CP violation in η muonic decays

In this study, we investigate the imprints of CP violation in certain η 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 η facility. After estimating the bounds that the neutron EDM places, we find still viable to discover signals of CP violation measuring the polarization of muons in η → μ+μ− decays, 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 required 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 the polarization of muons: REDTOP [1]. After estimating the impact of these 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].

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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 the η → µ + µ − , γµ + µ − , µ + µ − e + e − decays, accounting for the polarization of muons in dilepton and single-Dalitz decays. We provide the required 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, that sets stringent 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 here CP violation in three different sectors: that involving the hadronic part only, which we include in the CP H category; that mixing quark and leptons, which we include in the CP HL one; and that 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 ) , (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 section 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

Imc
(1)prst equ 2v 2 (ē p iγ 5 e r )(ū s u t )+(ē p e r )(ū s iγ 5 u t ) , Particularly, the µEDM [5] put tight constraints on these operators [6]. 2 We could have CP violation in the pure leptonic side via the O le operator [6,7]. However, this is irrelevant in what we find the best channel, η → µ + µ − , and should be negligible in other cases -see discussions later.

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].

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1. The unbarred(barred) variables are kept for the e + (e − ) and b ≡ b(x), with n(x) and b(x) defined below eq. (B.4). If integrating over d e ± , the terms in braces vanish, recovering the standard result for g 2 S → 0. For the CP HL scenario, g S = −C, and from eq. (2.4),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 section A, we findg S = (−0.87 − 5.5i) 1 + 0.66 2 , where large hadronic uncertainties are implied. In order to test our CP -violating scenarios, we define the following asymmetries
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 interference of the LO SM result with our

10F
ηγγ * (s) stands for the normalized transition form factor.

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While for the SM result [eq. (3.9)] these asymmetries vanish, in our CP -violating scenarios we find 11 , we obtain that the SM background for the asymmetry at the a 1σ level is of the order of 10 −5 for REDTOP statistics, which results in the following sensitivities: 1 ∼ 10 −2 and c 22st O ∼ 1. For completeness, we show in section D that the Z-boson parity-violating asymmetry is irrelevant for such statistics.

Classical channel:
The double Dalitz decay has been the standard way to test for CP -violation in pseudoscalar mesons decays, as it does not require to measure the polarization of the leptons [8][9][10]. In this study, we restrict ourselves to the η → µ + µ − e + e − decay 12 and study the interference terms alone. Concerning the SM result, notation, etc., we refer to ref. [20]. Regarding the CP H interaction, we recover the results in [20] with the addition of the second form factor that was omitted there, 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. Their contribution to the differential decay width reads

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As said, in these decays a polarization analysis is not required to test for CP violation; this is related to the lepton plane angular asymmetries. Defining we obtain Employing the form factors defined in section A, we obtain [20], we find the SM background at REDTOP at the 1σ level to be 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. Particularly, in their non-relativistic limit 14 Being suppressed in the SM, EDMs put severe constraints on CP -violating new physics scenarios [6]. In addition, the dipole moments of heavy atoms and molecules put strong constraints for contact CP -violating electron-quark D = 6 operators [22]. This is the reason for which we did not consider the electronic, but the muonic case -see also in this respect the implications of the recent ACME Coll. [2] results for the electron EDM in ref. [3]. In 13 In general, F1(0) = 1, except for a neutral fermion, such the neutron, where we take Qn = 1 and 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 units of e cm, such that it involves With these definitions, the dipole moments can be also obtained from the effective Lagrangian L = Q e 2ψ σ µν (µ + iγ 5 d)ψFµν . the sections below, it will be useful to employ projectors (in analogy to refs. [23] for the magnetic moment) for F E which, in D = 4 dimensions read 15

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(4.3) In the following, we discuss the bounds that the nEDM puts on our new physics scenarios, for which we employ the projector in the q → 0 limit, in which dipole moments are defined.

nEDM bounds on CP H scenario
As stated in section 2, there are a number of effective operators belonging to this case -each 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 figure 2, which amplitudes read 16 . (4.5) Regarding the Γ ν vertex, we take it to be given by the on-shell form factors F 1,2 in eq. (4.1), which closely follows the methodology in ref. [24]. 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 projector technique, we obtain 16 For the N N η coupling we take the results in ref. [24], where this was given by L ⊃  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 reach for 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's 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 figure 3. This involves the following operators For the nucleon, the CP -violating contribution to the electromagnetic vertex is ≃ + Figure 4. The generic contribution to a nucleon (N) EDM (additional counterterms need to be included as well). We approximate it as a low-energy (finite) contribution saturated via an intermediate nucleon (N) state (reversed diagrams implied) and a high-energy contribution (including counterterms) that mimics a contact term resulting from the OPE of the two currents.
where, for O equ( edq) , we have the combinations 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 figure 4. Above, we employ the operator product expansion (OPE) for large (euclidean) momentum k.
Concerning the low-energy part, we have two hadronic elements to be computed. For Γ = iγ 5 , we approximate such an interaction via an intermediate pseudo-Goldstone boson state (π 0 , η, η ) -similar to the CP H scenario. Regarding Γ = 1, we approximate it via the scalar form factor (see appendix E). For the electromagnetic form factors we use again ref. [26]. Regarding 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

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with h q P = 0|qiγ 5 q |P , q = {u, d, s} a flavor index, and the functions , (4.14) For the high-energy region, the OPE calculation for the two-currents (to be included in the final hadronic matrix element) parallels that at the quark level -the main difference being the scale. Assuming that the theory was renormalized at a scale close to the electroweak one and assuming the quark dipole moment negligible at such scale, 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 recent results in ref. [7]. 21 From the neutron matrix elements g q T ≡ n|qσ µν γ 5 q |n , obtained from lattice QCD at µ = 2 GeV [28], and using the renormalization scale µ 0 = 100 GeV, we obtain for the high-energy contribution d n E = Im(−0.59c (1) 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. equ -in agreement with ref. [27]. 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.

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Once more, we emphasize that large uncertainties are implied, thus these 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 .

Conclusions and outlook
In this study, we have examined different imprints of CP violation arising from the SMEFT in different η muonic decays, which are effectively encoded via CP -violating transition form factors or contact η-lepton interactions. Having in mind the REDTOP experiment -a proposed η factory with the ability to measure the polarization of muons -we have estimated the sensitivities that can be reached in each case. After computing the implications of these scenarios on the nEDM, we have found that only η-lepton interactions -particularly the O 2222 edq operator -might leave an imprint via the muons polarization in the η → µ + µ − decay. 22 This is complementary to first generation (electron) bounds from the EDMs of heavy atoms and molecules. Still, there would be possible ways to improve this study. They are beyond the scope of the present work, but we briefly comment on them in the following.
Regarding the 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 , and the hadronic model might have 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. Finally, one might wonder about 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 presumably setting stronger bounds [7].
Regarding additional decays, we did not discuss here the η → µ + µ − π + π − decay, especially in the CP H scenario. Yet the latter has a 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 would result in stronger bounds. For the CP HL scenario, on turn, we expect too small asymmetries as it happens for 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 isovector current, which would result in a similar diagram modulo photon propagator and form factors.

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Acknowledgments The author acknowledges C. Gatto for stimulating this work and comments concerning REDTOP experiment, J. Novotný for comments on two-loop renormalization, J. M. Alarcón and P. Masjuan for discussions regarding hadronic models, B. Kubis for discussions regarding σ terms, A. Pineda and A. Pomarol for discussions concerning the two-loops leading logs and K. Kampf

A The form factors parametrization
Here we describe the parametrizations employed for the TFFs appearing in eq. (2.1). Regarding the standard -CP conserving -one, F ηγ * γ * (q 2 1 , q 2 2 ), we employ the approach described in refs. [16,29] and stick to the simplest parametrization that implements precisely the low-energy behavior and respects the high-energy one [29] 24 where, F ηγγ = 0.2738 GeV −1 and Λ = 0.724 GeV [29,30], 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 is unknown. In the following, we assume the high-energy behavior from ref. [31], implying that F CP 1 ηγ * γ * (−Q 2 1 , −Q 2 2 ) and −Q 2 F CP 2 ηγ * γ * (−Q 2 1 , −Q 2 2 ) behave asQ −2 , where 2Q 2 = Q 2 1 + Q 2 2 . Thereby we choose .

(A.2)
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).
If only the imaginary parts are relevant for the asymmetry, we employ the TFF described in ref. [15] instead. This reads F P γ * γ (s) = F P γγ [c P ρ G ρ (s) + c P ω G ω (s) + c P φ G φ (s)], with G ρ,φ (s) Breit-Wigner functions, and the intermediate ππ rescattering modeled as in refs. [32,33] through and c η{ρ,ωφ} = {9, 1, −2}/8, which is -effectively -similar to the description in ref. [34]. 24 We checked that higher order approximants did not change significantly our results, so we stick to this for simplicity.

JHEP01(2019)031 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 Including phase-space and integrating over the neutrino spectra (we employ the muon rest frame), the result above reads 25 µ +m 2 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. (3.7) (Int X = 2 Re M SM M X ) yield 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 . This is, modulo radiative correction effects, the SM result from chapter 58: "Muon decay parameters" of ref. [5] and implemented in Geant4.

D Z boson contribution to Dalitz decay
In the SM, parity-violating contributions arise from an intermediate Z-boson state, . The term without the γ 5 is analogous to the QED result modulo form factor details and the s −1 factor; the γ 5 term induces a parity-violating interference
Again, the full decay width can be obtained to be With these results at hand, one finds that the Z-boson parity-violating contribution results in a non-vanishing asymmetry Regarding the form factor, we take its normalization from χPT at NLO (see refs. [11,12]) 28 where c 8 K ), and K 0T 3 2 = K 2 (m 2 K +m 2 π )/2 and the mixing parameters are taken from [12]. Finally, for its q 2 -dependence, we take a TFF analogous to that in section A with Λ = 0.57(7) GeV and 0.90(4) GeV for the η and η . This results from averaging the values that would be obtained from a BL-interpolation formula [36] and from a resonance saturation approach with weights given by mixing parameters similar to refs. [11,30,37]. As a result, we find A L = 6(1) × 10 −7 , 29 which is irrelevant for the expected statistics at REDTOP, and would require of the order of 10 16 η mesons for its observation.

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 average theoretical (if 27 Though at this step it cannot be compared to the results in ref. [35], that uses a different frame, we compared intermediate steps against their result in 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). 28 With this, we obtain FηγZ * (0, 0) = 0.074(6) GeV −1 and F η γZ * (0, 0) = 0.19(1) GeV −1 . 29 This might be compared to ref. [35] 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. [46]. We also include the isovector (blue) and strange (purple).

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.