,8].Therefore, the focus shall be laid on X → Y γ * → Y ℓ + ℓ − , where the off-shell photon decays subsequently into a pair of charged leptons.At the theoretical front, the investigation of this BSM one-photon exchange urgently requires an update [9,10] in comparison to analyses of the SM contribution, cf.Refs.[11][12][13][14][15][16][17], as well as studies of other BSM effects in these decays [8,18].From an experimental point of view, bounds on all the leptonic channels have already been set [19][20][21] and may become more stringent in future measurements [22][23][24][25][26][27][28].
We summarize the currently most stringent experimental upper bounds on the X → Y ℓ + ℓ − branching ratios in Table 1, and contrast them with the corresponding SM predictions based on the C-conserving two-photon mechanism [17].We observe that those predictions are below the current limits by large factors, between 5 × 10 3 (for η → π 0 e + e − ) and 5 × 10 6 (for η ′ → ηe + e − ).We will therefore perform the analysis in the following in the spirit that we assume the SM contribution to be small, and any observable or observed signal to be a sign of a C-odd single-photon mechanism. 1Note furthermore that C-even and C-odd amplitudes cannot interfere on the level of the branching ratio; such interference effects could only induce Dalitz-plot asymmetries (cf.Refs.[1][2][3]), which would only be sizable if both amplitudes are of comparable magnitude.We therefore refrain from discussing such interference effects in any detail.
We emphasize the special role of CP violation in flavor-conserving transitions such as all η and η ′ decays discussed here.In contrast to similar kaon decays K → πℓ + ℓ − [29,30], SM CP -odd contributions induced by the weak interactions are very strongly suppressed, as any weak phase cancels at one loop, and any such contribution only depends on the squared moduli of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements.This is not unlike the situation for electric dipole moments of nucleons, which are considered quasi-free of any CKM-matrix-induced SM background [31][32][33][34].
Assuming that the underlying new physics generating the C-and CP -odd decays X → Y ℓ + ℓ − originates from sources at some high-energy scale Λ, there are in principle three dominant mechanisms to consider: 1. short-distance contributions to the dilepton final state, 2. long-distance contributions caused by C-and CP -odd photon-hadron couplings, 3. long-distance contributions induced by hadronic intermediate states.
For the first two classes we rely on ToPeχPT as proposed in Ref. [4].One intricacy of the contribution by hadronic intermediate states is that the subsequent photon is allowed to have both isoscalar and isovector components.To predict the involved isovector transitions in a model-independent way, one can utilize the X → Y π + π − amplitudes derived nonperturbatively in the Khuri-Treiman framework [2,3] and establish dispersion relations for the respective transition form factors. 2 Analogous relations have previously been derived for the decays ω, ϕ, J/ψ → π 0 γ * [36][37][38][39][40], which are compatible with conservation of all discrete symmetries.In addition, we sketch an idea of how to evaluate the isoscalar contribution of the photon, employing a less sophisticated, but still symmetry-driven, vector-mesondominance (VMD) model for the decay X → Y γ * .By an analytic continuation of the three-body amplitudes X → Y π + π − to the second Riemann sheet we can extract ρY X couplings, which can be related to the relevant ones with the same total isospin in the VMD model using SU (3) symmetry and naive dimensional analysis (NDA).
To extract observables of the C-and CP -violating contribution in X → Y ℓ + ℓ − driven by a one-photon exchange we pursue the following strategy.First, we consider the phenomenology behind the three mechanisms mentioned above in Sect. 2. For this purpose we lay out the basic definitions of kinematics and relate the amplitude to the (differential) decay widths in Sect.2.1.We discuss the short-range semi-leptonic operators, the long-range direct photon-hadron couplings, and the long-range hadronic contributions on a general level in Sects.2.2-2.4,respectively.Subsequently, Sect.2.5 includes a discussion of the feasibility of these contributions.The remainder of the article solely focuses on long-distance describes a short-range semi-leptonic four-point vertex, the second one includes a longrange hadron-photon coupling, while the last two diagrams account for possible hadronic intermediate states.Among the latter, the pion loop corresponds to an isovector transition while the vector-meson conversion respects the isoscalar part of the virtual photon.The black dot, the red box, the gray circle, and the blue box refer to different C-and CPviolating vertices, while the white circle is C-and CP -conserving.
contributions with hadronic intermediate states.In Sect. 3 we investigate the isovector contributions to these hadronic long-range effects.For this purpose, we first sketch the C-and CP -odd contributions to the decays X → Y π + π − in Sect.3.1, which serve as input to the respective transition form factors.The computation of the latter is discussed in Sect.3.2.
In Sect.3.3, we extract the corresponding C-odd couplings of the ρ(770) resonance to ηπ 0 and η ′ η by analytic continuation in the complex-energy plane.Subsequently, we estimate the size of the hadronic long-range effects in the isoscalar parts in Sect. 4. Finally, we present the predicted upper limits on the branching ratios in Sect. 5 and close with a short summary and outlook in Sect.6.

Phenomenology
In this section we discuss the phenomenological importance of the three mechanisms driving X → Y ℓ + ℓ − and provide the model-independent expressions for them.As an illustration we depict the different contributions in Fig. 1.For simplicity we adapt the notation and conventions introduced for the construction of operators in ToPeχPT in Ref. [4] without further details.

Kinematics
Consider the transition amplitude Conventionally, we describe the three-body decay in terms of the Lorentz invariants With the electromagnetic quark current where Q f indicates the electric charge of the respective quarks with flavor f and is conventionally used in units of the proton charge e, the singularity-free electromagnetic transition form factor F XY (s) in X → Y γ * can be decomposed by Poincaré invariance and current conservation as [9,30] ⟨Y (p)|J µ (0)|X(P )⟩ = −i s(P + p) µ − (P 2 − p 2 )q µ F XY (s) ≡ −iQ µ F XY (s) . (2.3) Here we introduced the photon's momentum q µ = (P − p) µ .Note that F XY (s) thus defined is real at leading order in ToPeχPT. 3 Upon contraction with the lepton current the full decay amplitude becomes where the term proportional to q µ drops out due to current conservation [9].In the course of this work, we will see explicitly that the amplitude of each mechanism restores this functional form.Taking the squared absolute value and summing over the lepton spins, one may obtain the doubly differential decay width [41] dΓ in terms of the electromagnetic fine-structure constant α = e 2 /4π, the Källén function λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + xz + yz), and the Lorentz invariant τ = t ℓ − u ℓ .The τ -integration can be carried out analytically, giving where σ ℓ (s) = 1 − 4m 2 ℓ /s and the physical range is restricted to s ∈ 4m 2 ℓ , (M X − M Y ) 2 .Throughout, we use the masses m e = 0.51 MeV, m µ = 105.66MeV, M η = 547.86MeV, M η ′ = 957.78MeV, M π ± = 139.57MeV, and M π 0 = 134.98MeV [42].For later use, we also quote the vector-meson masses M ω = 782.66MeV [42] and M ρ = 763.7 MeV, where the latter is the real value of the ρ(770)-meson pole as determined in Ref. [43].The errors and additional decimal digits on all of these masses are negligible in our analysis.
2.2 Direct semi-leptonic contributions to X → Y ℓ + ℓ − In Ref. [4] it was shown that the only C-odd, P -even semi-leptonic four-point vertex to η → π 0 ℓ + ℓ − at lowest order in the QED fine-structure constant and soft momenta originates from the dimension-8 LEFT operator where c (u) ℓψ denotes flavor-dependent Wilson coefficients. 4The choice of the high-energy scale Λ depends on the interpretation of the ToPe operators: in the picture of LEFT, Λ can be in the order of the electroweak scale, while in the spirit of the Standard Model effective field theory Λ is a typical BSM scale.The respective leading ToPeχPT operators in the large-N c limit read [4] where we employ the simple single-angle η-η ′ mixing scheme [45], for which the singlet component corresponds to (2.9) The meson matrix in the large-N c limit is then given by (2.10) In both Eqs.(2.9) and (2.10), η and η ′ refer to the physical fields; see, e.g., Refs.[45][46][47] for detailed discussions on more elaborate mixing schemes.Furthermore, in relation (2.8) we have introduced the spurion matrices λ L,R in flavor space, which were defined in Ref. [4] and acquire the same physical values, namely λ L,R ∈ {diag(1, 0, 0), diag(0, 1, 0), diag(0, 0, 1)} for the quark flavor ψ = u, d, s, respectively.Besides, F 0 denotes the common meson decay constant in the combined chiral and large-N c limit, F 0 ≲ F π ≈ 92.3 MeV.Summing over ψ and only picking the interactions relevant for our interests, the operator X(u) ℓψ gives rise to the leading-order Lagrangians with the normalizations ℓs .(2.12) Both processes are uncorrelated as their normalizations are linearly independent; the flavor combinations reflect the isospin and SU (3) structure of the transitions.Making use of the Dirac equation for the leptons, the corresponding matrix element yields with In the last step we applied the NDA assumption N X→Y ℓ + ℓ − ∼ 4πF 4 0 .Note, however, that the sign of the normalization is not fixed by NDA.

Direct photonic contributions to X → Y γ *
The leading-order contribution to the effective Lagrangian of X → Y γ * reads [4] (2.15) We may access the normalization N X→Y γ * using NDA, by regarding the possible sources on the level of LEFT, cf.Ref. [4].In this discussion we can directly ignore LEFT sources whose leading-order contributions in ToPeχPT are proportional to the ϵ-tensor and can thus not generate an even number of pseudoscalars and at the same time preserve parity.The NDA estimate of N X→Y γ * for the chirality-breaking dimension-7 LEFT quark-quadrilinear [48][49][50][51][52]] which is in the focus of Ref. [4], yields evF 3 0 /4π, with Higgs vev v.For the C-and CP -odd dimension-8 operators listed in this reference with two quarks and two gluon field strengths, four quarks and one gluon field strength, four quarks and one photon field strength, we have N X→Y γ * ∼ eF 4 0 , eF 4 0 /(4π), F 4 0 , respectively.It has to be underlined that each of these estimates may differ by one order of magnitude, possibly rendering all of these operators to the same numerical size.However, in the scope of these NDA estimations, we assume the normalization of the dimension-7 LEFT operator to dominate the remaining ones.
Using L X→Y γ * to evaluate the C-odd vertex in the second diagram of Fig. 1, we obtain the matrix element [4] with Again, NDA does not provide any information on the sign of the amplitude.Comparing Eqs.(2.14) and (2.18), we note F 2 (s)/F 1 (s) ∼ αv/(4πF 0 ) ≈ 1.5, hence both contributions are really expected to be of comparable size.

Hadronic long-range effects
The hadronic long-range contributions to the transition form factor can be constructed with knowledge about ToPe forces in We consider in the following sections both the isovector and isoscalar part of the photon.

The isovector contribution
In this section we establish dispersion relations for hadronic contributions of the C-and CP -odd transition form factor F XY and restrict the calculation to the isovector part of the photon.The discontinuity of X → Y γ * , as depicted in Fig. 2, can be calculated by applying a unitarity cut on the dominant intermediate state, i.e., two charged pions, allowing us to access the transition form factor in a non-perturbative fashion.The first ingredient to the These amplitudes will be discussed in detail for the different cases in Sect.3. The remaining contribution is the pion vector form factor defined via the current Note that this equation and Eq. ( 2.3) differ, beside the respective momentum configuration, by an explicit imaginary unit as demanded by their different behavior under time reversal.With only elastic rescattering taken into account, the pion vector form factor obeys the discontinuity relation where δ 1 (s) denotes the P -wave ππ phase shift with two-body isospin I ππ = 1.The most general solution to this equation is given in terms of the Omnès function [53] with a real-valued subtraction polynomial P n of order n.The index of the Omnès function indicates the isospin I ππ of the dipion state.The pion vector form factor is expected to behave as F V π (s) ≍ 1/s for large energies [54][55][56][57][58][59][60][61] (up to logarithmic corrections), and to be free of zeros [61,62].Thus, P n is a constant and can be set to 1 due to gauge invariance, such that F V π (s) = Ω 1 (s).For consistency we waive the incorporation of inelastic effects, which we do not consider in X → Y π + π − either.In the region of the ρ(770) resonance dominating F V π (s), these are known to affect the form factor by no more than 6%, depending on the phase shift used as input [63].Given other sources of uncertainty in the present study, we consider this error negligible.When we cut the dipion intermediate state in Fig. 2, the discontinuity of the isovector contribution XY to the transition form factor F XY becomes ) where t = (P − p + ) 2 and u = (P − p − ) 2 .We find discF where σ π (s) = 1 − 4M 2 π /s.In this discontinuity relation the quantity f XY denotes the P -wave projection of the hadronic decay amplitude given by Adapting the high-energy behavior of f XY and δ 1 from Refs.[2,3], an unsubtracted dispersion relation is sufficient to ensure convergence of the remaining integral over the discontinuity, such that the form factor can be evaluated with (2.27)

The isoscalar contribution
In order to estimate the isoscalar contribution, we apply a VMD pole approximation and consider a vector-meson conversion of γ * to v µ , with v ∈ {ω, ϕ}, cf. the very right diagram in Fig. 1.While this strategy is not as model-independent and sophisticated as the dispersive analysis of the isovector part of γ * , it serves as a good approximation to at least estimate the relative size of this contribution, not least due to the narrowness of the ω and ϕ resonances dominating isoscalar vector spectral functions at low energies.Furthermore, this ansatz even correlates the decay η → π 0 γ * to η ′ → ηπ + π − and η ′ → ηγ * to η → π 0 π + π − by following the strategy sketched in Fig. 3 to relate the decays of same total isospin.
The combination of vector mesons with χPT was extensively worked out for instance in Refs.[64][65][66][67][68][69] and references therein.The number of free parameters can be reduced most efficiently, cf.Ref. [66], by coupling uncharged vector mesons to uncharged pseudoscalars via the field-strength tensor V µν L,R .The latter is the analog to the photonic one with the same discrete symmetries and transformations under SU (3) L × SU (3) R .If we only consider the relevant degrees of freedom, i.e., treating Ū and V µ L,R as diagonal matrices, we can effectively write (2.28) The physical value of this chiral building block can be evaluated with where the ellipsis indicates terms without vector mesons.At the mesonic level, we can deduce the desired interaction from the leading-order XY γ * operator, cf.Ref. [10], and hence write ) The ToPeχPT operators that can generate this mesonic interaction at leading order in the large-N c power counting, i.e., O(p 4 , δ 2 ) (see Ref. [4] for further details), and originate from the LEFT operator in Eq. (2.16) are Here, we only list the operators leading to distinct, non-vanishing, expressions after we set λ ( †) , λ L,R , and V µν L,R to their physical values and use the fact that in our application all appearing matrices are diagonal and therefore commute.Evaluating the flavor traces of the operator in the first line and labeling the vector-meson couplings with a corresponding superscript (V 1 ), we end up with ωηη ′ , g ρηη ′ , g (2.31) For the second operator in Eq. (2.30) we observe that the resulting vector meson couplings g (2.32)Both Eqs.(2.31) and (2.32) suggest that there is a correlation between g ρπη and g ωηη ′ as well as between g ρπη and g ωηη ′ , but none of g ϕπη or g ϕηη ′ with the ρ couplings.However, this observation does not necessarily hold for higher orders in ToPeχPT or for operators derived from other LEFT sources.We continue with the couplings in Eq. (2.32) as our central estimates and make use of the flavor relations implied therein.
Next, we consider the Lagrangian for the vector-meson conversion with known coupling constants g vγ .As Eq. (2.33) employs the photon field instead of the field strength tensor, it is not manifestly gauge invariant.
In the end, this is a necessity to implement strict VMD for the isoscalar part of the form factor, avoiding an additional direct photon coupling.We can now evaluate the isoscalar contribution illustrated on the very right in Fig. 1, which, in agreement with Eq. (2.4), gives rise to the matrix element The corresponding isoscalar form factor, which is consistent with the high-energy behavior of the isovector part in Eq. (2.27), finally reads where g equals g vπη for X = η, Y = π 0 and g vηη ′ for X = η ′ , Y = η.

Discussion
With the results worked out in the previous sections we can evaluate the full contribution of the X → Y γ * transition form factors by where each summand corresponds to one diagram in Fig. 1.Note that there is no way to distinguish between the four contributions in a sole measurement of the X → Y ℓ + ℓ − branching ratio.
Regarding F 1 and F 2 , we observe that their NDA estimates in Eqs.(2.14) and (2.18) yield roughly the same result, even without accounting for the uncertainty of NDA.Hence, there is no clear hierarchy between direct semi-leptonic contributions and C-and CPviolating photon-hadron couplings contributing to X → Y ℓ + ℓ − .In future analyses, the sum F 1 + F 2 (which does not depend on s at leading order in ToPeχPT) may be replaced by a single constant parameter in a regression to hypothetical measurements of respective singly-or doubly-differential momentum distributions.
We remark in passing that all transition form factor contributions could be expected to undergo further hadronic corrections due to "initial-state interactions" of ηπ and η ′ η Pwave type, respectively.However, all corresponding phase shifts are expected to be tiny and the resulting effects to be hence utterly negligible: the ηπ P -wave is strongly suppressed in the chiral expansion at low energies relative to ηπ S-wave or ππ rescattering [70,71], and resonances with quark-model-exotic quantum numbers J P C = 1 −+ due to their C-odd nature will have a rather large mass [72].We therefore do not consider any such corrections in this article.
Given the currently accessible experimental data and the missing information on the normalizations of F 1 and F 2 , we henceforth focus on the contributions of F XY .On the one hand, we are in a position to predict the latter with the input discussed in Sect.2.4.On the other hand, they provide new conceptual insights by directly relating ToPe forces in X → Y π + π − with X → Y γ * .Moreover, for simplicity we assume no significant cancellations among the individual contributions in Eq. (2.36) throughout this manuscript.We leave the study of such a more complicated interplay between different mechanisms for future analyses, when more rigorous experimental bounds may be exploited for correlated constraints between them.

Hadronic long-range effects: the isovector contribution
In this section we investigate the isovector contribution to the transition form factor X → Y γ * based on the dispersive representations derived in Sect.2.4.1.Thus, we focus on Cand CP -odd contributions of the lowest-lying hadronic intermediate state, i.e., on the decay chain The formalism, results, and most of the notation are adopted from Refs.[2,3].The latter uses a dispersive framework known as Khuri-Treiman equations [73] to access the threebody amplitude X → Y π + π − including its C-and CP -odd contributions.In this approach, a coupled set of integral equations is set up for the two-body scattering process and analytically continued to the physical realm of the three-body decay.
where the lower index denotes the total isospin of the three-body final state.Neglecting Dand higher partial waves, we can decompose these amplitudes in the sense of a reconstruction theorem [74][75][76] into single-variable functions G Iππ (s), H Iππ (s) with fixed two-body isospin I ππ and relative angular momentum ℓ of the π + π − state: with A Iππ ∈ {G Iππ , H Iππ } and the ππ scattering phase shift δ Iππ (s).The inhomogeneity ÂIππ (s) contains left-hand-cut contributions induced by crossed-channel rescattering effects.
In terms of the angular average ) For the A Iππ (s) we employ dispersion relations with a minimal number of subtractions to ensure convergence.Assuming that in the limit of infinite s the phase shifts scale like δ 1 (s) → π, δ 2 (s) → 0 and the single-variable functions as where here and in the following M π ≡ M π ± .The C-conserving SM amplitude for η → π + π − π 0 is similarly described in terms of Khuri-Treiman amplitudes; these have been discussed extensively in the literature, see Ref. [77] and references therein.The subtraction constants obtained by a fit to the Dalitz-plot distributions These subtraction constants give rise to the real-valued isoscalar and isotensor couplings With the A Iππ (s) defined above, the P -wave amplitude necessary to evaluate the η → π 0 γ * transition form factor is by definition, cf.Eq. (2.25), given as whose dependence on H 2 and Ĥ2 is rather subtle and enters the definition of Ĥ1 in Eq. (3.5).
The transition form factor is fully determined by knowledge about the partial-wave amplitude f XY (s) and the pion vector form factor F V π (s).These quantities are in turn fixed by the subtraction constants ε, ϑ, the S-wave ππ scattering phase shift δ 2 with isospin 2, and the P -wave ππ scattering phase shift δ 1 [80], respectively.The latter has to be used consistently in f XY (s) and F V π (s), i.e., we use the same continuation to asymptotic s and omit the incorporation of inelasticities, which is beyond the scope of this work.
In the C-and CP -violating contribution to η ′ → ηπ + π − the three-body final state carries total three-body isospin 1.The respective amplitude can be decomposed as where the cosine of the scattering angles in the s-channel is still given by the general expression Eq. (2.26), z s ≡ z, while the one in the t-channel reads (3.12) In these equations we used the notation We additionally introduced two new types of angular averages, namely Assuming the asymptotics δ ηπ (t) → π and G ππ (s) = O(1/s), G ηπ (t) = O(t 0 ), the singlevariable functions can be evaluated by Here, the ηπ S-wave phase shift from Refs.[82,83] has been employed.For the subtraction constants, the values Finally, the P -wave entering the η ′ → ηγ * transition form factor is given by The dependence of f η ′ η on the S-wave amplitude G ηπ is encoded in the angular averages in Eq. (3.11).

Computation of the isovector form factor X → Y γ *
When computing the transition form factors XY , it is advantageous to exploit the linearity of the three-body decay amplitudes M XY in the subtraction constants.As mentioned in Refs.[2,3], the solutions of the Khuri-Treiman amplitudes can be represented by so-called basis solutions, which are independent of the subtraction constants and can be fixed once and for all before even carrying out a regression to data.

η → π 0 γ *
The basis solutions for the P -wave amplitude f ηπ are defined by and illustrated in Fig. 4(top).The dimensionless f ε ηπ corresponds to the isoscalar amplitude M ηπ 0 , while f ϑ ηπ belongs to the isotensor one, i.e., to M ηπ 2 .The partial waves have a singular character at pseudothreshold, i.e., the upper limit in s of the physical region in the η → π 0 π + π − decay, which is contained in the inhomogeneities describing left-hand-cut contributions to the respective partial wave.Note that the form factor, after performing the dispersion integral over the discontinuity as in Eq. (2.27), is perfectly regular at that point.Based on Eq. (3.18), we can calculate the corresponding basis form factors which allow us to linearly decompose F (1) ηπ according to The F ν ηπ are pure predictions of our dispersive representation, independent of the subtraction constants.Our results for the basis solutions for the form factors are depicted in Fig. 4(bottom).ηπ .The plots in Fig. 4 show that the basis solutions for the isoscalar and isotensor contributions to the η → π 0 π + π − P -wave amplitude are of the same order of magnitude, and so are, as a result, the corresponding basis form factors.But due to the vast difference in their normalizing subtraction constants, the term ϑF ϑ ηπ (s) is negligibly small in comparison to εF ε ηπ (s).The origin of this discrepancy is well understood [1][2][3].The totally antisymmetric combination of P -wave single-variable functions in the isoscalar amplitude M ηπ 0 , cf.Eq. (3.2), leads to a strong kinematic suppression inside the Dalitz plot; for symmetry reasons alone, the amplitude is required to vanish along the three lines s = t, t = u, and u = s.As a result, the corresponding normalization ε is far less rigorously constrained from fits to experimental data [79] than the isotensor amplitude, which only vanishes for t = u.No such suppression occurs for the individual partial waves, or the transition form factors, be it in the ρ-resonance region or below, in the kinematic range relevant for the semi-leptonic decays studied here, where isoscalar and isotensor contributions show non-negligible, but moderate corrections to a ρ-dominance picture.We also remark that this subtle interplay demonstrates that the model-independent connection between Dalitz plots and transition form factors absolutely requires the use of dispersion-theoretical methods-a low-energy effective theory such as chiral perturbation theory is insufficient for such extrapolations.
For the numerical evaluation of F ηπ we only consider the by far dominant source of error, i.e., the uncertainty of the subtraction constants entering the partial wave.As their errors are of the same order of magnitude as their corresponding central values, it is a good approximation to neglect all other sources of uncertainties, such as the variation of phase-shift input.

η ′ → ηγ *
We now turn the focus on the transition form factor f η ′ η , whose basis solutions in terms of partial waves are defined as Using the f ν η ′ η we can define the basis from factors and finally obtain the complete isovector form factor in explicit dependence on the subtraction constants by means of The basis solutions for both partial waves and transition form factors are shown in Fig. 5.

Resonance couplings from analytic continuation
As both the partial waves f XY (s) and the resulting transition form factors F XY (s) have been constructed with the correct analytic properties, we can analytically continue them into the complex plane and onto the second Riemann sheet to extract resonance pole residues.The resonance in question is the ρ(770); its residues can be interpreted as model-independent definitions of C-violating ρ → XY coupling constants.To this end, we recapitulate aspects of Refs.[85,86].First, consider the discontinuity of the transition form factor in Eq. (2.24) on the first Riemann sheet The partial-wave amplitudes f ν η ′ η from Refs.[2,3] are depicted in the upper panel; again, the singularity at the pseudothreshold s = (M η ′ − M η ) 2 can be seen.These serve as an input to calculate the basis solutions of the transition form factors F ν η ′ η (s) as defined in Eq. (3.22) and shown in the lower panel.with Using that the pion vector form factor fulfills Schwarz' reflection principle and demanding continuity of the scattering amplitudes when moving from one Riemann sheet to another, i.e., we obtain Left-and right-hand side of Eq. (3.27) depend on the same argument, such that, by analytic continuation, this relation can be applied in the whole complex plane.In particular, in the vicinity of the ρ(770) pole, the transition form factors as well as the pion form factor on the second Riemann sheet behave as (3.28) The pole position s ρ has been determined most accurately in Ref. [43], using Roy-like equations for pion-pion scattering: M ρ = 763.7 MeV, Γ ρ = 146.4MeV (cf.also Ref. [87]); for later use, we also quote the coupling constant to ππ, |g ρππ | = 6.01, arg(g ρππ ) = −5.3• .We neglect the uncertainties in these parameters, as they are small compared to the ones fixing the partial waves f XY .While F V, II π is explicitly given in Ref. [86], we can match F (1), II XY to a VMD-type form factor similar to Eq. (2.35), but with g ρY X , g ργ , and M 2 ρ instead of g, g vγ , and M 2 v .Thus, in sufficient vicinity to the pole, we can write If we evaluate Eq. (3.27) near the pole s ρ and insert Eq. (3.29), we can compute the desired C-odd ρ-meson couplings by The problem is therefore reduced to evaluating the partial wave f I XY on the first Riemann sheet at the pole position, a task for which the dispersive representations are perfectly suited.To clarify the dependence on subtractions or effective coupling constants and therefore separate the uncertainty in these from the precisely calculable dispersive aspects, we will once more make use of the decomposition in terms of basis functions.
We begin with the η → π 0 transition form factor.The basis functions of the partial wave f ηπ , evaluated at the ρ pole, result in so that we obtain where we made use of Eq. (3.8).Employing of Eq. (3.30) and finally inserting the values for the coupling constants g 0 and g 2 as extracted from the η → π 0 π + π − Dalitz-plot asymmetry then yields g ρπη = (−0.089+ i 0.022) GeV 4 g 0 − (0.156 + i 0.007) g 2 = 0.25(0.40)− i 0.06(0.10)GeV −2 . (3.33) Note that the isotensor contribution g 2 is negligible in the coupling g ρπη .The analytic continuation of the basis partial wave for f η ′ η to the pole position of the ρ meson yields With Eq. (3.16) we can hence express the analytically continued partial wave at the ρ pole by where we considered correlated Gaussian errors for the couplings g 1 and δg 1 .
Note that the coupling constants in Eq. (3.30) become inevitably complex-valued, thus spoiling the well-defined transformation under time reversal when compared to the tree-level coupling constants from ToPeχPT.This is neither surprising nor specific to the context of symmetry violation studied here: in the strong interactions, resonance couplings that are real in the narrow-width limit necessarily turn complex when defined model-independently via pole residues in the complex plane.However, this points towards the reason why these complex phases will be irrelevant when using symmetry arguments to estimate isoscalar contributions in the next section: for the narrow ω and ϕ resonances, they are negligible to far better accuracy; symmetry arguments within the vector-meson nonet are not applicable to their total widths.We will therefore simply omit the imaginary parts in the next section and relate the C-odd ω couplings required for the model of the isoscalar parts of the form factors to the real parts of the ρ coupling (of the same total isospin) only.Note furthermore that Eqs.(3.33) and (3.36) still suggest the imaginary parts of g ρπη and g ρηη ′ to be rather small, such that the difference between real part and modulus, e.g., is negligible for our purposes.

Hadronic long-range effects: the isoscalar contribution
We now attempt to combine the findings of Sects.2.4.2 and 3.3.We wish to access the couplings g, cf.Eq. (2.35), by linking them to the g ρY X discussed in the last section.In Sect.2.4.2 we found a ToPeχPT operator that, when considered separately, allows usaccording to Eq. (2.32)-to relate these couplings by SU (3) symmetry.The vector-meson couplings with the same total isospin are found to be related by g ωπη = 1/ √ 3 g ρηη ′ and g ωηη ′ = √ 3 g ρπη , while g ϕπη = 0 and g ϕηη ′ does not correlate with respective ρ couplings.However, the predictive power of flavor symmetry arguments does not hold in general for all operators.This leads to the shortcoming that we cannot fix the relative sign of the couplings, which becomes evident when comparing Eqs.(2.31) and (2.32), and have to rely on NDA arguments to consider that there may be additional contributions to the couplings from linear combinations of Wilson coefficients, cf.Eq. (2.31).An alternative approach would be to use NDA right away and drop the relative factors of 1/ √ 3 and √ 3, respectively, but this still leads to the same caveats.
4.1 η → πℓ + ℓ − Possible contributions to the isoscalar form factor in η → π 0 ℓ + ℓ − can originate from an ω or a ϕ intermediate state.In accordance with Sect.2.4.2, these enter the form factor in the linear combination With our SU (3) estimate g ϕπη = 0 we can ignore the contribution of the ϕ.Dropping the latter is also justified from an NDA point of view: the difference of the two summands in Eq. (4.1) is negligible compared to the uncertainty of NDA if ηπ (s) is evaluated within the physical range.Therefore, we continue the estimation of the isoscalar contribution with the ω intermediate state only, for which we use |g ωγ | = 16.7(2)[86].
Relating g ωπη to the ρ coupling of the same total isospin I = 1 and omitting the imaginary part for the reasons given above, we find Throughout this manuscript we do not account for the numerically intangible uncertainties from our SU (3) estimates or NDA.As neither of the latter fixes the sign of |g ωπη |, we have to content ourselves with its absolute value.
Note that retaining the imaginary part of g ρηη ′ would have a negligible effect on the upper limit for |g ωπη |.
On the other hand, we can also place a bound on |g ωπη | using the upper limit on the branching ratio of ω → ηπ 0 as determined by the Crystal Ball multiphoton spectrometer at the Mainz Microtron (MAMI) [88] and the Lagrangian in Eq. (2.29).The partial decay width is found to be With BR(ω → ηπ 0 ) < 2.3 • 10 −4 [88] and Γ ω = 8.68 MeV [42], we obtain the bound which is significantly more restrictive than the theoretical estimate for the bound on the coupling inferred from g ρηη ′ .
4.2 η ′ → ηℓ + ℓ − Similarly to the previous section, the isoscalar part of the form factor in η ′ → ηℓ + ℓ − can be written as With the same reasoning as above we henceforth drop the contribution of the ϕ and only take the ω into account.The numerical result for the corresponding vector meson coupling, which has total isospin I = 0, is Once more, the imaginary part of g ρπη would yield just a minor contribution to the upper limit on |g ωηη ′ | and can be neglected.We furthermore remark that the ρπη coupling also has an isotensor component, which, however, has a negligible effect, cf.Eq. (3.33).

Results
With the theoretical apparatus at hand we are now able to predict upper limits on the decay widths (5.1) relying on the Dalitz-plot asymmetries in X → Y π + π − as the main input.As argued in Sect.2.5, we focus on the long-range contributions via hadronic intermediate states only, i.e., we set We disregard the contributions analyzed in Sects.2.2 and 2.3 according to the discussion in Sect.2.5: these do not show interesting correlations with other ToPe processes, and absent significant cancellations, we can study the consequences of limit setting for the long-range hadronic effects alone.The corresponding transition form factors for the isovector and isoscalar contributions to η → π 0 γ * read while the ones contributing to η ′ → ηγ * are F The subtraction constants fixing the F  is even smaller than the SM sensitivity for both decays η → π 0 ℓ + ℓ − , ℓ = e, µ.Therefore, these decays cannot be used to further constrain this parameter of C-odd, P -even η decays.When making such comparisons, it must be kept in mind that the calculations of the upper bounds and lower threshold values are based on the strict assumption that the other two couplings are negligible.Should one of these three coupling constants be different from  from the C-even two-photon process of the Standard Model.However, the gaps between the empirical values and lower thresholds are smaller than in the η cases, especially for g 1 .
most precise measurements of η → π 0 ℓ + ℓ − have a similar sensitivity to isoscalar ToPe interactions as the measured Dalitz-plot asymmetries in η → π 0 π + π − , despite their different scaling with small BSM couplings.As we found the experimental limits for η → π 0 ℓ + ℓ − to be more restrictive than our theoretically predicted ones, we were able to use the respective transition form factor as a constraint to sharpen the bounds on C violation in η → π 0 π + π − .On the other hand, both the isotensor coupling for C-odd η decays and the corresponding η ′ → ηℓ + ℓ − effects are more rigorously constrained indirectly from Dalitz-plot asymmetries.Finally, we have determined the best possible sensitivity to all C-odd hadronic couplings in semi-leptonic decays due to the Standard-Model background.
Given the relatively loose experimental bounds, we have largely eschewed concrete estimates of theoretical uncertainties in our limit setting.The dispersion relation used to connect hadronic and semi-leptonic decays are expected to work extremely well in the elastic approximation, with two-pion intermediate states only, for the relevant vector-isovector channel: comparable sum rules for SM processes typically work to better than 10% [37,63].This is the relevant accuracy for the relation between both types of C-odd effects.Neglected higher-order corrections in the chiral or large-N c expansions, which may easily amount to 30% or so, only affect the interpretation of effective coupling constants on the mesonic level in terms of underlying LEFT or SMEFT operators [4].
Further perspectives on the decays η → π 0 ℓ + ℓ − and η ′ → ηℓ + ℓ − could be opened by possible future measurements of the respective Dalitz-plot distributions.This would allow us to investigate actual C-and CP -odd observables, the Dalitz-plot asymmetries arising from the interference with the respective SM contributions.Such interference effects would, as the asymmetries in the hadronic η and η ′ decays, scale linearly with BSM couplings, however with likely less advantage in sensitivity due to the strongly suppressed SM amplitudes.Significant asymmetries can only be expected if, accidentally, C-even and -odd amplitudes happen to be of similar size.Still, due to synergy effects with other BSM searches in these decays, such as for weakly coupled light scalars [8], renewed experimental efforts are strongly encouraged.

Figure 1 :
Figure1: Contributions to the C-and CP -odd decay X → Y ℓ + ℓ − .The first diagram describes a short-range semi-leptonic four-point vertex, the second one includes a longrange hadron-photon coupling, while the last two diagrams account for possible hadronic intermediate states.Among the latter, the pion loop corresponds to an isovector transition while the vector-meson conversion respects the isoscalar part of the virtual photon.The black dot, the red box, the gray circle, and the blue box refer to different C-and CPviolating vertices, while the white circle is C-and CP -conserving.

Figure 4 :
Figure 4: Basis solutions for the partial waves and form factors for the η → π 0 transition.The partial-wave amplitudes f ν ηπ from Eq. (3.18) are depicted in the upper panel; the singularity at the pseudothreshold s = (M η − M π 0 ) 2 is clearly visible.These serve as an input to calculate the basis solutions of the transition form factors F ν ηπ (s) as defined in Eq. (3.19) and shown in the lower panel.

Figure 5 :
Figure 5: Basis solutions for the partial waves and form factors for the η ′ → η transition.The partial-wave amplitudes f ν η ′ η from Refs.[2, 3] are depicted in the upper panel; again, the singularity at the pseudothreshold s = (M η ′ − M η ) 2 can be seen.These serve as an input to calculate the basis solutions of the transition form factors F ν η ′ η (s) as defined in Eq. (3.22) and shown in the lower panel.

( 1 )
XY are given in Eqs.(3.7) and(3.15), the respective basis solutions F ν XY are depicted in Figs.4 and 5, and the coupling constants g ωY X entering the F (0) XY are quoted in Eqs.(4.4) and (4.6), respectively.We have pointed out above that we have no means to assess the relative sign of the isoscalar contribution.To determine theoretical upper bounds, we investigate the parameter space spanned by the coupling constants and their uncertainties fixing|F ηπ | 2 = |F (0) ηπ | 2 + |F (1) ηπ | 2 + 2 Re F (0) ηπ F (1) * ηπ (5.5)and similarly for F η ′ η .

Figure 6 :
Figure 6: Spectrum of the isovector contribution to the form factor (top) and the corresponding differential decay distribution (bottom) for η → π 0 ℓ + ℓ − .The dashed lines mark the respective upper and lower limits stemming from the uncertainties of the subtraction constants in Eq. (3.7).The physical ranges are in both cases restricted by 4m 2 ℓ ≤ s ≤ (M η − M π ) 2 .

Figure 7 :
Figure 7: Spectrum of the isovector contribution to the form factor (top) and the corresponding differential decay distribution (bottom) for η ′ → ηℓ + ℓ − .The dashed lines mark the respective upper and lower limits stemming from the uncertainties of the subtraction constants in Eq. (3.15).The physical ranges are in both cases restricted by 4m 2 ℓ ≤ s ≤ (M η ′ − M η ) 2 .