$C\!P$ violation in $\eta$ muonic decays

In this study, we investigate the imprints of $C\!P$ violation in certain $\eta$ muonic decays that could arise within the Standard Model effective field theory. In particular, we study the sensitivities that could be reached at REDTOP, a proposed $\eta$ facility. After estimating the bounds that the neutron EDM places, we find still viable to discover signals of $C\!P$ violation via the muons' polarization in $\eta\to\mu^+\mu^-$ decays at REDTOP, with a single effective operator as its plausible source.


Introduction
In this article, we investigate the signal of CP -violation in a set of η muonic decays. In doing so, we assume the signal as arising from heavy physics, so that the Standard Model effective field theory (SMEFT) can be applied. As a result, two different scenarios arise: that of CP -violating purely hadronic operators (CP H ) and that of CP -violating quark-lepton ones (CP HL ). We provide the necessary amplitudes for Monte Carlo (MC) generators and evaluate the impact of such operators in terms of certain asymmetries, using as a benchmark a proposed η factory with the ability of measuring muon polarization: REDTOP [1]. After estimating the impact of our operators on the neutron dipole moment (nEDM), we find that CP -violating quark-lepton interactions could be at the reach of REDTOP, while evading nEDM bounds. Dealing with muons, these bounds are complementary to those of the electron case which have been put recently after the ACME Collaboration results [2] in Ref. [3].
The article is organized as follows: in Section 2, we discuss the two CP violating scenarios arising from the SMEFT operators and their connection from quark to hadron degrees of freedom (e.g. with η-physics). Then, in Section 3, we compute η → {µ + µ − , γµ + µ − , µ + µ − e + e − } decays, accounting for muon polarization effects in dilepton and Dalitz cases. We provide necessary expressions for MC generators and compute different asymmetries that could be generated, providing the sensitivities at reach at REDTOP. Finally, in Section 4, we evaluate the impact of both CP -violating scenarios on the nEDM, which put powerful constraints.

The CP -violating scenarios
In our study, we assume that the CP violating new-physics effects are heavy enough to be described through the SMEFT. Following the operator basis in Ref. [4], we can find CP -violation of relevance in three different categories: those acting on the hadronic part only, which we include in the CP H category, those mixing quark and leptons, which we include in the CP HL one, and those affecting lepton-photon interactions (O eW,eB ), which we checked to be negligible and discard 1 for brevity. 2 Regarding the CP H category, since we are interested in decays connecting η to muons, these will result at low-energies in a CP -violating shift of the ηγ * γ * coupling, a scenario extensively discussed in the literature in the context of light pseudoscalar mesons [8][9][10]. In general, one has iM µν = ie 2 µνρσ q 1ρ q 2σ F ηγ * γ * (q 2 1 , q 2 2 ) + [g µν (q 1 · q 2 ) − q µ 2 q ν 1 ] F CP 1 ηγ * γ * (q 2 1 , q 2 2 ) + g µν q 2 1 q 2 2 − q 2 1 q µ 2 q ν 2 − q 2 2 q µ 1 q ν 1 + (q 1 · q 2 )q µ 1 q ν 2 F CP 2 ηγ * γ * (q 2 1 , q 2 2 ) , (1) where 0123 = +1, F ηγ * γ * is the standard transition form factor (TFF), and the latter two are CP -violating ones. For some details on our TFF description, we refer to Appendix A. Accounting for the hadronization details linking the SMEFT CP H operators to F CP 1,2 ηγ * γ * (q 2 1 , q 2 2 ) in a quantitative manner is a formidable task; however, this is enough to our purposes as we shall see. Coming back to the CP HL category, the relevant operators here are where v 2 √ 2G F , and {p, r, s, t} label flavor. Concerning the η, these contain CP -violating interactions of the kind L = −C(ηē p e r ), with 3

Muonic decays and asymmetries
Having discussed the relevant hadronic matrix elements, we are prepared to discuss the different muonic decays, which in the first two cases involve the muon polarization-a property that can be measured at REDTOP [1].
where-hereinafter-n(n) will refer to the polarization axis for the µ + (µ − ) (in its rest frame), thê z-axis will point along µ + direction, and β µ will refer to the µ + velocity in dimuon rest-framehere coinciding with the η one. This would suffice to produce a MC for polarized decays. 6 Still, in order to define the asymmetries and estimate their size (in vacuum), we need to suplement this with the polarized muon decay in Appendix B, that leads to 7 whereg S = −g S /(2m µ α 2 F ηγγ ) and d e ± = dΩdxdΩdx(4π) −2 n(x)n(x) refers to the e + e − differential spectra, which is normalized to BR(µ → eνν) 1. The unbarred(barred) variables are kept for the e + (e − ) and b ≡ b(x), with n(x) and b(x) defined below Eq. (57). If integrating over d e ± , the term in braces vanish, recovering the standard result forg 2 S → 0. For the CP HL scenario, g S = −C, and from Eq. (4),g S = (0.510(c . For the CP H one, the contribution is generated at the loop level and parallels the SM calculation. Defining q(l) = p µ − ± p µ + , we find Using the form factors description in Appendix A, we findg S = (−0.87 − 5.5i) 1 + 0.66 2 , where large hadronic uncertainties are implied. To test our CP -violating scenarios, we define the following asymmetries where the barred version is the A L asymmetry for the e − . As a result, we find 5 For that, we use the polarized spin projectors u(p, λn)ū(p, λn) = 1 2 1 + λγ 5 / n / p + m and v(p, λn)v(p, λn) = 1 2 1 + λγ 5 / n / p − m . Particularly, n µ = (0, n) → (γβnz, nT , γnz). 6 In a real experiment the muon trajectory, its polarization, and subsequent-polarized-decay are accounted through Geant4 [17]. 7 We use, as it is standard, Γγγ = |Fηγγ| 2 m 3 η α 2 π/4, to normalize the result. Although we giveg 2 S terms, in the following we will only consider the interference with SM terms. Figure 1: Left: the LO SM contribution to the Dalitz decay. Right: Momentum-labeling in the µ + µ − reference frame.
for the CP H and CP HL scenarions, respectively. From BR(η → µ + µ − ) = 5.8 × 10 −6 , and expected η mesons at REDTOP (2 × 10 12 ) [1], we obtain that the SM background for the asymmetry, at the a 1σ level, is of order of N −1/2 = 3 × 10 −4 . As a result, we find the following sensitivities: 3.2 The Dalitz decay: η → γµ + µ − In the following we introduce the-polarized-Dalitz decays: for simplicity we do not consider the most general amplitude, but the LO SM result and its interference with our CP -violating amplitudes. Concerning the SM, the LO amplitude arises from the diagram in Fig. 1 (left), whose amplitude reads 8 Employing the phase space description in terms of the dilepton invariant mass (q 2 = (p + + p − ) 2 ≡ s ≡ x µ m 2 η ) and polar angle (θ) (see Fig. 1 [right]), the differential decay width can be expressed as 9 The LO SM result for the polarized Dalitz decay results in |M(λn,λn)| 2 = 1 4 e 6 |F ηγγ * (s)| 2 2s (m 2 η − s) 2 2 − β 2 µ sin 2 θ + λλ β 2 µ sin 2 θ(n znz − n T ·n T ) + 2 n znz cos 2 θ + n yny sin 2 θ − 2 1 − β 2 µ sin θ cos θ(n zny +n z n y ) , (13) with similar conventions as in the previous section (note that we choose, again, the µ + to mark thê z direction and the γ to have an additional component along theŷ directions-see Fig. 1 [right]). Once more, we include the muon decay to estimate the asymmetries, obtaining 10 The Z boson contribution is discussed in Appendix D and do not affect the results here. 9 y = βµ cos θ and β 2 µ = 1 − 4m 2 µ /s. Integrating over d e ± , the terms in braces vanishes and we obtain the standard result [18,19]. Concerning our CP -violating scenarios, we give here the main results and relegate intermediate steps to Appendix C. The final result reads whereα R,I is obtained from the results in Appendix C upon α R,I → α R,I /m 3 η and, for α R , λn(λn) → bβ(bβ) while, forα I , λn(λn) → −bβ(+bβ). In the following, we introduce two additional asymmetries besides those in Section 3.1 While for our SM result these vanish, we find numerically 11 Using BR(η → µ + µ − γ) = 3.1 × 10 −4 [5], we obtain that the SM background for the asymmetry, at the a 1σ level, is of order of 10 −5 , finding the following sensitivities: 1 ∼ 10 −2 and c 22st O ∼ 1. Also, in Appendix D we find that the Z-boson parity asymmetry does not show up at this level of precision.

Classical channel:
The double Dalitz decay has been the standard way to test CP -violation of pseudoscalar mesons since polarization experiments are not required [8][9][10]. In this study, we restrict here to the η → µ + µ − e + e − decay 12 and study the interference terms alone. Concerning SM results, notation, etc., we refer to Ref. [20]. Regarding the CP H interaction, we recover the results in [20] with the addition of the the second form factor that was omitted there 11 For analytic results in terms of phase-space integrals, see Appendix C. In these results we use the form factors in Appendix A and assume the CP -violating form factors to be real. 12 The SMEFT operators involving electrons are tightly constrained as we shall see and we neglect them, while the purely muonic channel has a BR that is two orders of magnitude below, so we restrict to this channel.
Concerning the CP HL scenario, there are four different contributions. Those arising from the effective operators coupling to muons are while, if considering the coupling to electrons, the remaining two would be obtained upon 1(2) → 3(4) and µ → e exchange. Its contribution to the differential asymmetry reads As said, in these decays a polarization analysis is not requried to test for CP -violation; this is related to the lepton plane angular asymmetries. Defining we obtain Employing the form factors defined in Appendix A, we obtain From [20], BR(η → µ + µ − e + e − ) = 2.3 × 10 −6 , and expected η mesons at REDTOP, the 1σ SM background is 5 × 10 −4 . Consequently, we are sensitive to 1 ∼ 10 −3 and c O ∼ 40.

Bounds from neutron dipole moment
The interaction of a charged fermion with the electromagnetic current (j µ ) can be expressed with q(l) = p ∓ p. At low energies, F 2 and F E generate magnetic and electric dipole moments, respectively. Paricularly, in their non-relativistic limit 14 13 For a neutral fermion, such the neutron, we take Qn = 1, while F1(0) ≡ 0. 14 Usually µ is given in units of e /2m and F2(0) yields the anomalous magnetic moment. The electric dipole moment commonly refers to d in e cm units, such that involves ( c[GeV cm])/(2m c 2 [GeV])FE(0). Also, we take L =ψ(iγ µ Dµ − m)ψ and Dµ = ∂µ − ieQ Aµ so that iM = ieū p Γ µ up µ. With these definitions, the dipole moments can be also obtained from the effective lagrangians L = Q e 2ψ σ µν (µ + iγ 5 d)ψFµν . Being suppressed in the SM, EDMs put severe constraints on CP -violating new physics scenarios [6].
In addition, heavy atoms and molecules dipole moments put strong constraints for contact CPviolating electron-quark D = 6 operators [32], the reason for which we did not consider the electronic, but the muonic case-see also in this respect the recent implications from ACME Coll. [2] electron EDM result in Ref. [3]. In the sections below, it will be useful to employ projectors (in analogy to Refs. [33] for the magnetic moment) for F E which, in D = 4 dimensions read 15 In the following we discuss the bounds that nEDM puts on our new physics scenarios, for which we employ the projector in the q → 0 limit, where dipole moments are defined.

nEDM bounds on CP H scenario
As stated in Section 2, there might be a number of effective operators belonging to this caseeach of them contributing differently to the nEDM and posing an individual challenge. However, for our purpose-as we shall see-it will suffice to account that the CP -violating η TFF will generate a nEDM via the diagrams in Fig. 2, which amplitudes read 16 Regarding the Γ ν vertex [see Eq. (30)], we take F 1,2 on-shell form factors, which is rather similar to the methodology in [36]. Of course, this contribution is rather model dependent and there will be additional ones, but should be enough to provide an order of magnitude estimate. Using the 16 For the N N η coupling we take the results in Ref. [36], where this was given by L ⊃ g ηN N 2FηN γ µ γ 5 N ∂µη with gηNN = 0.673 and Fη = 1.37Fπ. projector technique, we obtain

Im{O
where β = (1 + Of course, this offers no bounds on 2 and an alternative would be to set F CP 1,2 ηγγ (0, 0) → 0-this is, a vanishing coupling to real photons. A tuning such that would be suspicious however without a dynamical origin and we conclude that CP -violating physics in the context of CP -violating ηγ * γ * interactions are out of any experiment so far.

nEDM bounds on CP HL scenario
Here, there is no mechanism inducing a dipole moment at one loop, which can be related to the fact that the Green function 0| T {V µ (x)S(P )(0)} |0 vanishes in QED+QCD due to charge conjugation. The first contribution appears at two-loops and requires renormalization, which is sketched at the quark level in Fig. 3. This involves the following operators ≃ + Figure 4: The generic contribution to a nucleon (N) EDM (additional counterterms need to be included as well). We approximate as a low-energy (finite) contribution saturated via an intermediate nucleon (N) state contribution at low energies (reversed diagrams implied) and a high-energy contribution, including counterterms, which mimics a contact term.
For the nucleon, the CP -violating contribution to the electromagnetic vertex is where for O equ( edq) , we have the 17 In the following, we will simplify the calculation to get an order of magnitude estimate as follows: in the low energy region (which we take below 2 GeV), we will assume the hadronic blob to be dominated by an intermediate neutron state, as shown in Fig. 4. Above, we employ the operator product expansion (OPE) for large (euclidean) k.
Concerning the low-energy part, we have two hadronic elements to be computed. For Γ = P , we approximate such an interaction via an intermediate pseudo-Goldstone boson state (π 0 , η, η )similar to the CP H scenario. Regarding Γ = S, we approximate it via the scalar form factor (see Appendix E), while for the electromagnetic form factors we use again Ref. [38]. Regarding the Π µν V V S(P ) (k, q), we provide them in the vanishing q → 0 limit up to O(q 2 ) corrections, where β 2 = 1 + 4m 2 K −2 and K 2 = −k 2 . We obtain 18 17 Note in particular that potential additional diagrams with a single photon attached to the lepton line will be related again to 0| T {V µ (x)S(P )(0)} |0 = 0. 18  with h q P = 0|qiγ 5 q |P , where q = {u, d, s} is a flavor index, and the functions Numerically we find 19 For the high-energy region the calculation parallels that of the quark level, the differences being the scale. Assuming that the theory was renormalized at a scale close to the electroweak one and assuming the resulting dipole moment negligible there, the result can be estimated by the large logs. To find these, we opt to use a cutoff regularization (Λ) for the quark diagram level leading to As a check, the ln 2 Λ terms reproduce the expectation from the one-loop RG equations. 20 Moreover, we find good agreement for the ln Λ term (which represents the leading log for O edq ) comparing to the results of the recent Ref. [7]. 21 From the neutron matrix elements g q T ≡ n|qσ µν γ 5 q |n obtained from lattice QCD in Ref. [40] at µ = 2 GeV, and using the renormalization scale µ 0 = 100 GeV we obtain d n E = Im(−0.59c an order of magnitude below the low-energy contribution. Summarizing, and assuming uncorrelated Wilson coefficients, we find that nEDM puts the following constraints Once more, we emphasize that large uncertainties are implied and that results should be taken as an order of magnitude estimate. As a conclusion, we find that η → µ + µ − decays are the only ones that might show CP -violating signatures for c 2222 edq 10 −2 . 19 We checked that the integral saturates at 2 GeV. The errors for the neutron are shown to illustrate the impact on the scalar form factor model only-dominated by the σπN term. 20 With our conventions, equ -in agreement with Ref. [39]. 21 In particular, with Eq. (2.35) in [7] one takes e ↔ d and Nc = 1. One also needs to take care of the sign conventions-which are essentially related to our opposite choice for the covariant derivative.

Conclusions and Outlook
In this study, we have examined different imprints of CP violation arising from the SMEFT in different muonic η decays, which are effectively encoded via CP -violating transition form factors or contact η-lepton interactions. With the mind in REDTOP experiment-a proposed η factory with the ability to measure muon polairaztion-we have estimated the sensitivities that can be reached for both scenarios. After computing the implication of these scenarios on the nEDM, we found that only η-lepton interactions-particularly the O 2222 edq operator-might leave an imprint via the muons' polarization in η → µ + µ − decay. 22 This is complementary then to first generation (electron) bounds from heavy atoms and molecules EDMs. Still, there would be possible ways to improve this study, but which were beyond the scope of the present work and we sketch below.

Regarding SMEFT operators
A possible extension would be an improved determination of nEDM bounds on O equ, edq operators. There are different lines that could be pursued: considering non-vanishing O (3) equ and O uW,uB,dW,dB operators and employ the full RG equations [6]; computing the full two-loop calculation; improving the hadronic model (with a serious estimate of uncertainties).
Also, one could estimate the impact on the same operators for the = τ case. Here the large-logs will become as important as hadronic effects, as they are ∝ m , but the hadronic model will-likely require to be improved up to higher scales.
Very differently, it might be interesting to check the induced O 11st equ(dq) operators that might appear at two loops from O 22st equ(dq) and to check whether these might allow to improve the bounds derived here.
Also, once could study the O le operator. As said, this does not produce an effect at LO in dilepton decays. In Dalitz decays, would be analogous (up to i factor) to the Z-boson contribution, which we found negligible. For double Dalitz decays it might appear as a loop contribution, so we expect this small, with lepton EDMS likely setting strong bounds [7] Regarding additional decays We did not discuss here η → µ + µ − π + π − decays, especially in the CP H scenario. Yet these have larger BR than the leptonic one, the nEDM contribution would be very similar (for the CP H scenario) to that in Section 4.1 up to an α −1 K 2 factor, 23 which results in stronger bounds. For the CP HL scenario, on turn, we expect too small asymmetries as in the leptonic case. Overall, we do not expect-in principle-any CP violation in these decays.
Finally, we did not discuss polarizations in the η → π 0 µ + µ − decay, that might be interesting to analyze [1], but are beyond the scope of this study. 22 Being this a potential channel to look for CP violation, one might wonder about its η counterpart. An analogous computation shows A L L = − Im(1.4(c (1)2211 equ + c 2211 edq ) + 2.9c 2222 edq ) × 10 −2 . Since BR(η → µ + µ − ) 1.4 × 10 −7 [15], this cannot place stronger bounds. 23 The π + π − state is essentially the low-energy manifestation of the vector isvocetor current, which would rsults in a similar diagram modulo photon propagator and form factors.

A The form factors parametrization
Here we describe the parametrizations employed for the TFFs appearing in Eq. (1). Regarding the standard-CP conserving-one, F ηγ * γ * (q 2 1 , q 2 2 ), we adopt the procedure described in Ref. [16,21] and stick to the simplest parametrization that implements precisely the low-energy behavior and respects the high-energy one [21] 24 where, F ηγγ = 0.2738 GeV −1 and Λ = 0.724 GeV [21,22], except when imaginary parts are relevant, which we postpone to the end of this section. For the CP -violating form factors there is of course no theoretical knowledge as they are speculative and its microscopic origin unknown.
In the following, we assume the high-energy behavior from Ref. [23], implying that .
We take the same value for Λ as before and introduce Λ H = 1.5 GeV inspired by heavier resonances (results are rather stable upon varying these masses).

B Polarized muon decay
In the effective Fermi theory, and using polarized spinor sums, we find for the µ ± → e ± (k)ν µ (q 1 )ν e (q 2 ) decay amplitude M µ ± , λn Including phase-space and integrating over the neutrino spectra (we employ the muon rest frame), the result above reads 2526 dBR(µ ± , λn) = dΩ 4π with n(x, e )/2m µ is the maximum positron energy, x = E e /W eµ the reduced positron energy, x 0 = m e /W eµ the minimum reduced positron energy and β = 1 − x 2 0 /x 2 has the usual meaning. Typically, the approximation m e /m µ → 0 is employed, that results in the simpler expression

C Results in Dalitz decays
The resulting amplitudes for our CP -violating scenarios read Their interference with the SM amplitude in Eq. (11) (Int X = 2 Re M SM M CP X ) yield where we introduced the following coefficients 25 In the second line, the result for integration over dΩdx has been employed, that introduces = m 2 e m 2 e (m 2 µ − m 2 e ) 2 + 6m 6 µ + 2m 2 e m 4 µ (1 + 6 ln(me/mµ)) (m 2 e + m 2 µ ) −4 . 26 The second line is, essentially, the SM result from Ref. [27] and is that implemented in Geant4, where the latter is also improved upon the inclusion of radiative corrections.
Finally, we give here the asymmetries in terms of the phase space where we have introduced the common paremeterC = C/(e 2 m η F ηγγ ) 27 and The latter is, up to an α factor, the dyd e ± -integrated version of Eq. (13).

D Z boson contribution to Dalitz decay
In the SM there are parity-violating contributions arising from an intermediate Z-boson state in Dalitz decays that reads where √ 2G F = g 2 /(4m 2 W ). The term without the γ 5 is analogous to the γγ * up to form factor details and the s −1 factor; the γ 5 one induces the a parity-violating interference with the QED contribution. Particularly, 28 2 Re M SM M Z, / P = 1 4 Again, the full decay width can be obtained to be 27 From Eq. (4),C = (1.142(cu + c d ) − 1.726cs) × 10 −4 . 28 Though at this step it cannot be compared to the results in Ref. [28] that uses a different frame, we compared intermediate steps to their expression in their Eq. (9). We found agreement up to a minus sign (we remark that they calculate the µ + polarization, lacking the necessary terms to compare to our full polarization amplitude).
With these results at hand, one finds that Z-boson contribution results in a non-vanishing asymmetry Regarding the form factor implementation, for the normalization we employ the η − η mixing at NLO, analogous to those in Refs. [11,12]. 29 where = K 2 (5m 2 π − 4m 2 K ), and K 0T 3 2 = K 2 (m 2 K + m 2 π )/2. Finally, for its q 2 -dependence, we take a TFF analog to that in Appendix A with Λ = 0.57(7) GeV and 0.90(4) GeV for the η and η . This results from the values that would be obtained from a BL-interpolation formula [29] and from a resonance saturation approach with weights given by mixing parameters similar to Refs. [11,22,30]. We find then A L = 6(1) × 10 −7 . 30 This is then irrelevant for the expected REDTOP statistics and would require of the order of 10 16 produced η.

E The nucleon scalar form factors
In this section, we introduce the nucleon scalar form factors F N ;q S (q 2 ) ≡ N p |qq |N p . At q 2 = 0, these are related to the σ-terms. In the following we combine theoretical (if available) and lattice results from Ref. [41] and average (enlarging errors if necessary). Regarding the theoreticla input, we take it from Ref. [42] for the isoscalar σ N ud = N p |m(ūu +dd) |N p , from Ref. [43] for the isovector σ N 3 = N p |m(ūu −dd) |N p and restrict to lattice results for the strange one σ s 3 = N p | m ss s |N p due to the large theoretical uncertainties. This results in σ p,n ud = 45(29), σ p 3 = 11(4), σ n 3 = 20(4), σ p,n s = 54(5), in MeV units. Regarding the q 2 -dependency, we use the half-width rule [44], which proved to provide excellent estimates for the form factors. Since at high energies F −Q 2 Q −6 [45,46], we use three resonances. Following [47], we employ for the isoscalar channel f 0 (500), f 0 (1370), f 0 (1500), for the isovector a 0 (980), a 0 (1450), a 0 (1950) and for the strange one f 0 (980), f 0 (1500), f 0 (1710). In the following we provide the central value [see ...] for the required form factors (x = −q 2 )  ) .
(80) 29 With this, we obtain FηγZ * (0, 0) = 0.074(6) GeV −1 and F η γZ * (0, 0) = 0.19(1) GeV −1 . 30 This might be compared to Ref. [28] results. Checking intermediate steps, we confirmed their results except for an overall sign. Still, we find 2 orders of magnitude supression. This is due to the relevant scale in the problem (2mµ rather than mη) and the resolution power function b(x) < 1.  Figure 5: Comparison of the half-width estimate for the-normalized-isovector form factor (orange band) to that in Ref. [48]. We also include the isovector (blue) and strange (purple).