- Email: [email protected]
Precision Measurement of the Mass of the τ Lepton at BESIII
Available online at www.sciencedirect.com
Nuclear and Particle Physics Proceedings 260 (2015) 23–26 www.elsevier.com/locate/nppp
Precision Measurement of the Mass of the τ Lepton at BESIII Tao Luo on behalf of BESIII Collaboration University of Hawaii, Honolulu, Hawaii 96822, USA
Abstract An energy scan near the τ pair production threshold has been performed using the BESIII detector. About 24 pb−1 of data, distributed over four scan points, was collected. This analysis is based on τ pair decays to ee, eμ, eh, μμ, μh, hh, eρ, μρ and πρ final states, where h denotes a charged π or K. The mass of the τ lepton is measured from a 2 maximum likelihood fit to the τ pair production cross section data to be mτ = (1776.91 ± 0.12 +0.10 −0.13 ) MeV/c , which is currently the most precise value in a single measurement. Keywords: τ mass, precision measurement, beam energy 1. Introduction The τ lepton mass, mτ , is one of the fundamental parameters of the Standard Model (SM). The relationship between the τ lifetime (ττ ), mass, its electronic branching fraction (B(τ → eν¯ν)) and weak coupling constant gτ is predicted by theory: g2 m5 B(τ → eν¯ν) = τ τ3 , ττ 192π
(1)
up to small radiative and electroweak corrections [1]. It appeared to be badly violated before the first precise mτ measurement of BES became available in 1992 [2]; this measurement was later updated with more τ decay channels [3] and confirmed by subsequent measurements from BELLE [4], KEDR [5], and BABAR [6]. The experimental determination of ττ , B(τ → eν¯ν) and mτ to the highest possible precision is essential for a high precision test of the SM. Currently, the mass precision for e and μ has reached Δm/m of 10−8 , while for τ it is 10−4 [7]. A precision mτ measurement is also required to check lepton universality. Lepton universality, a basic ingredient in the minimal standard model, requires that ∗ Email
address: [email protected] (Tao Luo)
http://dx.doi.org/10.1016/j.nuclphysbps.2015.02.006 2405-6014/© 2015 Elsevier B.V. All rights reserved.
the charged-current gauge coupling strengths ge , gμ , gτ should be identical: ge = gμ = gτ . Comparing the electronic branching fractions of τ and μ, lepton universality can be tested as: 2 τμ mμ 5 B(τ → eν¯ν) gτ = (1 + FW )(1 + Fγ ), (2) gμ ττ mτ B(μ → eν¯ν) where FW and Fγ are the weak and electromagnetic radiative corrections [1]. Note (gτ /gμ )2 depends on mτ to the fifth power. The testing of the Koide formula, which is an unexplained empirical equation discovered by Yoshio Koide in 1981 [8], also depends on the accuracy of mτ . The Koide formula is: me + mμ + mτ 2 Q= √ √ 2 = . √ 3 ( me + mμ + mτ )
(3)
It relates the masses of the three charged leptons so well that it predicted mτ . Furthermore, the precision of mτ will also restrict the ultimate sensitivity of mντ . The most sensitive bounds on the mass of the ντ can be derived from the analysis of the invariant-mass spectrum of semi-hadronic τ decays, e.g. the present best limit of mντ < 18.2 MeV/c2 (95% confidence level) was based on the kinematics of 2939 (52) events of τ− → 2π− π+ ντ (τ− → 3π− 2π+ (π− )ντ ) [9].
24
T. Luo / Nuclear and Particle Physics Proceedings 260 (2015) 23–26
This method depends on a determination of the kinematic end point of the mass spectrum; thus high precision on mτ is needed. So far, the pseudomass technique and the threshold scan method have been used to determine mτ . The former, which was used by ARGUS [10], OPAL [11], BELLE [4] and BABAR [6], relies on the reconstruction of the invariant-mass and energy of the hadronic system in the hadronic τ decay, while the latter, which was used in DELCO [12], BES [2, 3] and KEDR [5], is a study of the threshold behavior of the τ pair production cross section in e+ e− collisions and it is the method used in this paper. Extremely important in this approach is to determine the beam energy and the beam energy spread precisely. Here the beam energy measurement system (BEMS) [13] for BEPCII is used and will be described below. Before the experiment began, a study was carried out using Monte Carlo (MC) simulation and sampling to optimize the number and choice of scan points in order to provide the highest precision on mτ for a specified period of data taking time or equivalently for a given integrated luminosity [14]. The τ scan experiment was done in December 2011. The J/ψ and ψ resonances were each scanned at seven energy points, and data were collected at four scan points near τ pair production threshold with center of mass (CM) energies of 3542.4 MeV, 3553.8 MeV, 3561.1 MeV and 3600.2 MeV. The first τ scan point is below the mass of τ pair [7], while the other three are above. 2. Beam Energy Measurement System The BEMS is located at the north crossing point of the BEPCII storage ring. Here is a short introduction to the working principles of the BEMS. In the Compton scattering process, the maximal energy of the scattered photon Eγ is related to the electron energy Ee by the kinematics of Compton scattering [15, 16]: ⎡ ⎤ Eγ ⎢⎢⎢⎢ m2e ⎥⎥⎥⎥ ⎢⎢1 + 1 + ⎥⎥ , Ee = (4) 2 ⎢⎣ Eγ Eγ ⎥⎦ where Eγ is the energy of the initial photon, i.e. the energy of the laser beam in the BEMS. The scattered photon energy can be measured with high accuracy by the HPGe detector, whose energy scale is calibrated with photons from radioactive sources and the readout linearity is checked with a precision pulser. The maximum
energy can be determined from the fitting to the edge of the scattered photon energy spectrum. At the same time the energy spread of back-scattered photons due to the energy distribution of the collider beam is obtained from the fitting procedure [13]. Finally, the electron energy can be calculated by Eq. 4. Since the energy of the laser beam and the electron mass (me ) are determined with the accuracy at the level of 10−8 , Ee can be determined, utilizing Eq. 4, as accurately as Eγ . Another words, Ee is measured to an accuracy of 10−5 . The systematic error of the electron and positron beam energy determination in our experiment was tested through previous measurement of the ψ mass and was estimated to be 2 × 10−5 [13]; the relative uncertainty of the beam energy spread was about 6% [13]. More details can be found in the reference [13]. 3. Data Analysis 3.1. Luminosity at Each Scan Point For all scan points, the luminosity L is determined from L = Ndata /γγ σγγ , where Ndata is the number of selected two-gamma events in data and γγ and σγγ are the efficiency and the cross section determined by the Babayaga 3.5 MC, respectively. The measured integrated luminosities for the J/ψ, τ and ψ scan are calculated as 1505 nb−1 , 23261 nb−1 and 7526 nb−1 , respectively. The analysis using Bhabha events is done as a cross check of the two-gamma luminosity and gives consistent luminosity results within 2%. The Bhabha luminosities will also be used in the systematic error analyses. 3.2. J/ψ and ψ Hadronic Cross-section Line Shapes The number of hadronic events N h is fitted to the number of expected hadronic events N exp = σhad · L,
(5)
where σhad is the cross section of e+ e− → hadrons, which depends on ECM , and the energy spread, δw . The background cross section (σbg ) reconstruction efficiency (had ) M, and δw , are free parameters, obtained from minimizing χ2 =
N i=1
(Nih − σihad Li )2 Nih (1 + Nih (ΔLi /Li )2 )
,
(6)
where ΔLi /Li , Nih , σihad and Li are the relative luminosity error, the number of hadron events, the cross section of e+ e− → hadrons, and the luminosity at scan point i,
25
Cross Section (nb)
T. Luo / Nuclear and Particle Physics Proceedings 260 (2015) 23–26 2.0
is constructed from Poisson distributions, one at each of the four scan points, and takes the form [3]
1.5 1.0
L(mτ , RData/MC , σB ) =
0.5
4 μNi e−μi i
i=1 0.0 3540 3550 3560 3570 3580 3590 3600 3610
W (MeV)
Figure 1: The CM energy dependence of the τ pair cross section resulting from the likelihood fit (curve), compared to the data (Poisson errors).
Ni !
,
where Ni is the number of observed τ pair events at scan point i; μi is the expected number of events and calculated by i , mτ ) + σB ] × Li . μi = [RData/MC × i × σ(ECM
respectively. The mass difference with respect to nominal resonance mass, ΔM = M f it − M PDG , and the energy spread are obtained from the J/ψ and ψ fits. The values for δw agree well with those obtained from the BEMS. We extrapolate the results from the fits of the J/ψ and ψ line shapes to the τ-mass region in order to obtain the energy correction to the τ-mass. The systematic error associated with the energy scale is estimated by extrapolating under two assumptions: the first one is that the correction has a linear dependence on the energy and the second one assumes a constant shift. The linear fit between ΔM from the J/ψ and ψ fits gives the correction to the τ-mass of Δmτ = (0.054 ± 0.030) MeV/c2 ; the constant shift gives the correction of Δmτ = (0.043 ± 0.020) MeV/c2 , where both errors are statistical. The difference between these two methods, 0.011 MeV/c2 , is taken as the systematic uncertainty related to the energy determination. Overall, the systematic error in the energy determination is taken to be 0.012 MeV. The difference
mτ
will be taken into consideration in the measured τ-mass value.
RData/MC σB
In order to reduce the statistical error in the τ lepton mass, this analysis incorporates 13 two-prong τ pair final states, which are ee, eμ, eπ, eK, μμ, μπ, μK, πK, ππ, KK, eρ, μρ and πρ, with accompanying neutrinos implied. 3.3.1. Maximum Likelihood Fit to The Data The mass of the τ lepton is obtained from a maximum likelihood fit to the CM energy dependence of the τ pair production cross section. The likelihood function
(9)
In Eq. (9), mτ is the mass of the τ lepton, and RData/MC is the ratio of the overall efficiency for identifying τ pair events for data and for MC simulation, allowing for the difference of the efficiencies between the data and the corresponding MC sample. i is the efficiency at scan
point i, which is given by i = i Br j i j , where Br j is the branching fraction for the j − th final state and i j is the detection efficiency for the j − th final state at the i − th scan point. The efficiencies i , determined directly from the τ pair inclusive MC sample by applying the same τ pair selection criteria, are 0.065, 0.065, 0.069, 0.073 at the four scan points, respectively. σB is an effective background cross section, and it is assumed coni , covered stant over the limited range of CM energy, ECM by the scan. Li is the integrated luminosity at scan point i , mτ ) is the corresponding cross section for i, and σ(ECM τ pair production [3] The ML fit is performed on the selected τ pair candidate events. The fit yields
Δmτ = 0.054 ± 0.030(stat) ± 0.012(sys) MeV/c2 , (7)
3.3. τ Mass Measurement
(8)
= 1776.91 ± 0.12 MeV/c2 , = 1.05 ± 0.04, = 0+0.12 pb.
(10)
The fitted σB is zero, which indicates the selected τ pair candidate data set is very pure. The quality of the fit is shown explicitly in Fig. 1. The curve corresponds to the theoretical cross section with mτ = 1776.91 MeV/c2 ; the measured cross section at scan point i is given by σi =
Ni RData/MC i Li
.
(11)
The measured cross sections at different scan points are consistent with the theoretical values.
26
T. Luo / Nuclear and Particle Physics Proceedings 260 (2015) 23–26
ARGUS BES (96’) CLEO OPAL BELLE KEDR BABAR PDG12 This work
1766
1768
reference [1], the ratio of squared coupling constants is determined to be: 2 gτ = 1.0016 ± 0.0042, (15) gμ
1776.30+2.80 -2.80 1776.96+0.31 -0.28 1778.20+1.50 -1.50 1775.10+1.90 -1.90 1776.61+0.38 -0.38 1776.81+0.30 -0.28 1776.68+0.43 -0.43 1776.82+0.16 -0.16 1776.91+0.16 -0.18
1770 1772
1774 1776 2
τ mass (MeV/c )
1778
1780
Figure 2: Comparison of measured τ mass from this paper with those from the PDG. The vertical band corresponds to the 1 σ limit of the measurement of this paper
so that this test of lepton universality is satisfied at the 0.4 standard deviation level. The level of precision is compatible with previous determinations, which used the PDG average for mτ [18]. More details about this analysis can be found in the reference [17]. References
3.3.2. Systematic Error Estimation The systematic errors from the sources of theoretical accuracy, energy scale, energy spread, luminosity, event selection cuts, event selection efficiencies, background shape and fitted efficiency parameter are considered [17]. We assume that all systematical uncertainties are independent and add them in quadrature to obtain the total systematical uncertainty for τ mass measure2 ment, which is +0.10 −0.13 MeV/c . 4. Summary Using a maximum likelihood fit to the τ pair cross section data near threshold, the mass of the τ lepton has been measured as 2 mτ = (1776.91 ± 0.12+0.10 −0.13 ) MeV/c .
(12)
This value is in good agreement with the value mτ = 1776.97 MeV/c2 ,
(13)
which is predicted by the Koide formula (Eq. 3). Figure 2 shows the comparison of measured τ mass in this paper with values from the PDG [7]; our result is consistent with all of them, but with the smallest uncertainty. Using our τ mass value, together with the values of B(τ → eν¯ν) and ττ from the PDG [7], we can calculate gτ through Eq. 1: gτ = (1.1650 ± 0.0034) × 10−5 GeV−2 ,
(14)
which can be used to test the SM. Similarly, inserting our τ mass value into Eq. 2 , together with the values of τμ , ττ , mμ , mτ , B(τ → eν¯ν) and B(μ → eν¯ν) from the PDG [7] and using the values of FW (-0.0003) and Fγ (0.0001) calculated from
[1] W. Marciano, A. Sirlin, Electroweak Radiative Corrections to τ Decay, Phys.Rev.Lett. 61 (1988) 1815–1818. [2] J. Bai, et al., Measurement of the mass of the τ lepton, Phys.Rev.Lett. 69 (1992) 3021–3024. [3] J. Bai, et al., Measurement of the mass of the τ lepton, Phys.Rev. D53 (1996) 20–34. [4] K. Abe, et al., Measurement of the mass of the τ-lepton and an upper limit on the mass difference between τ+ and τ− , Phys.Rev.Lett. 99 (2007) 011801. arXiv:hep-ex/0608046. [5] V. Anashin, et al., Measurement of the lepton mass at the kedr detector, JETP Letters 85 (8) (2007) 347–352. [6] B. Aubert, et al., Measurements of the τ mass and the mass difference of the τ+ and τ− at BABAR, Phys.Rev. D80 (2009) 092005. arXiv:0909.3562. [7] K. Olive, et al., Review of Particle Physics, Chin.Phys. C38 (2014) 090001. [8] Y. Koide, A New View of Quark and Lepton Mass Hierarchy, Phys.Rev. D28 (1983) 252. [9] R. Barate, et al., An Upper limit on the τ-neutrino mass from three-prong and five-prong τ decays, Eur.Phys.J. C2 (1998) 395–406. [10] H. Albrecht, et al., A Measurement of the τ mass, Phys.Lett. B292 (1992) 221–228. [11] G. Abbiendi, et al., A Measurement of the τ mass and the first CPT test with τ leptons, Phys.Lett. B492 (2000) 23–31. arXiv:hep-ex/0005009. [12] W. Bacino, T. Ferguson, L. Nodulman, W. Slater, H. Ticho, et al., Measurement of the Threshold Behavior of τ+ τ− Production in e+ e− Annihilation, Phys.Rev.Lett. 41 (1978) 13. [13] E. Abakumova, M. Achasov, V. Blinov, X. Cai, H. Dong, et al., The Beam energy measurement system for the Beijing electron-positron collider, Nucl.Instrum.Meth. A659 (2011) 21– 29. arXiv:1109.5771. [14] M. Achasov, V. Blinov, X. Cai, C.-D. Fu, F. Harris, et al., A scenario for high accuracy τ mass measurement at BEPC-II, Chin.Phys. C36 (2012) 573–577. [15] P. Rullhusen, P. Dhez, X. Artru, Novel radiation sources using relativistic electrons, World Scientific, 1998. [16] V. Berestetskii, E. Lifshitz, L. Pitaevskii, Relativistic Quantum Mechanics ( Volume 4 part 1 of A Course of Theoretical Physics ), Pergamon, New York, 1971. [17] M. Ablikim, et al., Precision measurement of the mass of the τ lepton, Phys.Rev. D90 (1) (2014) 012001. arXiv:1405.1076. [18] A. Pich, Selected topics on τ physics, Acta Phys.Polon. B38 (2007) 3449–3458. arXiv:0711.0028.
Nuclear and Particle Physics Proceedings 260 (2015) 23–26 www.elsevier.com/locate/nppp
Precision Measurement of the Mass of the τ Lepton at BESIII Tao Luo on behalf of BESIII Collaboration University of Hawaii, Honolulu, Hawaii 96822, USA
Abstract An energy scan near the τ pair production threshold has been performed using the BESIII detector. About 24 pb−1 of data, distributed over four scan points, was collected. This analysis is based on τ pair decays to ee, eμ, eh, μμ, μh, hh, eρ, μρ and πρ final states, where h denotes a charged π or K. The mass of the τ lepton is measured from a 2 maximum likelihood fit to the τ pair production cross section data to be mτ = (1776.91 ± 0.12 +0.10 −0.13 ) MeV/c , which is currently the most precise value in a single measurement. Keywords: τ mass, precision measurement, beam energy 1. Introduction The τ lepton mass, mτ , is one of the fundamental parameters of the Standard Model (SM). The relationship between the τ lifetime (ττ ), mass, its electronic branching fraction (B(τ → eν¯ν)) and weak coupling constant gτ is predicted by theory: g2 m5 B(τ → eν¯ν) = τ τ3 , ττ 192π
(1)
up to small radiative and electroweak corrections [1]. It appeared to be badly violated before the first precise mτ measurement of BES became available in 1992 [2]; this measurement was later updated with more τ decay channels [3] and confirmed by subsequent measurements from BELLE [4], KEDR [5], and BABAR [6]. The experimental determination of ττ , B(τ → eν¯ν) and mτ to the highest possible precision is essential for a high precision test of the SM. Currently, the mass precision for e and μ has reached Δm/m of 10−8 , while for τ it is 10−4 [7]. A precision mτ measurement is also required to check lepton universality. Lepton universality, a basic ingredient in the minimal standard model, requires that ∗ Email
address: [email protected] (Tao Luo)
http://dx.doi.org/10.1016/j.nuclphysbps.2015.02.006 2405-6014/© 2015 Elsevier B.V. All rights reserved.
the charged-current gauge coupling strengths ge , gμ , gτ should be identical: ge = gμ = gτ . Comparing the electronic branching fractions of τ and μ, lepton universality can be tested as: 2 τμ mμ 5 B(τ → eν¯ν) gτ = (1 + FW )(1 + Fγ ), (2) gμ ττ mτ B(μ → eν¯ν) where FW and Fγ are the weak and electromagnetic radiative corrections [1]. Note (gτ /gμ )2 depends on mτ to the fifth power. The testing of the Koide formula, which is an unexplained empirical equation discovered by Yoshio Koide in 1981 [8], also depends on the accuracy of mτ . The Koide formula is: me + mμ + mτ 2 Q= √ √ 2 = . √ 3 ( me + mμ + mτ )
(3)
It relates the masses of the three charged leptons so well that it predicted mτ . Furthermore, the precision of mτ will also restrict the ultimate sensitivity of mντ . The most sensitive bounds on the mass of the ντ can be derived from the analysis of the invariant-mass spectrum of semi-hadronic τ decays, e.g. the present best limit of mντ < 18.2 MeV/c2 (95% confidence level) was based on the kinematics of 2939 (52) events of τ− → 2π− π+ ντ (τ− → 3π− 2π+ (π− )ντ ) [9].
24
T. Luo / Nuclear and Particle Physics Proceedings 260 (2015) 23–26
This method depends on a determination of the kinematic end point of the mass spectrum; thus high precision on mτ is needed. So far, the pseudomass technique and the threshold scan method have been used to determine mτ . The former, which was used by ARGUS [10], OPAL [11], BELLE [4] and BABAR [6], relies on the reconstruction of the invariant-mass and energy of the hadronic system in the hadronic τ decay, while the latter, which was used in DELCO [12], BES [2, 3] and KEDR [5], is a study of the threshold behavior of the τ pair production cross section in e+ e− collisions and it is the method used in this paper. Extremely important in this approach is to determine the beam energy and the beam energy spread precisely. Here the beam energy measurement system (BEMS) [13] for BEPCII is used and will be described below. Before the experiment began, a study was carried out using Monte Carlo (MC) simulation and sampling to optimize the number and choice of scan points in order to provide the highest precision on mτ for a specified period of data taking time or equivalently for a given integrated luminosity [14]. The τ scan experiment was done in December 2011. The J/ψ and ψ resonances were each scanned at seven energy points, and data were collected at four scan points near τ pair production threshold with center of mass (CM) energies of 3542.4 MeV, 3553.8 MeV, 3561.1 MeV and 3600.2 MeV. The first τ scan point is below the mass of τ pair [7], while the other three are above. 2. Beam Energy Measurement System The BEMS is located at the north crossing point of the BEPCII storage ring. Here is a short introduction to the working principles of the BEMS. In the Compton scattering process, the maximal energy of the scattered photon Eγ is related to the electron energy Ee by the kinematics of Compton scattering [15, 16]: ⎡ ⎤ Eγ ⎢⎢⎢⎢ m2e ⎥⎥⎥⎥ ⎢⎢1 + 1 + ⎥⎥ , Ee = (4) 2 ⎢⎣ Eγ Eγ ⎥⎦ where Eγ is the energy of the initial photon, i.e. the energy of the laser beam in the BEMS. The scattered photon energy can be measured with high accuracy by the HPGe detector, whose energy scale is calibrated with photons from radioactive sources and the readout linearity is checked with a precision pulser. The maximum
energy can be determined from the fitting to the edge of the scattered photon energy spectrum. At the same time the energy spread of back-scattered photons due to the energy distribution of the collider beam is obtained from the fitting procedure [13]. Finally, the electron energy can be calculated by Eq. 4. Since the energy of the laser beam and the electron mass (me ) are determined with the accuracy at the level of 10−8 , Ee can be determined, utilizing Eq. 4, as accurately as Eγ . Another words, Ee is measured to an accuracy of 10−5 . The systematic error of the electron and positron beam energy determination in our experiment was tested through previous measurement of the ψ mass and was estimated to be 2 × 10−5 [13]; the relative uncertainty of the beam energy spread was about 6% [13]. More details can be found in the reference [13]. 3. Data Analysis 3.1. Luminosity at Each Scan Point For all scan points, the luminosity L is determined from L = Ndata /γγ σγγ , where Ndata is the number of selected two-gamma events in data and γγ and σγγ are the efficiency and the cross section determined by the Babayaga 3.5 MC, respectively. The measured integrated luminosities for the J/ψ, τ and ψ scan are calculated as 1505 nb−1 , 23261 nb−1 and 7526 nb−1 , respectively. The analysis using Bhabha events is done as a cross check of the two-gamma luminosity and gives consistent luminosity results within 2%. The Bhabha luminosities will also be used in the systematic error analyses. 3.2. J/ψ and ψ Hadronic Cross-section Line Shapes The number of hadronic events N h is fitted to the number of expected hadronic events N exp = σhad · L,
(5)
where σhad is the cross section of e+ e− → hadrons, which depends on ECM , and the energy spread, δw . The background cross section (σbg ) reconstruction efficiency (had ) M, and δw , are free parameters, obtained from minimizing χ2 =
N i=1
(Nih − σihad Li )2 Nih (1 + Nih (ΔLi /Li )2 )
,
(6)
where ΔLi /Li , Nih , σihad and Li are the relative luminosity error, the number of hadron events, the cross section of e+ e− → hadrons, and the luminosity at scan point i,
25
Cross Section (nb)
T. Luo / Nuclear and Particle Physics Proceedings 260 (2015) 23–26 2.0
is constructed from Poisson distributions, one at each of the four scan points, and takes the form [3]
1.5 1.0
L(mτ , RData/MC , σB ) =
0.5
4 μNi e−μi i
i=1 0.0 3540 3550 3560 3570 3580 3590 3600 3610
W (MeV)
Figure 1: The CM energy dependence of the τ pair cross section resulting from the likelihood fit (curve), compared to the data (Poisson errors).
Ni !
,
where Ni is the number of observed τ pair events at scan point i; μi is the expected number of events and calculated by i , mτ ) + σB ] × Li . μi = [RData/MC × i × σ(ECM
respectively. The mass difference with respect to nominal resonance mass, ΔM = M f it − M PDG , and the energy spread are obtained from the J/ψ and ψ fits. The values for δw agree well with those obtained from the BEMS. We extrapolate the results from the fits of the J/ψ and ψ line shapes to the τ-mass region in order to obtain the energy correction to the τ-mass. The systematic error associated with the energy scale is estimated by extrapolating under two assumptions: the first one is that the correction has a linear dependence on the energy and the second one assumes a constant shift. The linear fit between ΔM from the J/ψ and ψ fits gives the correction to the τ-mass of Δmτ = (0.054 ± 0.030) MeV/c2 ; the constant shift gives the correction of Δmτ = (0.043 ± 0.020) MeV/c2 , where both errors are statistical. The difference between these two methods, 0.011 MeV/c2 , is taken as the systematic uncertainty related to the energy determination. Overall, the systematic error in the energy determination is taken to be 0.012 MeV. The difference
mτ
will be taken into consideration in the measured τ-mass value.
RData/MC σB
In order to reduce the statistical error in the τ lepton mass, this analysis incorporates 13 two-prong τ pair final states, which are ee, eμ, eπ, eK, μμ, μπ, μK, πK, ππ, KK, eρ, μρ and πρ, with accompanying neutrinos implied. 3.3.1. Maximum Likelihood Fit to The Data The mass of the τ lepton is obtained from a maximum likelihood fit to the CM energy dependence of the τ pair production cross section. The likelihood function
(9)
In Eq. (9), mτ is the mass of the τ lepton, and RData/MC is the ratio of the overall efficiency for identifying τ pair events for data and for MC simulation, allowing for the difference of the efficiencies between the data and the corresponding MC sample. i is the efficiency at scan
point i, which is given by i = i Br j i j , where Br j is the branching fraction for the j − th final state and i j is the detection efficiency for the j − th final state at the i − th scan point. The efficiencies i , determined directly from the τ pair inclusive MC sample by applying the same τ pair selection criteria, are 0.065, 0.065, 0.069, 0.073 at the four scan points, respectively. σB is an effective background cross section, and it is assumed coni , covered stant over the limited range of CM energy, ECM by the scan. Li is the integrated luminosity at scan point i , mτ ) is the corresponding cross section for i, and σ(ECM τ pair production [3] The ML fit is performed on the selected τ pair candidate events. The fit yields
Δmτ = 0.054 ± 0.030(stat) ± 0.012(sys) MeV/c2 , (7)
3.3. τ Mass Measurement
(8)
= 1776.91 ± 0.12 MeV/c2 , = 1.05 ± 0.04, = 0+0.12 pb.
(10)
The fitted σB is zero, which indicates the selected τ pair candidate data set is very pure. The quality of the fit is shown explicitly in Fig. 1. The curve corresponds to the theoretical cross section with mτ = 1776.91 MeV/c2 ; the measured cross section at scan point i is given by σi =
Ni RData/MC i Li
.
(11)
The measured cross sections at different scan points are consistent with the theoretical values.
26
T. Luo / Nuclear and Particle Physics Proceedings 260 (2015) 23–26
ARGUS BES (96’) CLEO OPAL BELLE KEDR BABAR PDG12 This work
1766
1768
reference [1], the ratio of squared coupling constants is determined to be: 2 gτ = 1.0016 ± 0.0042, (15) gμ
1776.30+2.80 -2.80 1776.96+0.31 -0.28 1778.20+1.50 -1.50 1775.10+1.90 -1.90 1776.61+0.38 -0.38 1776.81+0.30 -0.28 1776.68+0.43 -0.43 1776.82+0.16 -0.16 1776.91+0.16 -0.18
1770 1772
1774 1776 2
τ mass (MeV/c )
1778
1780
Figure 2: Comparison of measured τ mass from this paper with those from the PDG. The vertical band corresponds to the 1 σ limit of the measurement of this paper
so that this test of lepton universality is satisfied at the 0.4 standard deviation level. The level of precision is compatible with previous determinations, which used the PDG average for mτ [18]. More details about this analysis can be found in the reference [17]. References
3.3.2. Systematic Error Estimation The systematic errors from the sources of theoretical accuracy, energy scale, energy spread, luminosity, event selection cuts, event selection efficiencies, background shape and fitted efficiency parameter are considered [17]. We assume that all systematical uncertainties are independent and add them in quadrature to obtain the total systematical uncertainty for τ mass measure2 ment, which is +0.10 −0.13 MeV/c . 4. Summary Using a maximum likelihood fit to the τ pair cross section data near threshold, the mass of the τ lepton has been measured as 2 mτ = (1776.91 ± 0.12+0.10 −0.13 ) MeV/c .
(12)
This value is in good agreement with the value mτ = 1776.97 MeV/c2 ,
(13)
which is predicted by the Koide formula (Eq. 3). Figure 2 shows the comparison of measured τ mass in this paper with values from the PDG [7]; our result is consistent with all of them, but with the smallest uncertainty. Using our τ mass value, together with the values of B(τ → eν¯ν) and ττ from the PDG [7], we can calculate gτ through Eq. 1: gτ = (1.1650 ± 0.0034) × 10−5 GeV−2 ,
(14)
which can be used to test the SM. Similarly, inserting our τ mass value into Eq. 2 , together with the values of τμ , ττ , mμ , mτ , B(τ → eν¯ν) and B(μ → eν¯ν) from the PDG [7] and using the values of FW (-0.0003) and Fγ (0.0001) calculated from
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