An improved search for the electric dipole moment of the τ lepton

We report a measurement of the electric dipole moment of the τ lepton (dτ) using an 833 fb−1 data sample collected near the Υ(4S) resonance, with the Belle detector at the KEKB asymmetric-energy e+e− collider. Using an optimal observable method, we obtain the real and imaginary parts of dτ as Re(dτ) = (−0.62 ± 0.63) × 10−17ecm and Im(dτ) = (−0.40 ± 0.32) × 10−17ecm, respectively. These results are consistent with null electric dipole moment at the present level of experimental sensitivity and improve the sensitivity by about a factor of three.

The electric dipole moment (EDM) of the τ lepton is a fundamental parameter that parameterizes time-reversal (T) or charge-conjugation-parity (CP) violation at the γτ τ vertex.In the Standard Model (SM), CP violation arises due to an irreducible phase in the CKM matrix [1], which predicts an unobservably small τ -lepton EDM (d τ ) of order 10 −37 ecm [2].Hence, observation of a nonzero d τ value would be a clear sign of new physics.Some new physics models indicate a larger EDM of order 10 −19 ecm [3].
The most sensitive previous measurement set an upper limit on the EDM of order 10 −17 ecm [4]; the results were obtained by the Belle collaboration [5] using 29.5 fb −1 of data collected at the KEKB collider [6]  In this paper, we present updated results on d τ using a much larger sample of 833 fb −1 Belle data, of which 571 fb −1 collected at the Υ(4S) resonance; 74 fb −1 collected 60 MeV below it; and 188 fb −1 collected near the Υ(1S), Υ(2S), Υ(3S), and Υ(5S) resonances.These samples are independent from the one used in the previous Belle result.The sensitivity for Re(d τ ) and Im(d τ ) has improved by about a factor of three, due to the increase of the data statistics and improved analysis strategy.
The effective Lagrangian for τ -pair production including the EDM term in the vertex is Including the EDM term, the squared spin-density matrix (χ prod ) for the production vertex in the process e + e − → τ + τ − is given by [7] where χ SM is the SM term, and χ Re and χ Im are the interference terms between the SM and the EDM for the real and imaginary parts of d τ .Here, χ d 2 is a higher-order EDM term, which we can neglect since d τ is small.The matrix elements in Eq. ( 2) can be expressed using the momenta of the electron beam and the τ lepton, and the spins of τ + and τ − in the e + e − CM frame.The interference terms are proportional to CP-odd spin-momentum correlation terms where k 0 is the energy of the τ ± , m τ is the τ mass, p is the three-momentum of the e + , k is the three-momentum of the τ + , S ± are the spin vectors for the τ ± , and hats denote unit momenta.In Eqs. ( 3) and (4) above, χ Re is T-odd and χ Im is T-even.A more detailed discussion is given in Ref. [4].Several CP-violating observables have been proposed in the literature [3].For this study, we use the so-called optimal observable method [8] to obtain the d τ values.The optimal observables are They maximize sensitivity to the τ EDM. since where the integration is performed over the available phase space φ and The expression for O Im is identical with the exchange of "Re" and "Im" in Eqs.(7) and (8).The cross-term containing the integral of the product of χ Re and χ Im drops out because of their different symmetry properties.To determine the coefficients, we have performed the integration using Monte Carlo (MC) samples in order to account for detector effects.In principle, the constant term (b Re/Im ) should be zero as χ Re/Im is symmetric, but can be nonzero owing to nonuniform acceptance of the detector.We therefore add this term.Using linear relations, Re(d τ ) and Im(d τ ) can be obtained from the measured values of O Re/Im .We have used the data collected by the Belle detector for this analysis.Belle is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector, a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrellike arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter (ECL) comprised of CsI(Tl) crystals; all located inside a superconducting solenoid coil that provides a 1.5 T magnetic field.An iron flux-return yoke located outside of the coil is instrumented to detect K 0 L mesons and muons (KLM).The detector is described in detail elsewhere [5].
The MC event generators KKMC and TAUOLA [9] are used for τ -pair production and decays, respectively.Detector simulation is performed by a GEANT3 [10] based program.We use a sample of MC events corresponding to about five times the data luminosity.In order to study the background contamination arising from non τpair events, we generate MC samples for the e + e − → q q (q = u, d, c, s) continuum and e + e − → Υ(4S) → B B events using the EVTGEN [11] program, and for twophoton mediated processes (e + e − → e + e − ℓ + ℓ − , e + e − q q) using the AAFH [12] program.
We use τ -pair events with a 1-prong versus 1-prong topology in which the particles are selected by the following criteria.Charged tracks are required to have a transverse momentum of p T > 0.1 GeV/c and an impact parameter along the positron beam and in the transverse plane less then 3.0 cm and 1.0 cm, respectively.An ECL cluster not matching any track is identified as a photon candidate.Photon candidates should deposit an energy of E > 0.1 GeV in the ECL.Each charged particle is identified using a likelihood ratio formed combining the ionization energy loss in the CDC, the ratio of energy deposited in the ECL and momentum measured in the CDC, the shower shape in the ECL, the position matching of the ECL cluster and CDC track, the range and hit pattern in the KLM, the time-of-flight information from the TOF, and the light yield of the ACC.Electron and muon candidates are selected by requiring the likelihood ratios P(e) [13] and P(µ) [14] to exceed 0.9 and 0.95, respectively.The corresponding identification efficiency is above 90% with a pion misidentification rate less than 2%.Charged pions and kaons are distinguished using likelihood ratios, P(i/j) = L i /(L i + L j ), where L i is the likelihood for a track to be identified as i.Pion candidates for the τ → πν mode are selected by requiring P(K/π) < 0.8, P(µ) < 0.05, P(e) < 0.01, and an electron likelihood ratio obtained by combining information from the ACC and CDC less than 0.9 to reduce electron backgrounds, which do not interact in the ECL.The requirement P(K/π) < 0.8 rejects 78% of kaons, while 94% of muons are rejected by the requirement P(µ) < 0.05 and 98% of electrons are rejected by the requirement P(e) < 0.01.A ρ ± is reconstructed from a charged track and a π 0 , requiring the opening angle between them to be less than 90 • in the CM frame and the charged track not to be an electron or a muon.The ρ ± candidates include higher ρ resonances since no mass cut is applied.The π 0 candidates, reconstructed from γγ combinations, should have an invariant mass between 110 and 150 MeV/c 2 and a momentum of p > 0.2 GeV/c.
In order to suppress two-photon mediated background contributions, we require the missing momentum vector not to point along the beam pipe, −0.950 < cos θ miss < 0.985.To reject Bhabha scattering and µµ backgrounds, we require the sum of the charged track momenta in the CM frame be less than 9.0 GeV/c.For the eπ mode, we remove events if at least one of the following conditions is satisfied: the opening angle between the two charged particles in the plane perpendicular to the beam axis is greater than 175 • , the sum of the charged track momenta in the CM frame is greater than 6.0 GeV/c, or the E ECL /p of pion is larger than 1.05.These criteria are required because of the large contamination from radiative Bhabha events.In addition, for the eρ mode, we require that the electron momentum in the CM frame be less than 5 GeV/c to suppress the same background.
These selection criteria are similar to those required in the previous analysis, with some changes following updates to the reconstruction software, and updated detector calibration.
To maintain consistency between the data and simulation, the effect of the trigger [15] should be taken into account.A hardware trigger simulator is used for MC samples.We also reject events in which the τ flight direction cannot be kinematically reconstructed in the observable calculation discussed below.
The obtained signal yield, purity, and dominant background are listed in Table I.The purity and the background are estimated using MC samples.In some modes, τ decays with additional π 0 mesons contribute a significant background due to low energy photons that escape detection.
The momentum and cos θ distributions for the obtained samples are shown in Figs. 1 and 2. In order to calculate the observables, we need to determine the τ spin vectors and flight direction.The quantities used in the following calculation are obtained in the CM frame.The spin vectors, which give the most probable direction of the spin, are reconstructed using the momenta of τ and its decay products [16,17].For example, the spin vectors for τ ± → π ± ν τ are given by where p π ± and E π ± are the π ± momentum and energy, respectively.(See the Appendix for other decays.)Although the τ flight direction k is necessary to calcu- late the spin vector and observables [17], experimentally the τ direction cannot be uniquely determined due to the presence of two or more missing neutrinos.In the reactions where both τ leptons decay semileptonically, e + e − → τ + τ − → A + B − ν τ ντ without initial-state radiation (ISR), the two possible solutions for the unit vector of the τ + flight direction, k+ and k− , are given by where p A (p B ) are the sum of three-momentum vectors in the decay products, A + (B − ).The parameters u, v, and w are where θ A (θ B ) are the angles between the momenta of the decay product A + (B − ) and the τ momentum: where i = A or B. In this case, the τ direction can be obtained with a twofold ambiguity.Experimentally this ambiguity cannot be resolved.Therefore, we take an average of the two possible solutions in the calculation of the observables.In the case of leptonic τ decays, one more ambiguity in the invariant mass of two neutrinos from the same τ , m νν , arises as We then take an average over multiple solutions using the MC method by varying m νν uniformly within the possible kinematical range.For each event, we make 100 trials using a "hit-and-miss" approach while varying the effective mass m νν randomly.With N hit successful trials in which the τ direction can be constructed kinematically, the average value of the observable is obtained for each event.In the case where both τ 's decay leptonically, the m νν is varied for each τ .In the calculation, we require w in Eq. ( 13) be real and cos θ j (j = A, B, ℓ) in Eqs. ( 14) and ( 15) be within the range [−1, 1], therefore we removed the cases when the above requriements were not satisfied.In the analysis, we neglect the effect of ISR for the calculation of the observables, and treat it as a systematic source.The distributions of observables for the obtained samples are shown in Fig. 3, along with those obtained from MC simulations with no EDM.We calculate the mean value of each observable using the data in the full range including events beyond the range shown in Fig. 3.
To obtain the EDM values from the observables, we must determine the relation between the EDM and the mean value of the observables shown in Eq. ( 6).In order to take into account the finite detector acceptance, the use of the most probable (rather than actual) spin direction, the ambiguity from the resolution, the unknown τ direction, and missing neutrinos, the relation between the EDM and the mean value of the observables, O Re/Im , is evaluated using MC simulation for various values of the EDM.By fitting the relation with a linear function in Eq. ( 6), as shown in Fig. 4, the coefficients a Re/Im and the offsets b Re/Im are obtained, which are plotted in Fig. 5.As seen from the values of the coefficients, a Re/Im , the πρ and ρρ modes have the highest sensitivities for Re(d τ ) and Im(d τ ), thanks to the high spin analyzing power for π and ρ modes.Nonzero offsets seen for the imaginary part, b Im , are due to a forward-backward asymmetry in the detector acceptance.The effects of the background are also taken into account in these coefficients.The coefficients are corrected for by the purity and the coefficients obtained using background samples.
We examine a number of possible systematic effects on the EDM measurements.The corresponding results are listed in Table II.Differences between the data and simulation result in systematic uncertainties.To check for an asymmetry in the tracking systems, we analyze e + e − → µ + µ − events.We measure the difference of the polar and azimuthal angle of the tracks between µ + and µ − , and then find shifts from the back-to-back direction of −0.67 mrad for the polar angle and −0.03 mrad for the azimuthal angle.By applying an artificial rotation to one of the charged tracks, we obtain residual values of the observables and find the results to be less than 10% of the statistical uncertainties.
There are small data-MC differences in the ρ and π 0 mass distributions.These can be caused by an imperfect momentum reconstruction resulting in a systematic offset of the observables.We check the effect of a momentum shift of the charged tracks by applying a momentum scaling factor of 1.0026, which is estimated from the peak position of the ρ mass distribution.We also check the effect of a π 0 momentum shift by applying the same factor assuming that the ρ mass difference is due to π 0 momentum shift.
In the π 0 invariant mass distribution, we observe a difference of 0.3% in the mass resolution between data and MC samples.This is due to data-MC difference in the reconstructed photon energy.We check the effect by changing the photon energies according to the difference found in a D * 0 → D 0 γ study.The change of the observables is obtained by conservatively varying the photon energy to E γ ± σ for both photons from π 0 , however the change is smaller than the other uncertainties.
The detector response depends on the particle charge, especially for electrons and pions.However, we know that the MC simulation does not exactly reproduce this difference.We compare the ratios of the yield N (A + B − )/N (A − B + ), where A and B denote the finalstate particles, e, µ, π, and ρ, between the data and simulation, and find the difference in ratios to be about 1%.We apply the observed shift of the charge asymmetry on the yield to the efficiency, and find a large systematic uncertainty on the offset of the imaginary part at the same level as the statistical uncertainty.(See the entries for Im(d τ ) for the eπ and µπ modes in Table II.)The changes in other parameters are negligible.
We have checked the polar angle dependence of the charge asymmetry.Although the data-MC consistency seems satisfactory, there are some differences.We also find a small difference between data and simulation in Figs. 1 and 2, where the momentum and polar angle distributions of the decay product are plotted.These differences are probably due to the reconstruction efficiency, which causes a systematic offset.This effect is checked by re-weighting the MC samples with the weight functions constructed bin-by-bin from the data-MC ratio for the momentum and cos θ distributions, and independently of the charge.This is the largest source of systematic uncertainty for Re(d τ ) in the high-sensitivity πρ and ρρ modes.
In this analysis, the purity is obtained from simulation.Any data-MC difference in purity could lead to a bias in the sensitivities and offsets.In order to take into account these possibilities, we include any difference of yields between the data and simulation as a systematic uncertainty on the background level.The resulting systematic uncertainties are about 10% for the sensitivities and about the same order of statistical uncertainties of the observables for the offsets.
In addition, we check the effect of ISR by introducing it into the calculation.We obtain the momenta of the ISR photons randomly from the KKMC generator, then, boost all momenta of the final-state particles into the τpair rest frame assuming that the ISR is coming from the e + e − beam.We calculate the observables in this frame.We iterate this process 100 times using the same hit-andmiss approach as in the nominal analysis.For successful trials, we obtain the mean of the observables.The shifts and fluctuations with the ISR effect give estimates of the systematic effects of ignoring it.
We calculate the final EDM values using the 833 fb −1 data sample, the results of which are listed in Table III for each mode.We obtain the mean values of the electric dipole moment weighted by a quadrature sum of statistical and systematic uncertainties, for the real and imaginary parts, Re(d τ ) = (−0.62± 0.63) × 10 −17 ecm, Im(d τ ) = (−0.40± 0.32) × 10 −17 ecm.
The 95% confidence intervals become Compared to the previous analysis [4], the obtained statistical uncertainties are reduced in proportion to the increase in the data size.The systematic uncertainties are improved because of the improved simulation, corrections and the larger statistics of the MC samples.The sensitivity for Re(d τ ) and Im(d τ ) has improved by about a factor of three.The systematic uncertainty from the detector modeling limits our result and needs to be controlled for future analysis.0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 Kinematic dependence of reconstruction efficiency 3.2 4.8 3.8 0.9 2.2 0.9 0.9 3.6 Data-MC diffedence in backgrounds 1.6 0.
FIG.1: Momentum distributions of (a) electrons, (b) muons, (c) pions, and (d) ρ mesons for the samples obtained after all event selections in each mode.The points with error bars are the data, the solid histograms are the MC expectation, and the gray shaded histograms are the contribution from misidentification for each particle species.

FIG. 2 :
FIG.2:The cos θ distributions of (a) electrons, (b) muons, (c) pions, and (d) ρ mesons for the samples obtained after all event selections in each mode.The points with error bars are the data, the solid histograms are the MC expectation, and the gray shaded histograms are the contribution from misidentification for each particle species.

FIG. 3 :
FIG.3: Distributions of optimal observable for each mode.The upper (a)-(h) plots are ORe and the lower (a')-(h') plots are OIm for each mode.The points with error bars are the data and the solid histograms are the MC expectation with zero EDM.The gray shaded histograms are the background contribution estimated from simulation.

FIG. 4 :FIG. 5 :
FIG.4: Relation of Re(dτ ) and ORe for the ρρ mode obtained by the MC simulation.The line shows the fitted function.Other modes also show a similar linear dependence; the non-linearity is negligible for all modes.

TABLE I :
Yield, purity, and dominant backgrounds for each selected mode.The values in square brackets indicate the expected background rates in %.

TABLE II :
Systematic uncertainties for Re(dτ ) and Im(dτ ) in units of 10 −17 e cm.

TABLE III :
Results on the τ electric dipole moment obtained using 833 fb −1 of data.The first uncertainties are statistical and the second ones are systematic.Mode Re(dτ )(10 −17ecm) Im(dτ )(10 −17 ecm) eµ −3.2 ± 2.5 ± 3.6 0.6 ± 0.4 ± 1.8 eπ 0.7 ± 2.3 ± 4.8 2.4 ± 0.5 ± 2.2 µπ 1.0 ± 2.2 ± 4.3 2.4 ± 0.5 ± 2.6 (EMSL) computing group for strong computing support; and the National Institute of Informatics, and Science Information NETwork 5 (SINET5) for valuable network support.We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS) including in particular the Grant-in-Aid for Scientific Research (A) 19H00682, and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Research Council including grants DP180102629, DP170102389, DP170102204, DP150103061, FT130100303; Austrian Federal Ministry of Education, Science and Research (FWF) and FWF Austrian Science Fund No. P 31361-N36; the National Natural Science Foundation of China under Contracts No. 11435013, No. 11475187, No. 11521505, No. 11575017, No. 11675166, No. 11705209; Key Research Program of Frontier Sciences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; the CAS Center for Excellence in Particle Physics (CCEPP); the Shanghai Pujiang Program under Grant No. 18PJ1401000; the Shanghai Science and Technology Committee (STCSM) under Grant No. 19ZR1403000; the Ministry of Education, Youth and Sports of the Czech Republic under Contract No. LTT17020; Horizon 2020 ERC Advanced Grant No. 884719 and ERC Starting Grant No. 947006 "InterLeptons" (European Union); the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster Universe, and the