- Email: [email protected]
Study of high precision τ mass measurement at BESIII
Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139 www.elsevierphysics.com
Study of high precision τ mass measurement at BESIII X.H. Moa∗ a
Institute of High Energy Physics, CAS – P.O. Box 918-1, Beijing 100049, China
The statistical and systematic uncertainties of mτ measurement at the forthcoming detector BESIII are studied. Monte Carlo simulation and sampling techniques are adopted to simulate various data taking strategies, from which the optimal scheme is determined. It is found that the optimal position of data taking is located at the energy point with large cross section derivative near τ + τ − production threshold; one point in the optimal position with luminosity of 63 pb−1 is sufficient for a statistical accuracy better than 0.1 MeV. In addition, if Compton backscattering technique can be used at BESIII to measure the absolute energy, the final systematic uncertainty around 0.09 MeV could be expected.
1. Introduction
2. Statistical Optimization
The mass of the τ lepton is an fundamental parameter in standard model, many experiments have been performed to determine the τ mass accurately, some measurements [1–10] are displayed in Figure 1. Recently, the developments of experimental techniques [12] and theoretical calculations [13–16] provide the possibility to measure the τ mass with unprecendented accuracy. Moreover, large τ data are expected from the forthcoming detector BESIII [17], therefore it is of great interest to know what accuracy of τ mass we can look forward to in the near future. Usually, the pseudomass and threshold-scan methods are employed to measure the τ mass, where the latter is adopted by BES collaboration to achieve the accurately measured value [6]:
Within a specified period of data taking time or equivalently for a given integrated luminosity, we try to find out the scheme which can provide the highest accuracy. We utilize sampling technique to simulate various data taking possibilities among which we chose the optimal one. For certain simulation, the likelihood function is constructed as follows [6]:
mτ = 1776.96+0.18+0.25 −0.21−0.17 MeV .
(1)
We note that the relative statistical (1.6 × 10−4 ) and systematic (1.7 × 10−4 ) uncertainties have comparable magnitude, so the following research is divided into two parts, one about the statistical and the other about the systematic uncertainties.
∗ Supported by National Natural Science Foundation of China (10491303) and 100 Talents Programme of CAS (U25).
0920-5632/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2007.02.107
LF (mτ ) =
n μNi e−μi i
i
Ni !
,
(2)
where Ni is the observed number of τ + τ − events for eμ final state at scan point i; μi is the expected number of events and calculated by i μi (mτ ) = [ · Beμ · σobs (mτ , Ecm ) + σBG ] · Li .(3)
In Eq. (3), Li is the integrated luminosity at ith point; is the overall efficiency of eμ final state for identifying τ + τ − events, which includes trigger efficiency and event selection efficiency; Beμ is the combined branching ratio for decays τ + → e+ νe ν τ and τ − → μ− ν μ ντ , or the corresponding charge conjugate mode; σBG is the efi fective background cross section; σobs (mτ , Ecm ) is the observed cross section measured at energy i with τ mass (mτ ) as a parameter. Ecm In this part of study, we take = 14.2%, Δ (energy spread) = 1.4 MeV [18], B = 0.06194 [11], and neglect corresponding uncertainties which are
133
X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139
DASP
1807 ± 20
SPEC
1787 +10
MarkII
1787±10
2.1. Data taking point As a tentative beginning, the energy interval needed to be studied is divided evenly, viz.
-18
Ei = E0 + i × δE,
ARGUS 1776.3±2.8 1778.2±1.4
OPAL
1775.1±1.9
Belle
1776.77±0.35
BES
1776.96-0.27
KEDR
1776.80-0.27
+0.31
+0.29
n
mτ = 1700
1725
1750
(4)
where the initial point E0 = 3.545 GeV, the final point Ef = 3.595 GeV, and the fixed step δE = (Ef − E0 )/Npt with Npt being the number of energy points. For a given integrated luminosity, Ltot , it is also apportioned evenly at each point, i.e. Li = Ltot /Npt . In order to suppress or lessen the statistical fluctuation, sampling is repeated many times (say 500 times) for each scheme (say for each Npt ), the averaged fit mτ and corresponding variance are worked out as follows [19] :
+2
DELCO 1783-7
CLEO
(i = 1, 2, ..., Npt )
1775
1800
1825
mτ (MeV) Figure 1. Comparison of the different measurements of the τ mass. The vertical line indicates the current world average value: 1776.99+0.29 −0.26 MeV [11], which is the averaged result by virtue of the measurements from Refs. [4–8].
left to be considered in systematic study. we also set σBG = 0, which means that our study is background free. As to theoretical calculated cross section σobs , we adopt the improved Voloshin’s formulas [13]. Since the accuracy of mτ is mainly concerned, we assume mτ itself is known then attempt to find out: 1. What is the optimal energy point distribution for data taking; 2. How many energy points are needed for the scan in the vicinity of threshold; 3. How much luminosity is required for certain precision expectation.
1 mτ i , n i=1
2 (mτ ) = Sm τ
(5) n
1 (mτ i − mτ i )2 . n − 1 i=1
(6)
Without special declaration, in this part of study all the fit masses and variances are the averaged values as calculated by Eqs. (5) and (6). Using the fore-given experiment parameters, such as , Δ, and Beμ , setting Ltot = 30 pb−1 , and Npt ranging from 3 to 20, the fitted results are shown in Figure 2, where Δmτ = mτ − m0τ , the difference between the fitted mτ and the input one (m0τ = 1776.99 MeV according to PDG06 [11]), and Smτ the corresponding uncertainty. Here two points should be noted: first, Smτ is much larger than |Δmτ |, so from point view of accuracy, the former is much more crucial than the latter. Therefore, in the following study Smτ is used as the goodness of the fit. The smaller Smτ the better is the fit. Second, it is prominent that too few the data taking points lead to large uncertainty while too many points have no contribution for accuracy improvement either. As indicated in Figure 2, Npt = 5 is taken as the optimized number of points for the evenlydivided-distribution scheme. 2.2. Optimal distribution With five points, we want to further search for the distribution which can afford us the small fit
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X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139 1.4
1
1.2 1
0.6 0.4 0.2
0.8
σ (nb)
Smτ (MeV)
|Δmτ| (MeV)
0.8
0.6 0.4 0.2 0 -0.2
0 2
4
6
8 10 12 14 16 18 20 Npt
Figure 2. The variation of |Δmτ | (the absolute value of Δmτ ) and Smτ against the number of points Npt . The diamond denotes |Δmτ | and the cross Smτ .
uncertainty. As without any theoretical implication, the sampling technique is employed and the energy points is taken randomly in the chosen interval. For 200 times sampling, singled out are two fit results with the smallest (Smτ = 0.152 MeV) and greatest (Smτ = 1.516 MeV) fit uncertainties; their distributions are shown in Figure 3, on the strength of which it is obvious when the points crowd near the threshold the uncertainty is small; on the contrary, when the points are far from the threshold the uncertainty becomes large. More mathematically, it is found that the smallest uncertainty is acquired when points gather at the region with the large derivative of cross section to energy. So this result implies that the region with large derivative is presumably the optimal position for data taking. 2.3. Optimal region To hunt for the sensitive position, two the regions are selected: the region I (Ecm ⊂ (3.553, 3.557) GeV ) is selected with the derivative falls to 75% of its maxinum while the region II (Ecm ⊂ (3.565, 3.595) GeV ) is selected with
-0.4 3.54
3.56
3.58
3.6
Ecm (GeV)
Figure 3. The distributions for 5 scan points with the small and large Smτ . The diamonds indicate energy points; the solid line is the calculated observed cross section, and the dashed line the corresponding derivative of cross section to energy with a scale factor 10−2 .
the variation of derivative is comparative smooth than that in region I. To confirm the speculation mentioned in the previous subsection, two schemes are designed. For the first scheme, two points are taken in the region I, one at 3.55398 GeV as the threshold point and the other at 3.5548 GeV corresponding to the largest derivative point. As in the region II, the number of points Npt increases from 1 to 20, with each point having the luminosity 5 pb−1 . The fit results are displayed in Figure 4 by the crosses. Clearly, the increase of points in the region II hardly has contribution for accuracy improvement (Smτ = 0.25 MeV almost remains the same with the increasing number of points). That is to say, the region I is the sensitive region so far as fit uncertainty is concerned while the region II is not. To prove this point further, for the second scheme, merely the points in the region II are taken, Npt also increases from 1 to 20. The fit results are displayed in Figure 4 by the diamonds. As expected, with the increas-
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X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139 2.25
0.2
2
0.18 0.16 0.14
1.5
Smτ (MeV)
Smτ (MeV)
1.75
1.25 1
0.12 0.1 0.08
0.75
0.06
0.5
0.04
0.25
0.02
0
0
0
5
10
15
20
Npt
Figure 4. The fit uncertainties for different schemes, the cross and the diamond denote respectively the results for the first and second schemes as depicted in the text.
ing number of points, Smτ decreases as well, but even with 20 points in the region II the value of Smτ = 0.7256 MeV is still much larger than that of Smτ with solely two points in the region I. So it is concluded that the points within the region I are more useful for optimal data taking. 2.4. Optimal position In this subsection we first want to know how many points are optimal in the region with larger derivative. As the procedure in subsection 2.1, the total luminosity Ltot = 45 pb−1 is rationed averagely into Npt points (Npt = 1, 2, · · · , 6) within the energy region from 3.553 to 3.557 GeV. The results are shown in Figure 5, according to which the number of points has weak effect on the final uncertainty. In another word, within the large derivative region, one point suffices to give rise to small uncertainty. Then an immediate question is where the most optimal point should locate? To answer it, we do scan with one point with the luminosity Ltot = 45 pb−1 . The scan results are shown in Figure6. Just as previous study indicated, the small uncertainty is
0
1
2
3
4
5
6
7
Npt
Figure 5. The relation between Smτ and the number of points within the energy region from 3.553 to 3.557 GeV.
achieved near the peak of the derivative. If looking into the region from 3.552 to 3.5565 GeV, it is found that the smallest Smτ = 0.105 MeV is obtained near the mτ threshold (Ecm = 3.55398 MeV), which has a deviation from the position (Ecm = 3.55484 MeV) with the greatest cross section derivative where Smτ = 0.122 MeV. In addition, the study also indicates that within 2 MeV region the variation of Smτ is fairly smooth (from 0.105 to 0.127 MeV), which is very favorable for actual data taking. 2.5. Luminosity and uncertainty The last question we are concerned with is the relation between uncertainty and luminosity. For the fit with one point at the large derivative region, some special results are listed in Table 1. The second and third columns present the results for one point at Ecm = 3.55398 GeV which corresponds to the threshold while the last two columns give the results at Ecm = 3.55484 GeV which corresponds to the point with the greatest derivative of cross section. In the light of Table 1, first the accuracy is inversely proportional to the luminosity; second the luminosity of 63 pb−1 is
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X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139
Table 1 The relation between luminosity and uncertainty.
1
S mτ (MeV)
Ltot -1
10
-2
10
3.54
3.56
3.58
3.6
Ecm (GeV)
(pb−1 )
Ecm =3.55398 GeV Δmτ Smτ (keV) (keV)
Ecm =3.5548 GeV Smτ Δmτ (keV) (keV)
9 18 27 36 45 54 63 72 100 1000 10000
248.74 169.26 140.24 121.30 106.53 97.83 90.35 84.24 67.81 21.46 6.84
292.40 196.35 156.70 143.84 127.17 107.14 99.23 92.97 78.76 25.15 8.05
29.31 15.50 12.34 8.12 8.24 7.17 7.26 5.20 1.29 0.16 0
21.14 7.56 4.75 5.04 2.92 −0.37 −0.03 0.08 −0.02 0 0
Figure 6. The variation of Smτ against energy for one point fit with Ltot = 45 pb−1 .
sufficient to provide the accuracy less than 0.1 MeV. 3. Systematic uncertainty In this section the following effects on mτ measurements will be examined, including theoretical accuracy, energy spread, energy scale, luminosity, efficiency, background analysis, and so forth. 3.1. Theoretical accuracy The experimentally measured cross section has the form [21] ∞ √ √ √ σexp (mτ , s, Δ) = d s G( s , s) 0
×
0
4m2 1− sτ
dxF (x, s )
σ ¯ (s (1 − x), mτ ) , |1 − Π(s (1 − x))|2
(7)
2 . Here F (x, s) is the initial state with s = Ecm radiation factor [21], Π is the√vacuum √ polarization factor [16,22,23], and G( s , s), which is usually treated as a Gaussian distribution [24], depicts the energy spread (Δ) of the e+ e− collider. As mentioned in section 1, the high accurate calculations for the production cross sec-
tion (¯ σ ) are available recently. Anyway, the improved Voloshin’s formulas [13] are adopted, since the previous production cross section (denoted as σ ¯ ∗ ) used in BES mτ fit is calculated based on Voloshin’s formulas [20]. The relative differences of cross sections calculated with two sets of Voloshin’s formulas are drawn in Figure 7. To estimate the effect due to theoretical calculation accuracy, the fitted τ masses by using two kinds of formulas are compared. The comparison shows the uncertainty due to this effect is at the level of 10−3 MeV. 3.2. Energy spread As indicated in Eq. (7), the experimentally measured cross section σexp depends on the energy spread (Δ), whose effect on mτ fit is to be considered here. In fact, Δ at τ mass region is often deduced by virtue of the energy spreads measured at the J/ψ (ΔJ/ψ ) and ψ (Δψ ) regions; and derived according to the interpolation formula Δ − ΔJ/ψ f (E) − f (EJ/ψ ) , = Δψ − ΔJ/ψ f (Eψ ) − f (EJ/ψ )
(8)
with assumption Δ ∝ f (E). Here f (E) denotes the dependence of Δ on energy Ecm (subscript “cm” has been suppressed for brevity). Never-
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X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139 0.7
where Es indicates the scaled energy value. If the difference between measured and nominal quantities is small, that is E = M + δ, where δ is a small variable and estimated at the level of 10−4 . Under such case, the relation is readily obtained
0.6
0.4
Es − MJ/ψ δs − δJ/ψ = . Mψ − MJ/ψ δψ − δJ/ψ
0.3 0.2 0.1 0 3.54 3.55 3.56 3.57 3.58 3.59
3.6
Ecm (GeV)
Figure 7. The relative differences of cross sections calculated with different sets of Voloshin’s formulas [18]. Here Δσ = σ − σ ∗ , where σ and σ ∗ can be σexp (denoted by the solid line) or σ ¯ (denoted by the dashed line).
theless, since the knowledge of concrete form of function f is lack , a generic form is assumed: f (E) = a · E + b · E 2 + c · E 3 ,
(9)
and the linear, quadric, cubic or mixing type of energy-dependence form is tested to estimate the uncertainty, that is the following parameter values are taken: a = 1, b = 0, c = 0; a = 0, b = 1, c = 0; (10) a = 0, b = 0, c = 1; a = 1, b = 1, c = 1. The fit results give δmτ < 1.5 × 10−3 MeV. Even if Δ is artificially changed to 3Δ, and the repeat fits indicate that δmτ < 6 × 10−3 MeV.
(12)
Similar to the uncertainty study of the energy spread, assuming δ ∝ f (E), where f (E) expressed by Eq. (9) and parameters take values given in Eq. (10), it is found that the uncertainty due to the energy scale is about 8 × 10−3 MeV. Anyway, the relation δ ∝ f (E) is merely a theoretical surmise, not an empirical formula. The most direct approach to determine the energy scale is to measure it experimentally. At present, two methods have used to determine the absolute energy value accurately, one is depolarization method and the other is Compton backscattering method, both of which have been utilized by KEDR group [10].
0.2 0.1
Δmτ (MeV)
Δσ/σ (%)
0.5
0 -0.1 -0.2 -0.2
-0.1
0
0.1
ΔE (MeV)
0.2
3.3. Energy scale As to the energy scale, PDG mass values of the J/ψ (MJ/ψ ) and the ψ (Mψ ) are adopted as the standard to scale the energies measured at J/ψ (EJ/ψ ), ψ (Eψ ), and τ (Eτ ) mass regions, viz.
Figure 8. The effect due to the uncertainty of absolute energy calibration on mτ measurement.
Es − MJ/ψ Eτ − EJ/ψ = , Eψ − EJ/ψ Mψ − MJ/ψ
It is proposed to adopt Compton backscattering technique to measure the energy at BE-
(11)
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X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139
SIII, the relative accuracy of such technique is expected at the level of 5 × 10−5 . If this technique can be realized at BESIII, the systematic uncertainty can be expected at this level, that is 5 × 10−5 (relative error) or around 9 × 10−2 MeV (absolute error), which is one order of magnitude larger than 8 × 10−3 MeV. Since the uncertainty of energy scale will transfer to the final fit result of mτ directly and linearly as displayed in Figure 8, the absolute calibration of energy scale is actually a bottleneck problem in accuracy improvement for mτ measurement. 3.4. Other factors The systematic uncertainties due to other experimental factors are quantified by reasonable variations of the corresponding quantities (2% variation for luminosity L, 2% for efficiency , 0.5% for branching fraction Beμ , 10% for background σBG ), the total uncertainty due to all these factors is estimated to be less than 2.02 × 10−2 MeV. The summary of possible systematic uncertainties are presented in Table 2, in the light of which the summed uncertainty is 2.3×10−2 MeV, or the relative one is 1.3 × 10−5 . Here the uncertainties due to luminosity and efficiency dominate.
Table 2 Some possible systematic uncertainties. Source
δmτ ( 10−3 MeV)
δmτ /mτ ( 10−6 )
Theoretical accuracy Energy spread Energy scale Luminosity Efficiency Branching Fraction Background
3 6 8† 14 14 3.5 1.7
1.7 3.4 4.5† 7.9 7.9 2.0 1.0
Summation 22.7 12.7 †: based on the estimation of Eq. (12).
4. Discussion At BESIII the designed peak luminosity is around 1 nb−1 s−1 , if the average efficiency of luminosity is taken as 50% of the peak value, then two days data taking time will lead to the statistic uncertainty less than 0.1 MeV. Notice this evaluation is solely for eμ-tagged event, if more channels are utilized to tag τ -pair final state, such as ee, eμ, eh, μμ, μh, hh (h : hadron), and so on, more statistics can be expected (according to previous BES analyses [6,25] experience, the number of multi-channel-tagged events is at least 5 times more than that of eμ-tagged event). Therefore at BESIII, one week data taking time will lead to the statistical uncertainty less than 0.017 MeV. Comparatively, even if the Compton backscattering technique is adopted at the new detector, the systematic uncertainty is merely up to 0.09 MeV. On this extent, the final accuracy of mτ measurement at BESIII will be mainly determined by the systematic uncertainty. At last, a few words about the data taking design for BESIII. To measure the mτ , at least three points are needed. One is at the large derivative region, which is the optimal position as our study indicates; the other is below the threshold, which can be used for background study; and the third one is above the threshold, which can be used to determine the event selection criteria for data analysis at the optimal point. In fact, the data below and above threshold can also be used for R-value measurement and background study for J/ψ and ψ data analyses. So it is reasonable to consider the mτ measurement between the period of data taking for the J/ψ and ψ . 5. Summary The possible statistical and systematic uncertainties of mτ measurement at the forthcoming detector BESIII are studied based on previous experiences and recent theoretical calculations. Monte Carlo simulation and sampling techniques are employed to obtain optimal data taking point for high accurate mτ measurement. It is found: 1. The optimal position is located at the region with the large derivative of cross sec-
X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139
tion to energy near production threshold; 2. One point is enough to accommodate small error within the optimal region; 3. The luminosity of 63 pb−1 suffices the statistical accuracy better than 0.1 MeV. In addition, many factors have been taken into account to estimate possible systematic uncertainties, the total relative error is at the level of 1.3 × 10−5 . However, the absolute calibration of energy scale may be a crux for further accuracy improvement of mτ measurement. If the new technique could be adopted at BESIII, the final systematic uncertainty around 0.09 MeV could be expected. Acknowledgments The author would like to acknowledge help from his graduated student Y.K. Wang who work hard on actual fits. Thanks are also due to Prof. C.Z. Yuan who give many valuable suggestions and comments for improvement of the proceedings. REFERENCES 1. R. Brandelik et al., DASP Collaboration, Phys. Lett. B 73 (1978) 109. 2. W. Bartel et al., Phys. Lett. B 77 (1978) 331. 3. C.A. Blocker, Ph.D. Thesis, LBL-Report 10801 (1980). 4. W. Bacino et al., Phys. Rev. Lett. 41 (1978) 13. 5. H. Albrecht et al., ARGUS Collaboration, Phys. Lett. B 292 (1992) 221. 6. J.Z. Bai et al., BES Collaboration, Phys. Rev. D 53 (1996) 20. 7. A. Anastassov et al., CLEO Collaboration, Phys. Rev. D 55 (1997) 2559; Phys. Rev. D 58 (1998) 119904 (Erratum). 8. G. Abbiendi et al., OPAL Collaboration, Phys. Lett. B 492 (2000) 23. 9. M. Shapkin, “Measurement of the mass of the τ -lepton”, these proceedings. 10. B. Shwartz, “New precise determination of the τ lepton mass at the KEDR detector”, these proceedings.
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11. W.-M. Yao et al., Particle Data Group, J. Phys. G 33 (2006) 1. 12. A. Bogomyagkov et al., “Research of Possibility to use Beam Polarization for Absolute Energy Calibration in High-precision Measurement of Tau Lepton Mass at VEPP-4M.”, Presented at the 9th European Particle Accelerator Conference (EPAC 2004), Lucerne, Switzerland, July 5-9, 2004. 13. M. B. Voloshin, Phys. Lett. B 556 (2003) 153. 14. B. H. Smith and M.B. Voloshin, Phys. Lett. B 324 (1994) 117; Phys. Lett. B 333 (1994) 564, Erratum. 15. A. Pich, “Perspectives on Tau-Charm Factory Physics” in Proceedings of the 3rd Workshop on the Tau-Charm Factory, Marbella, Spain, 1993, edited by J. Kirkby and R. Kirkby (Editions Frontieres, Gif-sur-Yvete, 1994), p. 767. 16. P. Ruiz-Femen´ıa and A. Pich, Phys. Rev. D 64 (2001) 053001. 17. Y.F. Wang, “BESIII status”, talk at Charm2006, Beijing, China, June 5-7, 2006. 18. Y.K. Wang, X.H. Mo, C.Z. Yuan and J.P. Liu, “Measurement of τ mass based on improved theoretical calculations”, to be published in HEP& NP. 19. S. Brandt, Data Analysis (3rd ed.), SpringerVerlag, New York, 1999. 20. J.Z. Bai et al., BES Collaboration, HEP& NP, 16 (1992) 343. 21. E.A. Kuraev and V.S. Fadini, Sov. J. Nucl. Phys. 41 (1985) 466; G. Altarelli and G. Martinelli, CERN 86-02 (1986) 47; O. Nicrosini and L. Trentadue, Phys. Lett. B 196 (1987) 551; F.A. Berends, G. Burgers and W.L. Neerven, Nucl. Phys. B 297 (1988) 429; Nucl. Phys. B 304 (1988) 921. 22. F.A. Berends, K.J.F. Gaemers and R. Gastmans, Nucl. Phys. B 57 (1973) 381; F.A. Berends and G.J. Komen, Phys. Lett. B 63 (1976) 432. 23. G. Rodrigo, A. Pich and A. Santamaria, hepph/97077474. 24. P. Wang, C.Z. Yuan, X.H. Mo and D.H. Zhang, Phys. Lett. B 593 (2004) 89. 25. J.Z. Bai et al., BES Collaboration, Phys. Rev. Lett. 69 (1992) 3021.
Study of high precision τ mass measurement at BESIII X.H. Moa∗ a
Institute of High Energy Physics, CAS – P.O. Box 918-1, Beijing 100049, China
The statistical and systematic uncertainties of mτ measurement at the forthcoming detector BESIII are studied. Monte Carlo simulation and sampling techniques are adopted to simulate various data taking strategies, from which the optimal scheme is determined. It is found that the optimal position of data taking is located at the energy point with large cross section derivative near τ + τ − production threshold; one point in the optimal position with luminosity of 63 pb−1 is sufficient for a statistical accuracy better than 0.1 MeV. In addition, if Compton backscattering technique can be used at BESIII to measure the absolute energy, the final systematic uncertainty around 0.09 MeV could be expected.
1. Introduction
2. Statistical Optimization
The mass of the τ lepton is an fundamental parameter in standard model, many experiments have been performed to determine the τ mass accurately, some measurements [1–10] are displayed in Figure 1. Recently, the developments of experimental techniques [12] and theoretical calculations [13–16] provide the possibility to measure the τ mass with unprecendented accuracy. Moreover, large τ data are expected from the forthcoming detector BESIII [17], therefore it is of great interest to know what accuracy of τ mass we can look forward to in the near future. Usually, the pseudomass and threshold-scan methods are employed to measure the τ mass, where the latter is adopted by BES collaboration to achieve the accurately measured value [6]:
Within a specified period of data taking time or equivalently for a given integrated luminosity, we try to find out the scheme which can provide the highest accuracy. We utilize sampling technique to simulate various data taking possibilities among which we chose the optimal one. For certain simulation, the likelihood function is constructed as follows [6]:
mτ = 1776.96+0.18+0.25 −0.21−0.17 MeV .
(1)
We note that the relative statistical (1.6 × 10−4 ) and systematic (1.7 × 10−4 ) uncertainties have comparable magnitude, so the following research is divided into two parts, one about the statistical and the other about the systematic uncertainties.
∗ Supported by National Natural Science Foundation of China (10491303) and 100 Talents Programme of CAS (U25).
0920-5632/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2007.02.107
LF (mτ ) =
n μNi e−μi i
i
Ni !
,
(2)
where Ni is the observed number of τ + τ − events for eμ final state at scan point i; μi is the expected number of events and calculated by i μi (mτ ) = [ · Beμ · σobs (mτ , Ecm ) + σBG ] · Li .(3)
In Eq. (3), Li is the integrated luminosity at ith point; is the overall efficiency of eμ final state for identifying τ + τ − events, which includes trigger efficiency and event selection efficiency; Beμ is the combined branching ratio for decays τ + → e+ νe ν τ and τ − → μ− ν μ ντ , or the corresponding charge conjugate mode; σBG is the efi fective background cross section; σobs (mτ , Ecm ) is the observed cross section measured at energy i with τ mass (mτ ) as a parameter. Ecm In this part of study, we take = 14.2%, Δ (energy spread) = 1.4 MeV [18], B = 0.06194 [11], and neglect corresponding uncertainties which are
133
X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139
DASP
1807 ± 20
SPEC
1787 +10
MarkII
1787±10
2.1. Data taking point As a tentative beginning, the energy interval needed to be studied is divided evenly, viz.
-18
Ei = E0 + i × δE,
ARGUS 1776.3±2.8 1778.2±1.4
OPAL
1775.1±1.9
Belle
1776.77±0.35
BES
1776.96-0.27
KEDR
1776.80-0.27
+0.31
+0.29
n
mτ = 1700
1725
1750
(4)
where the initial point E0 = 3.545 GeV, the final point Ef = 3.595 GeV, and the fixed step δE = (Ef − E0 )/Npt with Npt being the number of energy points. For a given integrated luminosity, Ltot , it is also apportioned evenly at each point, i.e. Li = Ltot /Npt . In order to suppress or lessen the statistical fluctuation, sampling is repeated many times (say 500 times) for each scheme (say for each Npt ), the averaged fit mτ and corresponding variance are worked out as follows [19] :
+2
DELCO 1783-7
CLEO
(i = 1, 2, ..., Npt )
1775
1800
1825
mτ (MeV) Figure 1. Comparison of the different measurements of the τ mass. The vertical line indicates the current world average value: 1776.99+0.29 −0.26 MeV [11], which is the averaged result by virtue of the measurements from Refs. [4–8].
left to be considered in systematic study. we also set σBG = 0, which means that our study is background free. As to theoretical calculated cross section σobs , we adopt the improved Voloshin’s formulas [13]. Since the accuracy of mτ is mainly concerned, we assume mτ itself is known then attempt to find out: 1. What is the optimal energy point distribution for data taking; 2. How many energy points are needed for the scan in the vicinity of threshold; 3. How much luminosity is required for certain precision expectation.
1 mτ i , n i=1
2 (mτ ) = Sm τ
(5) n
1 (mτ i − mτ i )2 . n − 1 i=1
(6)
Without special declaration, in this part of study all the fit masses and variances are the averaged values as calculated by Eqs. (5) and (6). Using the fore-given experiment parameters, such as , Δ, and Beμ , setting Ltot = 30 pb−1 , and Npt ranging from 3 to 20, the fitted results are shown in Figure 2, where Δmτ = mτ − m0τ , the difference between the fitted mτ and the input one (m0τ = 1776.99 MeV according to PDG06 [11]), and Smτ the corresponding uncertainty. Here two points should be noted: first, Smτ is much larger than |Δmτ |, so from point view of accuracy, the former is much more crucial than the latter. Therefore, in the following study Smτ is used as the goodness of the fit. The smaller Smτ the better is the fit. Second, it is prominent that too few the data taking points lead to large uncertainty while too many points have no contribution for accuracy improvement either. As indicated in Figure 2, Npt = 5 is taken as the optimized number of points for the evenlydivided-distribution scheme. 2.2. Optimal distribution With five points, we want to further search for the distribution which can afford us the small fit
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1
1.2 1
0.6 0.4 0.2
0.8
σ (nb)
Smτ (MeV)
|Δmτ| (MeV)
0.8
0.6 0.4 0.2 0 -0.2
0 2
4
6
8 10 12 14 16 18 20 Npt
Figure 2. The variation of |Δmτ | (the absolute value of Δmτ ) and Smτ against the number of points Npt . The diamond denotes |Δmτ | and the cross Smτ .
uncertainty. As without any theoretical implication, the sampling technique is employed and the energy points is taken randomly in the chosen interval. For 200 times sampling, singled out are two fit results with the smallest (Smτ = 0.152 MeV) and greatest (Smτ = 1.516 MeV) fit uncertainties; their distributions are shown in Figure 3, on the strength of which it is obvious when the points crowd near the threshold the uncertainty is small; on the contrary, when the points are far from the threshold the uncertainty becomes large. More mathematically, it is found that the smallest uncertainty is acquired when points gather at the region with the large derivative of cross section to energy. So this result implies that the region with large derivative is presumably the optimal position for data taking. 2.3. Optimal region To hunt for the sensitive position, two the regions are selected: the region I (Ecm ⊂ (3.553, 3.557) GeV ) is selected with the derivative falls to 75% of its maxinum while the region II (Ecm ⊂ (3.565, 3.595) GeV ) is selected with
-0.4 3.54
3.56
3.58
3.6
Ecm (GeV)
Figure 3. The distributions for 5 scan points with the small and large Smτ . The diamonds indicate energy points; the solid line is the calculated observed cross section, and the dashed line the corresponding derivative of cross section to energy with a scale factor 10−2 .
the variation of derivative is comparative smooth than that in region I. To confirm the speculation mentioned in the previous subsection, two schemes are designed. For the first scheme, two points are taken in the region I, one at 3.55398 GeV as the threshold point and the other at 3.5548 GeV corresponding to the largest derivative point. As in the region II, the number of points Npt increases from 1 to 20, with each point having the luminosity 5 pb−1 . The fit results are displayed in Figure 4 by the crosses. Clearly, the increase of points in the region II hardly has contribution for accuracy improvement (Smτ = 0.25 MeV almost remains the same with the increasing number of points). That is to say, the region I is the sensitive region so far as fit uncertainty is concerned while the region II is not. To prove this point further, for the second scheme, merely the points in the region II are taken, Npt also increases from 1 to 20. The fit results are displayed in Figure 4 by the diamonds. As expected, with the increas-
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X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139 2.25
0.2
2
0.18 0.16 0.14
1.5
Smτ (MeV)
Smτ (MeV)
1.75
1.25 1
0.12 0.1 0.08
0.75
0.06
0.5
0.04
0.25
0.02
0
0
0
5
10
15
20
Npt
Figure 4. The fit uncertainties for different schemes, the cross and the diamond denote respectively the results for the first and second schemes as depicted in the text.
ing number of points, Smτ decreases as well, but even with 20 points in the region II the value of Smτ = 0.7256 MeV is still much larger than that of Smτ with solely two points in the region I. So it is concluded that the points within the region I are more useful for optimal data taking. 2.4. Optimal position In this subsection we first want to know how many points are optimal in the region with larger derivative. As the procedure in subsection 2.1, the total luminosity Ltot = 45 pb−1 is rationed averagely into Npt points (Npt = 1, 2, · · · , 6) within the energy region from 3.553 to 3.557 GeV. The results are shown in Figure 5, according to which the number of points has weak effect on the final uncertainty. In another word, within the large derivative region, one point suffices to give rise to small uncertainty. Then an immediate question is where the most optimal point should locate? To answer it, we do scan with one point with the luminosity Ltot = 45 pb−1 . The scan results are shown in Figure6. Just as previous study indicated, the small uncertainty is
0
1
2
3
4
5
6
7
Npt
Figure 5. The relation between Smτ and the number of points within the energy region from 3.553 to 3.557 GeV.
achieved near the peak of the derivative. If looking into the region from 3.552 to 3.5565 GeV, it is found that the smallest Smτ = 0.105 MeV is obtained near the mτ threshold (Ecm = 3.55398 MeV), which has a deviation from the position (Ecm = 3.55484 MeV) with the greatest cross section derivative where Smτ = 0.122 MeV. In addition, the study also indicates that within 2 MeV region the variation of Smτ is fairly smooth (from 0.105 to 0.127 MeV), which is very favorable for actual data taking. 2.5. Luminosity and uncertainty The last question we are concerned with is the relation between uncertainty and luminosity. For the fit with one point at the large derivative region, some special results are listed in Table 1. The second and third columns present the results for one point at Ecm = 3.55398 GeV which corresponds to the threshold while the last two columns give the results at Ecm = 3.55484 GeV which corresponds to the point with the greatest derivative of cross section. In the light of Table 1, first the accuracy is inversely proportional to the luminosity; second the luminosity of 63 pb−1 is
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Table 1 The relation between luminosity and uncertainty.
1
S mτ (MeV)
Ltot -1
10
-2
10
3.54
3.56
3.58
3.6
Ecm (GeV)
(pb−1 )
Ecm =3.55398 GeV Δmτ Smτ (keV) (keV)
Ecm =3.5548 GeV Smτ Δmτ (keV) (keV)
9 18 27 36 45 54 63 72 100 1000 10000
248.74 169.26 140.24 121.30 106.53 97.83 90.35 84.24 67.81 21.46 6.84
292.40 196.35 156.70 143.84 127.17 107.14 99.23 92.97 78.76 25.15 8.05
29.31 15.50 12.34 8.12 8.24 7.17 7.26 5.20 1.29 0.16 0
21.14 7.56 4.75 5.04 2.92 −0.37 −0.03 0.08 −0.02 0 0
Figure 6. The variation of Smτ against energy for one point fit with Ltot = 45 pb−1 .
sufficient to provide the accuracy less than 0.1 MeV. 3. Systematic uncertainty In this section the following effects on mτ measurements will be examined, including theoretical accuracy, energy spread, energy scale, luminosity, efficiency, background analysis, and so forth. 3.1. Theoretical accuracy The experimentally measured cross section has the form [21] ∞ √ √ √ σexp (mτ , s, Δ) = d s G( s , s) 0
×
0
4m2 1− sτ
dxF (x, s )
σ ¯ (s (1 − x), mτ ) , |1 − Π(s (1 − x))|2
(7)
2 . Here F (x, s) is the initial state with s = Ecm radiation factor [21], Π is the√vacuum √ polarization factor [16,22,23], and G( s , s), which is usually treated as a Gaussian distribution [24], depicts the energy spread (Δ) of the e+ e− collider. As mentioned in section 1, the high accurate calculations for the production cross sec-
tion (¯ σ ) are available recently. Anyway, the improved Voloshin’s formulas [13] are adopted, since the previous production cross section (denoted as σ ¯ ∗ ) used in BES mτ fit is calculated based on Voloshin’s formulas [20]. The relative differences of cross sections calculated with two sets of Voloshin’s formulas are drawn in Figure 7. To estimate the effect due to theoretical calculation accuracy, the fitted τ masses by using two kinds of formulas are compared. The comparison shows the uncertainty due to this effect is at the level of 10−3 MeV. 3.2. Energy spread As indicated in Eq. (7), the experimentally measured cross section σexp depends on the energy spread (Δ), whose effect on mτ fit is to be considered here. In fact, Δ at τ mass region is often deduced by virtue of the energy spreads measured at the J/ψ (ΔJ/ψ ) and ψ (Δψ ) regions; and derived according to the interpolation formula Δ − ΔJ/ψ f (E) − f (EJ/ψ ) , = Δψ − ΔJ/ψ f (Eψ ) − f (EJ/ψ )
(8)
with assumption Δ ∝ f (E). Here f (E) denotes the dependence of Δ on energy Ecm (subscript “cm” has been suppressed for brevity). Never-
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where Es indicates the scaled energy value. If the difference between measured and nominal quantities is small, that is E = M + δ, where δ is a small variable and estimated at the level of 10−4 . Under such case, the relation is readily obtained
0.6
0.4
Es − MJ/ψ δs − δJ/ψ = . Mψ − MJ/ψ δψ − δJ/ψ
0.3 0.2 0.1 0 3.54 3.55 3.56 3.57 3.58 3.59
3.6
Ecm (GeV)
Figure 7. The relative differences of cross sections calculated with different sets of Voloshin’s formulas [18]. Here Δσ = σ − σ ∗ , where σ and σ ∗ can be σexp (denoted by the solid line) or σ ¯ (denoted by the dashed line).
theless, since the knowledge of concrete form of function f is lack , a generic form is assumed: f (E) = a · E + b · E 2 + c · E 3 ,
(9)
and the linear, quadric, cubic or mixing type of energy-dependence form is tested to estimate the uncertainty, that is the following parameter values are taken: a = 1, b = 0, c = 0; a = 0, b = 1, c = 0; (10) a = 0, b = 0, c = 1; a = 1, b = 1, c = 1. The fit results give δmτ < 1.5 × 10−3 MeV. Even if Δ is artificially changed to 3Δ, and the repeat fits indicate that δmτ < 6 × 10−3 MeV.
(12)
Similar to the uncertainty study of the energy spread, assuming δ ∝ f (E), where f (E) expressed by Eq. (9) and parameters take values given in Eq. (10), it is found that the uncertainty due to the energy scale is about 8 × 10−3 MeV. Anyway, the relation δ ∝ f (E) is merely a theoretical surmise, not an empirical formula. The most direct approach to determine the energy scale is to measure it experimentally. At present, two methods have used to determine the absolute energy value accurately, one is depolarization method and the other is Compton backscattering method, both of which have been utilized by KEDR group [10].
0.2 0.1
Δmτ (MeV)
Δσ/σ (%)
0.5
0 -0.1 -0.2 -0.2
-0.1
0
0.1
ΔE (MeV)
0.2
3.3. Energy scale As to the energy scale, PDG mass values of the J/ψ (MJ/ψ ) and the ψ (Mψ ) are adopted as the standard to scale the energies measured at J/ψ (EJ/ψ ), ψ (Eψ ), and τ (Eτ ) mass regions, viz.
Figure 8. The effect due to the uncertainty of absolute energy calibration on mτ measurement.
Es − MJ/ψ Eτ − EJ/ψ = , Eψ − EJ/ψ Mψ − MJ/ψ
It is proposed to adopt Compton backscattering technique to measure the energy at BE-
(11)
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SIII, the relative accuracy of such technique is expected at the level of 5 × 10−5 . If this technique can be realized at BESIII, the systematic uncertainty can be expected at this level, that is 5 × 10−5 (relative error) or around 9 × 10−2 MeV (absolute error), which is one order of magnitude larger than 8 × 10−3 MeV. Since the uncertainty of energy scale will transfer to the final fit result of mτ directly and linearly as displayed in Figure 8, the absolute calibration of energy scale is actually a bottleneck problem in accuracy improvement for mτ measurement. 3.4. Other factors The systematic uncertainties due to other experimental factors are quantified by reasonable variations of the corresponding quantities (2% variation for luminosity L, 2% for efficiency , 0.5% for branching fraction Beμ , 10% for background σBG ), the total uncertainty due to all these factors is estimated to be less than 2.02 × 10−2 MeV. The summary of possible systematic uncertainties are presented in Table 2, in the light of which the summed uncertainty is 2.3×10−2 MeV, or the relative one is 1.3 × 10−5 . Here the uncertainties due to luminosity and efficiency dominate.
Table 2 Some possible systematic uncertainties. Source
δmτ ( 10−3 MeV)
δmτ /mτ ( 10−6 )
Theoretical accuracy Energy spread Energy scale Luminosity Efficiency Branching Fraction Background
3 6 8† 14 14 3.5 1.7
1.7 3.4 4.5† 7.9 7.9 2.0 1.0
Summation 22.7 12.7 †: based on the estimation of Eq. (12).
4. Discussion At BESIII the designed peak luminosity is around 1 nb−1 s−1 , if the average efficiency of luminosity is taken as 50% of the peak value, then two days data taking time will lead to the statistic uncertainty less than 0.1 MeV. Notice this evaluation is solely for eμ-tagged event, if more channels are utilized to tag τ -pair final state, such as ee, eμ, eh, μμ, μh, hh (h : hadron), and so on, more statistics can be expected (according to previous BES analyses [6,25] experience, the number of multi-channel-tagged events is at least 5 times more than that of eμ-tagged event). Therefore at BESIII, one week data taking time will lead to the statistical uncertainty less than 0.017 MeV. Comparatively, even if the Compton backscattering technique is adopted at the new detector, the systematic uncertainty is merely up to 0.09 MeV. On this extent, the final accuracy of mτ measurement at BESIII will be mainly determined by the systematic uncertainty. At last, a few words about the data taking design for BESIII. To measure the mτ , at least three points are needed. One is at the large derivative region, which is the optimal position as our study indicates; the other is below the threshold, which can be used for background study; and the third one is above the threshold, which can be used to determine the event selection criteria for data analysis at the optimal point. In fact, the data below and above threshold can also be used for R-value measurement and background study for J/ψ and ψ data analyses. So it is reasonable to consider the mτ measurement between the period of data taking for the J/ψ and ψ . 5. Summary The possible statistical and systematic uncertainties of mτ measurement at the forthcoming detector BESIII are studied based on previous experiences and recent theoretical calculations. Monte Carlo simulation and sampling techniques are employed to obtain optimal data taking point for high accurate mτ measurement. It is found: 1. The optimal position is located at the region with the large derivative of cross sec-
X.H. Mo / Nuclear Physics B (Proc. Suppl.) 169 (2007) 132–139
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