Novel method for the direct measurement of the tau lepton dipole moments

A novel method for the direct measurement of the elusive magnetic and electric dipole moments of the tau lepton is presented. The experimental approach relies on the production of tau+ leptons from Ds+ ->tau+ nu_tau decays, originated in fixed-target collisions at the LHC. A sample of polarized tau+ leptons is kinematically selected and subsequently channeled in a bent crystal. The magnetic and electric dipole moments of the tau+ lepton are measured by determining the rotation of the spin-polarization vector induced by the intense electromagnetic field between crystal atomic planes. The experimental technique is discussed along with the expected sensitivities.

A novel method for the direct measurement of the elusive magnetic and electric dipole moments of the τ lepton is presented. The experimental approach relies on the production of τ + leptons from D + s → τ + ντ decays, originated in fixed-target collisions at the LHC. A sample of polarized τ + leptons is kinematically selected and subsequently channeled in a bent crystal. The magnetic and electric dipole moments of the τ + lepton are measured by determining the rotation of the spinpolarization vector induced by the intense electromagnetic field between crystal atomic planes. The experimental technique is discussed along with the expected sensitivities. The measurements of the electromagnetic dipole moments for common particles like the electron, muon and nucleons, combined with precise theoretical calculations, provide stringent tests of physics within and beyond the Standard Model (SM) [1][2][3][4][5][6][7][8]. For short-lived particles like heavy baryons and the τ lepton, the short lifetime (∼ 10 −13 s) prevents the use of the spin-precession technique adopted in the muon g − 2 experiment [3,4]. Recently, the possibility of measuring directly the electromagnetic dipole moments of short-lived baryons, produced in fixed-target collisions at the Large Hadron Collider (LHC) and channeled in bent crystals [9][10][11][12][13][14], has been considered. For the τ lepton, the use of B + → τ + ν τ decays was suggested [15] and more recently the D + s → τ + ν τ process with higher yield has been explored [16]. In this Letter, a novel method that fully exploits the polarization properties of τ + leptons produced in D + s decays is proposed. The magnetic (MDM) and the electric (EDM) dipole moments are defined as µ = ge /(2m τ c)s/2 and δ = de /(2m τ c)s/2, respectively, where m τ is the τ mass, g (d) is the gyromagnetic (gyroelectric) factor, and s is the spin-polarization vector [17]. In the SM, the τ anomalous MDM is expected to be a = (g − 2)/2 ≈ 10 −3 [18], and its EDM, d, to be minuscule [19]. However, the dipole moments can be largely enhanced in presence of physics beyond the SM [20,21]. Methods based on precise measurements of τ + τ − pair production cross section in e + e − annihilations set indirect limits on a at few percent level [22], still above the SM prediction, and lead limits on δ at 10 −16 e cm level [23]. Other indirect measurements have been suggested to improve the precision [20,24,25].
The proposed solution to provide direct measurements of the τ dipole moments, illustrated in Fig. 1, is based on the large production cross section of high-energy polarized τ + leptons, originated in proton fixed-target col- 1: Sketch of the fixed-target setup along with the τ + production and decay processes (not to scale). The crystal frame (X,Y ,Z) is tilted in the laboratory frame (x, y, z) by θy to avoid channeling of non-interacting protons.
lisions at the LHC. The τ + → π + π − π + ν τ (3πν τ ) decay is considered. A bent crystal is employed to exploit the channeling phenomenon of positively-charged particles aligned with the crystal atomic planes within few µrad. Angular momentum conservation selects negative helicity τ + leptons in the D + s rest frame. The τ + leptons emitted at relatively large θ y,Dsτ angles with respect to the D + s flight direction in the yz plane show enhanced polarization along the Y axis, perpendicular to the crystal plane. The Lorentz boost, making larger acceptance for forward-than for backward-emitted τ + , induces a polarization of approximately −β/β ≈ −10% along the crystal Z axis, where β (β ) is the velocity of the D + s (τ + ) in the laboratory (D + s rest) frame. Thus, the selection of the highest momentum candidates enhances the Z polarization. The MDM (EDM) signature is given by the spin rotation in the Y Z bending plane (appearance of a spin component along the X axis) induced by the interaction with the crystal electric field. A novel analysis technique based on multivariate classifiers is employed to determine the rotation of the spin-polarization vector.
In a reference frame defined by the crystal edges and comoving with the channeled particle, the τ + initial polarization s 0 is given by the unit vector along the D + s momentum in the τ + rest frame [30,31], where p (q) is the momentum of the τ + (D + s ) and p 0 (q 0 ) its energy in the laboratory frame, ω = (m 2 Ds − m 2 τ )/2, and m Ds is the D + s mass. The projections of s 0 along the crystal frame axes are: where θ x,Dsτ is the angle between the D + s and the τ + in the xz plane. All angles are O(10 −3 ) rad due to the highly boosted D + s mesons and the small D s -τ mass difference. Rotational invariance and the unconstrained θ x,Dsτ in the crystal XZ plane imply a zero s 0,X average. Very large samples of fixed-target D + s → τ + ν τ events have been produced using Pythia [32], EvtGen [33], and a fast simulation that generates phase-space kinematics. The τ + channeling has been simulated using the parameterization and procedures described in Refs. [13,34]. A polarized sample is obtained by selecting channeled τ + and imposing kinematic requirements, as illustrated in Fig. 2 for the optimal Ge crystal layout described later. For example, by requiring the 3π system momentum to exceed 1 TeV a s 0,Z polarization of about −20% or higher is achieved. Instead, selecting regions of positive or negative θ y,Dsτ angles, in the following referred to as θ y tagging, a large s 0,Y polarization can be obtained. The spin-polarization precession induced by the interaction of the MDM and EDM of a relativistic charged particle channeled in a bent crystal is derived elsewhere [10,13]. The spin-polarization projections after precession in the crystal read: where a = a + 1 For small Φ (as γθ C ∼ 10 and a d ∼ 10 −3 ) and s 0,Z initial polarization, the statistical uncertainties on a and d are estimated from Eq. (4) as where N rec τ + is the number of channeled and reconstructed τ + , and S i is the average event information [36] along a given i (= X, Y, Z) crystal axis, as discussed later. For s 0,Y initial polarization, which show comparable sensitivity to a but disfavoured by a factor 1/(γθ C a ) ∼ 100 to d with respect to Eq. (5) for initial s 0,Z polarization.
An optimization of the layout has been performed for initial s 0,Z polarization to determine the region of minimal uncertainty on a and d using a scan in the (θ C , L, θ y , L tar , p 3π ) parameter space, where θ C (L) is the crystal bending angle (length), L tar the distance between the target and crystal, and p 3π the momentum of the 3π system. The following kinematic requirements have been imposed: p 3π > 800 GeV/c, the D + s decay position before the crystal, while for the channeled τ + after the crystal. For a Ge (Si) crystal tilted by θ y = 0.1 mrad, the optimal parameters θ C ≈ 16 mrad, L ≈ 8 (11) cm, and L tar ≈ 12 cm are obtained (see supplemental material [35]). Recently, crystal prototypes with similar parameters have been tested on beam at the CERN SPS [37]. The Ge and Si crystals show identical s 0,Z ≈ −18%, s 0,Y ≈ 0% polarizations and average Lorentz factor γ ≈ 800, while the Ge channeling efficiency, ≈ 6.3 × 10 −6 , is a factor three higher than for Si. A s 0,Y ≈ ∓40% polarization can be achieved with a θ y -tagging that discriminates between positive and negative θ y,Dsτ angles. Information statistically correlated with θ y,Dsτ is required for θ y -tagging. A possible strategy could be the exploitation of the global event topology, e.g. kinematic distributions of particles associated with the interaction point where the D + s is produced. The relatively large separation between the target and the crystal would allow to allocate additional instrumentation, e.g. several layers of pixel rad-hard diamond sensors could be used to reconstruct the D + s trajectory. Another possibility would be to place a second bent crystal to channel the D + s using a layout similar to that suggested in Ref. [16], inducing s 0,Y ≈ ∓60% for tagged events with an efficiency of a few percent. The initial polarization can be accurately determined by reconstructing unchanneled τ + decays with kinematic properties similar to the signal and by using detailed simulations of the experimental setup calibrated with data.
A technique based on multivariate classifiers is explored to extract the τ + polarization vector without apriori knowledge of the detailed decay dynamics and the (unknown) τ + energy. A classifier discriminating between τ + with full positive (+1) and negative (−1) polarization along each crystal frame axis is built. The classifiers are trained on simulated events passing the kinematic requirements of the optimal layout and are based upon variables describing the decay distribution. The used variables, referred to with the symbol ζ, are: twoand three-pion invariant masses, the angles between the 3π momentum in the τ + rest frame and the crystal frame axes, and the angles describing the 3π decay plane orientation in the 3π rest frame with respect to the crystal frame axes. The τ + average momentum is estimated by applying corrections, determined from simulated events, to the measured average p 3π vector as a function of its magnitude and direction. In absence of the τ + production vertex, the flight direction is assumed to be that connecting the D + s production vertex and the τ + decay vertex, lying in the crystal channeling plane. The vertex positions are smeared according to Gaussian distributions to mimic experimental resolutions, assumed to be 13 µm (70 µm) for the production vertex in the longitudinal (transverse) direction with respect to the beam, and 100 µm (1 mm) for the decay vertex. The polarization component s i for the i-th crystal frame axis is extracted by fitting the classifier distribution W i (η) on data, where η ≡ η(ζ) is the classifier response, and W ± i (η) the templates representing the response for ±1 polarizations, The statistical separation between templates also represents the squared average event information of the polarization (at s i = 0) [38], where σ i is the uncertainty on s i . The template fit results for s Y polarization are shown in Fig. 3, while for s X and s Z are shown in the supplemental material [35]. The estimated average event information is S X ≈ S Y ≈ 0.42 and S Z ≈ 0.23, using either Multilayer Perceptron Networks or Boosted Decision Trees [39], to be compared to the ideal value of 0.58 reached in case the complete kinematics of the τ + decay is reconstructed [38]. The difficulty in determining the τ + momentum, due to the invisible ν τ , affects mainly the determination of the s Z polarization.
The a and δ sensitivities are assessed from a large number of pseudoexperiments generated and fit using a probability density function based on the spin precession equation of motion reported in Eq. (4), and the classifier distributions in Eq. (7). Figure 4 illustrates the estimated sensitivities as a function of the number of impinging protons for a Ge crystal with optimal parameters. Sensitivities for other configurations with maximum average event information, θ y -tagging based on a discrimination between positive and negative θ y,Dsτ and ideal efficiency of 100%, and the double crystal (DC) option proposed in Ref. [16], are also shown for comparison. A detector reconstruction efficiency of 40% is assumed. The corresponding sensitivities for Si are about a factor two worse. The channeling process keeps the high momentum unchanged while deflecting the τ + at the bending angle θ C ≈ 16 mrad. This signature can be identified in the 3πν τ decays through the reconstruction of the 3π vertex and momentum. For highly-boosted particles with γ ≈ 800 the latter defines the τ + direction with an uncertainty of ≈ 0.5 mrad, mainly due to the missing ν τ . The contribution of non-channeled particles is reduced to a negligible level < 0.3% using the following selection criteria: p 3π ≥ 800 GeV/c, 3π momentum direction consistent with θ C within 1.5 mrad, and the 3π vertex located after the crystal, at a distance L + L tar > ∼ 20 cm from the interaction point. A potential source of background comes from τ + that are channeled through a fraction of the crystal length. These are mainly events in which the D + s decays inside the crystal or the τ + does not reach the end of the crystal, either because it decays or is dechanneled. They compose 28% of the τ + candidates. Nevertheless, only τ + particles that travel almost through the entire crystal are selected with the applied cuts and experience very similar electromagnetic field, inducing a relatively small bias on the spin preces-sion angle Φ of 1.4% that can be corrected. Background contributions from channeled hadron decays with 3π in the final state, e.g. D + , D + s mesons, Λ + c baryons, can be vetoed using the reconstructed invariant mass and event information from a dedicated detecting apparatus. Systematic effects due to the experimental technique can be controlled using up-and down-bending crystals, inducing opposite spin precession [13]. Possible effects due to τ + weak interactions with the crystal are estimated to be negligible [14] compared to the sensitivity and can be removed by using different crystal bending orientations.
In summary, a novel method for the direct measurement of the τ MDM and EDM has been presented with interesting perspective for a stringent test of the SM and search of new physics. The fixed-target setup and the analysis technique have been discussed along with sensitivity projections for possible future scenarios. The SM prediction for the τ MDM could be verified experimentally with a statistics around 10 18 PoT, whereas at the same time a search for the τ EDM at the level of 10 −17 e cm or below could be performed. In preparation of a possible future experiment at the LHC, this method could be tested using the fixed-target setup proposed for the study of heavy baryons [10,12,13] with the LHCb apparatus. The possibility of a test or an experiment at the CERN SPS will also be explored.