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

TL;DR

This paper introduces a new experimental method to directly measure the magnetic and electric dipole moments of the tau lepton using polarized tau leptons produced at the LHC and channeled through bent crystals.

Contribution

It presents a novel approach combining fixed-target collisions, polarization selection, and crystal channeling to measure tau lepton dipole moments directly.

Findings

Method enables direct measurement of tau dipole moments.

Expected sensitivities discussed for the experimental setup.

Potential to improve understanding of tau lepton properties.

Abstract

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.

Equations

$$\sigma [pp\to D_s^{+}(\to {\tau }^{+}{\nu }_{\tau })X]N_A\frac{\rho T{A}_N}{A_T}\mathcal{B}({\tau }^{+}\to 3\pi {\overline{\nu }}_{\tau })$$

$$\approx 2.1\times 10^{-6},$$

$${\mathbf{s}}_0=\frac{1}{\omega }\left(m_{\tau }\mathbf{q}-q_0\mathbf{p}+\frac{\mathbf{q}\cdot \mathbf{p}}{p_0+m_{\tau }}\mathbf{p}\right),$$

$$s_{0,X}$$

$$s_{0,Y}$$

$$s_{0,Z}$$

$$s_X$$

$$s_Y$$

$$s_Z$$

$${\mathcal{W}}_i(\eta )$$

$$S_i^2=\frac{1}{N_{{\tau }^{+}}^{\mathrm{rec}}{\sigma }_i^2}=⟨{\left(\frac{{\mathcal{W}}_i^{+}(\eta )-{\mathcal{W}}_i^{-}(\eta )}{{\mathcal{W}}_i^{+}(\eta )+{\mathcal{W}}_i^{-}(\eta )}\right)}^2⟩,$$

$${\sigma }_a\approx \frac{1}{S_Y{s}_{0,Z}\gamma {\theta }_C}\frac{1}{\sqrt{N_{{\tau }^{+}}^{\mathrm{rec}}}},{\sigma }_d\approx \frac{2}{S_X{s}_{0,Z}\gamma {\theta }_C}\frac{1}{\sqrt{N_{{\tau }^{+}}^{\mathrm{rec}}}}.$$

$${\sigma }_a\approx \frac{1}{S_Z{s}_{0,Y}\gamma {\theta }_C}\frac{1}{\sqrt{N_{{\tau }^{+}}^{\mathrm{rec}}}},{\sigma }_d\approx \frac{2}{S_X{s}_{0,Y}(\gamma {\theta }_C{)}^2{a}^{\prime }}\frac{1}{\sqrt{N_{{\tau }^{+}}^{\mathrm{rec}}}},$$

$$s_X$$

$$s_Y$$

$$s_Z$$

$$s_X$$

$$s_Y$$

$$s_Z$$

Full text

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) Mohr et al. (2016); Andreev et al. (2018); Bennett et al. (2006, 2009); Pendlebury et al. (2015); Schneider et al. (2017); Sahoo (2017); Graner et al. (2016). For short-lived particles like heavy baryons and the $\tau$ lepton, the short lifetime ($\sim 10^{-13}\mathrm{{\,s}}$) prevents the use of the spin-precession technique adopted in the muon $g-2$ experiment Bennett et al. (2006, 2009). Recently, the possibility of directly measuring the electromagnetic dipole moments of short-lived baryons, produced in fixed-target collisions at the Large Hadron Collider (LHC) and channeled in bent crystals Baryshevsky (2016); Botella et al. (2017); Baryshevsky (2017); Fomin et al. (2017); Bagli et al. (2017); Baryshevsky (2018), has been considered. For the $\tau$ lepton, the use of ${{{B}^{+}}}\rightarrow{\tau^{+}}{{\nu}_{\tau}}$ decays was suggested Samuel et al. (1991) and more recently the ${{D}^{+}_{s}}\rightarrow{\tau^{+}}{{\nu}_{\tau}}$ process with higher yield has been explored Fomin et al. (2019). In this Letter, a novel method that fully exploits the polarization properties of $\tau^{+}$ leptons produced in ${D}^{+}_{s}$ decays is proposed. The magnetic (MDM) and the electric (EDM) dipole moments are defined as ${\bm{\mu}}=ge\hbar/(2m_{\tau}c){\mathbf{s}}/2$ and ${\bm{\delta}}=de\hbar/(2m_{\tau}c){\mathbf{s}}/2$, respectively, where $m_{\tau}$ is the $\tau$ mass, $g$ ($d$) is the gyromagnetic (gyroelectric) factor, and ${\mathbf{s}}$ is the spin-polarization vector Leader (2011). In the SM, the $\tau$ anomalous MDM is expected to be ${a}=(g-2)/2\approx 10^{-3}$ Eidelman and Passera (2007), and its EDM, $d$, to be minuscule Pospelov and Ritz (2014). However, the dipole moments can be largely enhanced in the presence of physics beyond the SM Pich (2014); Dekens et al. (2019). Methods based on precise measurements of the ${\tau^{+}}{\tau^{-}}$ pair production cross section in ${e^{+}e^{-}}$ annihilations set indirect limits on $a$ at the few percent level Abdallah et al. (2004), still above the SM prediction, and lead to limits on $\delta$ at $10^{-16}~{}e\mathrm{\,cm}$ level Inami et al. (2003). Other indirect measurements have been suggested to improve the precision Pich (2014); Hayreter and Valencia (2015); Chen and Wu (2018).

The proposed solution to provide direct measurements of the $\tau$ dipole moments, illustrated in Fig. 1, is based on the large production cross section of high-energy polarized $\tau^{+}$ leptons, originating in proton fixed-target collisions at the LHC. The ${\tau^{+}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\overline{\nu}}_{\tau}}$ ($3\pi{{\overline{\nu}}_{\tau}}$) decay is considered. A bent crystal is employed to exploit the channeling phenomenon of positively-charged particles aligned with the crystal atomic planes within a few ${\,\upmu\mathrm{rad}}$. Angular momentum conservation selects negative helicity $\tau^{+}$ leptons in the ${D}^{+}_{s}$ rest frame. The $\tau^{+}$ leptons emitted at relatively large $\theta_{{\sc y},{D}_{s}\tau}$ 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 $\tau^{+}$ leptons, induces a polarization of approximately $\beta^{\star}/\beta\approx 10\%$ anti-aligned with the crystal $Z$ axis, where $\beta$ ($\beta^{\star}$) is the velocity of the ${D}^{+}_{s}$ ($\tau^{+}$) 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.

The vast majority of $\tau^{+}$ leptons produced in proton fixed-target collisions at \sqrt{s}\approx 115$$\mathrm{\,Ge\kern-1.00006ptV} comes from ${{D}^{+}_{s}}\rightarrow{\tau^{+}}\nu_{\tau}$ decays. The corresponding production cross section \sigma[pp\rightarrow{{D}^{+}_{s}}(\rightarrow{\tau^{+}}{{\nu}_{\tau}})X]\approx 1.96$${\mathrm{\,\upmu b}} is estimated using the rescaled charm production cross section measured by the LHCb experiment in proton-helium collisions at $\sqrt{s}=86.6\mathrm{\,Ge\kern-1.00006ptV}$ Aaij et al. (2018), the $c$ quark to ${D}^{+}_{s}$ fragmentation fraction Lisovyi et al. (2016); Gladilin (2015), and the ${{D}^{+}_{s}}\rightarrow{\tau^{+}}\nu_{\tau}$ branching fraction Patrignani et al. (2016). The conversion factor for a 7 $\mathrm{\,Te\kern-1.00006ptV}$ proton on a $T=1\mathrm{\,cm}$ thick tungsten ($\rm W$) target to produce a ${\tau^{+}}\rightarrow 3\pi{{\overline{\nu}}_{\tau}}$ final state is estimated

[TABLE]

where $N_{A}$ is the Avogadro number, $\rho$ the target density, $A_{T}$ ($A_{N}$) its atomic mass (mass number), and ${\cal B}({\tau^{+}}\rightarrow 3\pi{{\overline{\nu}}_{\tau}})$ the $\tau^{+}$ branching fraction Patrignani et al. (2016).

In a reference frame defined by the crystal edges and comoving with the channeled particle, the $\tau^{+}$ initial polarization ${\mathbf{s}}_{0}$ is given by the unit vector along the ${D}^{+}_{s}$ momentum in the $\tau^{+}$ rest frame Halzen and Martin (1984); Berestetskii et al. (1982),

[TABLE]

where ${\mathbf{p}}$ (${\mathbf{q}}$) is the momentum of the $\tau^{+}$ (${D}^{+}_{s}$) and $p_{0}$ ($q_{0}$) its energy in the laboratory frame, $\omega=(m^{2}_{{D}_{s}}-m^{2}_{\tau})/2$, and $m_{{D}_{s}}$ is the ${D}^{+}_{s}$ mass. The projections of ${\mathbf{s}}_{0}$ along the crystal frame axes are:

[TABLE]

where $\theta_{{\sc x},{D}_{s}\tau}$ is the angle between the ${D}^{+}_{s}$ and the $\tau^{+}$ momenta in the $\sc x$ $\sc z$ plane. All angles are ${\cal O}(10^{-3})$ $\mathrm{\,rad}$ due to the highly boosted ${D}^{+}_{s}$ mesons and the small ${D}_{s}$-$\tau$ mass difference. Rotational invariance and the unconstrained $\theta_{{\sc x},{D}_{s}\tau}$ in the crystal $X$ $Z$ plane imply a zero $s_{0,{X}}$ average.

Very large samples of fixed-target ${{D}^{+}_{s}}\rightarrow{\tau^{+}}{{\nu}_{\tau}}$ events are produced using Pythia Sjostrand et al. (2006), EvtGen Lange (2001), and a fast simulation that generates phase-space kinematics. The $\tau^{+}$ channeling is simulated using the parameterization and procedures described in Refs. Biryukov et al. (1997); Bagli et al. (2017). A polarized sample can be obtained by selecting channeled $\tau^{+}$ and imposing kinematic requirements, as illustrated in Fig. 2 for the optimal experimental layout described later. For example, by requiring the $3\pi$ system momentum, ${p_{\mathrm{3}\pi}}$, to exceed 1$\mathrm{\,Te\kern-1.00006ptV}$ a $s_{0,{Z}}$ polarization of about $-20\%$ or higher is achieved. Instead, selecting regions of positive or negative $\theta_{{\sc y},{D}_{s}\tau}$ angles, in the following referred to as $\theta_{{\sc y}}$-tagging, a large $s_{0,{Y}}$ polarization can be obtained.

The interaction of the MDM (EDM) of a relativistic charged particle channeled in a bent crystal induces spin precession Botella et al. (2017); Bagli et al. (2017) in the bending plane (perpendicular to the bending plane). By measuring the spin-polarization components $s_{Y}$ and $s_{Z}$ ($s_{X}$ component), it is possible to extract the MDM (EDM) information. In particular, the appearance of an $s_{X}$ component represents the EDM signature. The spin-polarization projections after precession in the crystal read:

[TABLE]

where ${a^{\prime}}={a}+\frac{1}{1+\gamma}$, ${d^{\prime}}={d}/2$, ${a^{\prime}_{d}}=\sqrt{{a^{\prime}}^{2}+{d^{\prime}}^{2}}$, and $\Phi=\gamma\theta_{C}{a^{\prime}_{d}}$ is the precession angle, which is proportional to the $\tau^{+}$ Lorentz factor $\gamma$ and the crystal bending angle $\theta_{C}$. Equation (4) holds at ${\cal O}(10^{-2})$ precision, while expressions at ${\cal O}(10^{-5})$ are reported in the supplemental material sup .

A technique based on multivariate classifiers is explored to extract the $\tau^{+}$ polarization vector without prior knowledge of the detailed decay dynamics and of the $\tau^{+}$ energy. A classifier discriminating between $\tau^{+}$ with full positive ($+1$) and negative ($-1$) polarization along each crystal frame axis is built. The classifiers are trained on simulated events and are based upon variables describing the decay distribution. The used variables that provide sensitivity to the $\tau^{+}$ spin polarization, referred to with the symbol ${\bm{\zeta}}$, are: the angles between the ${3\pi}$ momentum in the $\tau^{+}$ rest frame and the crystal frame axes, the angles describing the $3\pi$ decay plane orientation in the $3\pi$ rest frame with respect to the crystal frame axes, and two- and three-pion invariant masses. The $\tau^{+}$ momentum is estimated by applying kinematic corrections, determined from simulated events, to the measured $p_{\mathrm{3}\pi}$ vector as a function of its magnitude and direction. In absence of the $\tau^{+}$ production vertex, the flight direction is assumed to be that connecting the ${D}^{+}_{s}$ production vertex and the $\tau^{+}$ 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${\,\upmu\mathrm{m}}$ (70${\,\upmu\mathrm{m}}$) for the production vertex in the longitudinal (transverse) direction with respect to the beam, and 100${\,\upmu\mathrm{m}}$ (1$\mathrm{\,mm}$) for the decay vertex.

The polarization component $s_{i}$ along the $i$-th crystal frame axis $(i={X},{Y},{Z}$) is extracted by fitting the classifier distribution ${\cal W}_{i}(\eta)$ on data,

[TABLE]

where $\eta\equiv\eta({\bm{\zeta}})$ is the classifier response, and ${\cal W}_{i}^{\pm}(\eta)$ the templates representing the response for $\pm 1$ polarizations.

The statistical separation between templates also represents the squared average event information Kendall et al. (1983) of the polarization (at $s_{i}=0$) Davier et al. (1993),

[TABLE]

where $\sigma_{i}$ is the uncertainty on $s_{i}$, and $N_{\tau^{+}}^{\rm rec}$ is the number of channeled and reconstructed $\tau^{+}$. The template fit results for $s_{{Y}}$ polarization are shown in Fig. 3, while those for $s_{{X}}$ and $s_{{Z}}$ are shown in the supplemental material sup . The estimated average event information is $S_{X}\approx S_{Y}\approx 0.42$ and $S_{Z}\approx 0.23$, using either Multilayer Perceptron Networks or Boosted Decision Trees Voss et al. (2007), to be compared to the ideal value of $0.58$ reached in case the complete kinematics of the $\tau^{+}$ decay is reconstructed Davier et al. (1993). The difficulty in determining the $\tau^{+}$ momentum, due to the undetected ${\overline{\nu}}_{\tau}$, affects mainly the determination of the $s_{{Z}}$ polarization.

For small $\Phi$ (as $\gamma\theta_{C}\sim 10$ and ${a^{\prime}_{d}}\sim 10^{-3}$) and $s_{0,{Z}}$ initial polarization, the statistical uncertainties on $a$ and $d$ are estimated from Eq. (4) as

[TABLE]

For $s_{0,{Y}}$ initial polarization,

[TABLE]

which show comparable sensitivity to $a$ but disfavored by a factor $1/(\gamma\theta_{C}{a^{\prime}})\sim 100$ to $d$ with respect to Eq. (7) for initial $s_{0,{Z}}$ polarization.

The optimization of the experimental layout is performed using simulated events for the case of initial $s_{0,{Z}}$ polarization. The region of minimal uncertainty on $a$ and $d$ is determined using a scan in the $({\theta_{C}},{L},{\theta_{{\sc y}}},{L_{\mathrm{tar}}})$ parameter space, where $\theta_{C}$ ($L$) is the crystal bending angle (length) and $L_{\mathrm{tar}}$ the distance between the target and crystal. Channeled $\tau^{+}$ are required to have ${p_{\mathrm{3}\pi}}>800{\mathrm{\,Ge\kern-1.00006ptV\!/}c}$ to enhance $s_{0,{Z}}$ polarization, and to originate before the crystal and to decay after the crystal to insure maximum $\Phi$ precession angle. For a $\rm Ge$ ($\rm Si$) crystal tilted by ${\theta_{{\sc y}}}=0.1\mathrm{\,mrad}$, the optimal parameters ${\theta_{C}}\approx 16\mathrm{\,mrad}$, ${L}\approx 8~{}(11)\mathrm{\,cm}$, and ${L_{\mathrm{tar}}}\approx 12\mathrm{\,cm}$ are obtained (see supplemental material sup ). The $\rm Ge$ crystal provides relatively high channeling efficiency, $\approx 6.3\times 10^{-6}$, a factor of three higher than for $\rm Si$. Recently, crystal prototypes with similar length and bending angle have been tested on beam at the CERN SPS Mazzolari (2018). The selected $\tau^{+}$ sample has $s_{0,{Z}}\approx-18\%$, $s_{0,{Y}}\approx 0\%$ polarization, and average Lorentz factor $\gamma\approx 800$. A $s_{0,{Y}}\approx\mp 40\%$ polarization can be achieved with a $\theta_{{\sc y}}$-tagging that discriminates between positive and negative $\theta_{{\sc y},{D}_{s}\tau}$ angles. Information statistically correlated with $\theta_{{\sc y},{D}_{s}\tau}$ is required for $\theta_{{\sc y}}$-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 for additional instrumentation, e.g. several layers of pixel radiation-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. Fomin et al. (2019), inducing $s_{0,{Y}}\approx\mp 60\%$ for tagged events with an efficiency of a few percent.

Dipole moment 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. (5). Figure 4 illustrates the estimated sensitivities as a function of the number of impinging protons for a $\rm Ge$ crystal with optimal parameters (thick solid red line). Sensitivities for other configurations with maximum average event information $S_{i}=0.58$ (thick dotted red line), $\theta_{{\sc y}}$-tagging based on a discrimination between positive and negative $\theta_{{\sc y},{D}_{s}\tau}$ with ideal tagging efficiency of 100% (thick dashed and hatched blue lines), and the double crystal (DC) option proposed in Ref. Fomin et al. (2019) (thin solid and dotted black lines), are also shown for comparison. A detector reconstruction efficiency of 40% is assumed. The corresponding sensitivities for $\rm Si$ are about a factor two worse.

The channeling process keeps the high momentum unchanged while deflecting the $\tau^{+}$ at the bending angle $\theta_{C}\approx 16\mathrm{\,mrad}$. This signature can be identified in the $3\pi{{\overline{\nu}}_{\tau}}$ decays through the reconstruction of the $3\pi$ vertex and momentum. For highly-boosted particles with $\gamma\approx 800$ the latter defines the $\tau^{+}$ direction with an uncertainty of $\approx 0.5\mathrm{\,mrad}$, mainly due to the missing ${\overline{\nu}}_{\tau}$. The contribution of non-channeled leptons is reduced to a negligible level $<0.3\%$ using the following selection criteria: ${p_{\mathrm{3}\pi}}\geq 800{\mathrm{\,Ge\kern-1.00006ptV\!/}c}$, $3\pi$ momentum direction consistent with $\theta_{C}$ within 1.5$\mathrm{\,mrad}$, and the $3\pi$ vertex located after the crystal, at a distance {L}+{L_{\mathrm{tar}}}{~{}\raise 1.49994pt\hbox{>}\kern-8.50006pt\lower 3.50006pt\hbox{\sim}~{}}20\mathrm{\,cm} from the interaction point. With these requirements, 28% of the $\tau^{+}$ candidates are channeled through a fraction of the crystal length. These are mainly events in which the ${D}^{+}_{s}$ decays inside the crystal or the ${\tau^{+}}$ does not reach the end of the crystal, either because it decays or is dechanneled. Nevertheless, only $\tau^{+}$ particles that travel almost through the entire crystal are selected. They experience a very similar electromagnetic field, inducing a relatively small bias on the spin precession angle $\Phi$ of $1.4\%$ that can be corrected. Background contributions from channeled hadron decays with $3\pi$ in the final state, e.g. ${D}^{+}$, ${D}^{+}_{s}$ mesons, ${\Lambda}^{+}_{c}$ baryons, can be vetoed using the reconstructed invariant mass and event information from a dedicated detecting apparatus. Systematic effects could arise from the limited knowledge of the crystal position and orientation, the initial polarization, and the $\tau^{+}$ momentum. Those uncertainties can be controlled using up- and down-bending crystals, inducing opposite spin precession Bagli et al. (2017), by reconstructing unchanneled $\tau^{+}$ decays with kinematic properties similar to the signal, and by using detailed simulations of the experimental setup calibrated with data. Possible effects due to $\tau^{+}$ weak interactions with the crystal are estimated to be negligible Baryshevsky (2018) 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 $\tau$ 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 $\tau$ MDM could be verified experimentally with a sample of around 10^{17}$$\mathrm{\,PoT}, whereas at the same time a search for the $\tau$ EDM at the level of $10^{-17}~{}e\mathrm{\,cm}$ or below could be performed. This would require about 10% of the protons storaged during a decade of LHC operation Apollinari et al. (2017). In preparation of a possible future experiment this method could be tested using the fixed-target setup proposed for the study of heavy baryons Botella et al. (2017); Fomin et al. (2017); Bagli et al. (2017) with the LHCb apparatus. The possibility of a test or an experiment at the CERN SPS will also be explored.

We express our gratitude for stimulating discussions to V. Baryshevsky, F.J. Botella, G. Cavoto, A. S. Fomin, A. Mazzolari, A. Pich and J. Walsh. We acknowledge support from INFN (Italy), MINECO and GVA (Spain), the Severo Ochoa excellence certification SEV-2014-0398-01, and the ERC Consolidator Grant SELDOM G.A. 771642.