Measurement of the east–west asymmetry of the cosmic muon flux in hanoi

Nuclear Physics B 678 (2004) 3–15 www.elsevier.com/locate/npe Measurement of the east–west asymmetry of the cosmic muon flux in Hanoi Pham Ngoc Diep a , Pham Ngoc Dinh a , Nguyen Hai Duong a,b , Pham Thi Tuyet Nhung a , Pierre Darriulat a , Nguyen Thi Thao a , Dang Quang Thieu a , Vo Van Thuan a a VATLY, Institute for Nuclear Science and Technology, 5T-160 Hoang Quoc Viet, Nghia Do, Cau Giay, Hanoi, Viet Nam b Department of Nuclear Physics, University of Natural Sciences, Vietnam National University, 227 Nguyen Van Cu, Dist 5, Ho Chi Minh, Viet Nam Received 21 November 2003; accepted 24 November 2003 Abstract The east–west asymmetry of the cosmic muon flux has been measured in Hanoi where the geomagnetic rigidity cut-off reaches its maximal value of 17 GV The measurement was made using an orientable scintillator telescope that had been previously used to measure the zenith angle distribution of the cosmic muon flux in the direction of the geographic north The data exhibit a clear east–west asymmetry in good agreement with the predictions of a model widely used in the analysis of atmospheric neutrino oscillation studies  2003 Elsevier B.V All rights reserved Introduction In recent publications [1,2] measurements of the cosmic muon flux performed in Hanoi have been reported Its dependence over zenith angle θ was found to be well described by a form Φ(θ ) = a cos2 θ × (1 − b sin2 θ ) with a = 72.0 ± 1.6 m−2 sr−1 s−1 and b = 0.108 ± 0.011 Moreover the results of these measurements were found in excellent agreement with the predictions of a one-dimensional model [3] commonly used in the analysis of the Super-Kamiokande data that have revealed the existence of atmospheric neutrino oscillations [4] E-mail address: darriulat@mail.vaec.gov.vn (P Darriulat) 0550-3213/$ – see front matter  2003 Elsevier B.V All rights reserved doi:10.1016/j.nuclphysb.2003.11.035 RAPID COMMUNICATION P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 As primary cosmic rays and atmospheric nuclei are both positively charged, a charge asymmetry exists among the hadronic constituents of atmospheric cosmic showers and therefore among the muons into which they may decay The earth magnetic field (Hanoi is located in the region of the geomagnetic equator, 21 ◦ N, 105.5 ◦E, the telescope is located 12 m above sea level) points to the south and bends primary positive particles toward the east, resulting in an east–west asymmetry of the muon flux itself This asymmetry being sensitive to the momentum spectra of positive and negative cosmic muons, its measurement is of obvious interest The same orientable scintillator telescope that was used in the measurements reported in [2] has now been used to study the azimuthal dependence of the cosmic muon flux As the hardware and the method of data analysis are identical in both sets of measurements, it will be sufficient to briefly recall the main features in Section where the measurement of the cosmic muon rate is presented The evaluation of the muon flux is discussed in Section and comparisons with earlier flux measurements [1,2] and with the prediction of a new (three-dimensional) version of the Honda model [5] are made in Section Measurement of the cosmic muon rate 2.1 Detection Data were collected over a period of three months between February 10 and May 9, 2003 They include measurements of the zenith angle distributions in the geographic west (ϕ = 270◦ ) and east (ϕ = 90◦ ) directions and measurements of the azimuthal distributions at zenith angles of 50◦ and 65◦ where large asymmetries are observed Each measurement, corresponding to a fixed direction of the telescope axis, extended over a full 24 hours day, from one morning to the next A number of additional measurements were made that served as consistency checks The telescope (Fig 1) is made of three coaxial pairs of scintillator plates, the front pair being separated from the back pairs by a distance of 190 cm The back pairs sandwich a cm thick iron plate used to help electron rejection The scintillators of a same pair are each cm thick and are viewed by a 2′′ photomultiplier tube through a square (40 × 40 cm2 ) lucite light guide from opposite ends The scintillator plates are 40 × 80 cm2 in area, corresponding to typical apertures of θ = ±4.5◦ and ϕ = ±9◦ / sin θ Data are collected whenever a fourfold coincidence between the signals of the four front scintillator plates (excluding the pair located behind the iron plate) is recorded The low value of the discriminator thresholds (about 0.1 × minimum ionization) and the relatively broad discriminator signals (15 ns) allow for full trigger efficiency while keeping accidental rates to a negligible level The integrated charges and times of arrival of the phototube signals are recorded for each event 2.2 Data reduction and analysis The detected cosmic muon rate Rµ is calculated as the product of the trigger rate Rtrig by the fraction of trigger events identified as genuine muons The trigger rate is obtained from the time intervals separating successive triggers, measured using a 10 kHz clock RAPID COMMUNICATION P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 (a) (b) Fig Schematic telescope assembly: (a) artist view of the orientable ensemble and (b) schematic scintillator arrangement Within statistical errors their distributions are observed to be exponential with equal mean and rms values The selection of genuine muon events is made in three steps • In a first step a small background contamination associated with particles slightly outside the telescope acceptance but splashing from the back (or front) scintillator pair onto the front (or back) pair, is eliminated by the requirement that the time of flight between the front and back pairs be more than 3.5 ns (the average time of flight for genuine muons is ∼ ns) in association with the requirement that the mean pulse height in the four front scintillator plates should not exceed 1.5 minimum ionizing equivalent (mip) The fraction ρgood of trigger events retained as good events is of the order of 90%; • In a second step untagged good events, namely good events having no significant signals in the back plates (behind the iron converter), are rejected The fraction ρtag of good events retained as tagged events is of the order of 78%; • In a third step tagged good events are required to have a mean pulse height in the back plates (behind the iron converter) smaller than mip The fraction ρsel of tagged events obeying this selection criterion is of the order of 96% A fourth step is necessary to correct the selected event samples for a small remaining electron and/or hadron contamination associated with mean pulse heights in the back plates smaller than mip The procedure followed in the evaluation of this correction is described in [2] It uses the result of a measurement at vertical incidence made with a 10 cm thick lead filter in front of the back scintillator pairs The correction, simply evaluated from ρsel as δel = 3.6 × (0.974 − ρsel ), increases from ∼ 6% at small angles (θ < 45◦ ) to ∼ 9% at θ = 65◦ The resulting muon rates, Rà = Rtrig ì good × ρtag × ρsel × (1 − δel ), are listed in Table RAPID COMMUNICATION P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 Table Summary of the measurements θ (deg) ϕ (deg) Rµ (Hz) Jexp (θ, ϕ) ◦ Zenith angle distribution at ϕ = 90 (east) JH (θ, ϕ) 10 15 20 25 30 35 40 45 50 50 60 65 90 90 90 90 90 90 90 90 90 90 90 90 90 1.429 1.346 1.288 1.181 1.080 0.985 0.854 0.713 0.607 0.515 0.509 0.309 0.223 70.02 ± 0.58 68.51 ± 0.59 68.21 ± 0.60 66.59 ± 0.60 65.88 ± 0.61 66.18 ± 0.63 65.05 ± 0.73 63.91 ± 0.69 62.92 ± 0.80 64.57 ± 0.79 64.34 ± 0.80 64.57 ± 0.97 65.08 ± 1.19 70.86 69.45 68.79 68.16 67.57 67.03 66.59 66.28 66.13 66.16 66.16 66.84 67.47 0 10 15 20 25 30 35 45 50 65 270 270 270 270 270 270 270 270 270 270 270 Zenith angle distribution at ϕ = 270◦ 1.412 1.453 1.397 1.395 1.271 1.217 1.121 0.979 0.721 0.604 0.275 (west) 69.05 ± 0.62 71.68 ± 0.61 71.29 ± 0.61 74.00 ± 0.63 72.03 ± 0.64 72.94 ± 0.80 75.46 ± 0.72 73.47 ± 0.80 73.90 ± 0.91 75.96 ± 1.05 78.37 ± 1.58 70.86 70.86 72.94 73.99 75.02 76.01 76.94 77.81 79.29 79.85 80.41 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 15 30 45 60 75 90 90 105 105 120 135 150 165 180 195 210 225 240 Azimuth distribution at θ = 50◦ 0.566 70.99 ± 0.97 0.553 69.35 ± 0.94 0.535 66.70 ± 0.86 0.552 69.23 ± 0.94 0.515 65.17 ± 0.81 0.520 65.24 ± 0.81 0.509 64.34 ± 0.80 0.515 64.57 ± 0.79 0.511 64.49 ± 0.80 0.522 64.76 ± 0.81 0.514 64.98 ± 0.81 0.530 66.73 ± 0.83 0.530 67.33 ± 0.84 0.548 69.09 ± 0.86 0.564 70.55 ± 0.88 0.575 71.82 ± 0.90 0.585 72.78 ± 0.92 0.586 73.07 ± 0.90 0.593 74.56 ± 0.98 75.55 73.68 71.69 69.74 68.06 66.82 66.16 66.16 66.16 66.16 66.81 68.02 69.63 71.46 73.32 75.08 76.61 77.85 78.81 (continued) RAPID COMMUNICATION P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 Table (continued) θ (deg) ϕ (deg) Rµ (Hz) Jexp (θ, ϕ) JH (θ, ϕ) 50 50 50 50 50 50 50 255 270 285 300 315 330 345 0.604 0.604 0.601 0.613 0.603 0.589 0.565 75.25 ± 0.97 75.96 ± 1.05 75.35 ± 0.93 75.99 ± 0.99 74.81 ± 0.91 73.19 ± 0.90 72.37 ± 1.05 79.47 79.85 79.97 79.78 79.27 78.40 77.15 65 65 65 65 65 65 65 65 65 75 90 105 135 165 195 225 255 270 Azimuth distribution at θ = 65◦ 0.224 64.92 ± 1.19 0.223 65.08 ± 1.19 0.222 64.17 ± 1.18 0.222 64.67 ± 1.19 0.240 69.90 ± 1.27 0.250 72.35 ± 1.27 0.257 74.01 ± 1.34 0.261 75.11 ± 1.46 0.275 78.37 ± 1.58 67.92 67.47 67.60 69.47 72.71 76.11 78.71 80.14 80.41 The experimental and predicted values of J , measured in units of m−2 sr−1 s−1 , are shown as a function of zenith angle θ and azimuth ϕ The quoted uncertainties are point-to-point uncertainties including the angular setting error, the statistical error and a 0.8% error added in quadrature (see text) Additional uncertainties of ±2.2% (global) and of ±(1 − cos θ ) × 2.3% (point-to-point) are not listed (see text) 2.3 Uncertainties As the solid angle acceptance of the telescope is virtually independent of the direction in which it is pointing, flux asymmetry measurements are in principle affected by only minor sources of uncertainties, namely statistical fluctuations and inaccurate settings of the direction of the telescope axis Between 30 and 200 kevents were recorded at each angular setting and the direction of the telescope axis was measured with an accuracy of ±0.2◦ in zenith angle and ±1◦ in azimuth, resulting in point-to-point uncertainties at the percent level In particular at vertical incidence where the angular setting uncertainty is negligible and where the statistical uncertainty is very small (largest counting rate) the point-to-point uncertainty is of the order of only permil Yet, the reproducibility of the measurements is not that good For example, the χ describing the consistency of the three vertical incidence measurements listed in Table is 21 for degrees of freedom (dof), suggesting that the uncertainties taken into account are too low by a factor of three, namely that there exists a source of uncertainties at the to permil level that was not taken in consideration Such uncertainties may be instrumental or reflect real fluctuations in the muon rate One such source of fluctuations is the variation of the atmospheric pressure with time Its effect on the cosmic muon rate has been studied using earlier data [2] with the result [6] that the cosmic muon rate decreases by 1.5 permil when the ground atmospheric pressure increases by mbar Dependence on ground temperature was found to be nearly fully accounted for by the correlation that exists between temperature and pressure As the RAPID COMMUNICATION P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 rms deviation of typical atmospheric pressure fluctuations is of the order of mbar one expects muon rate fluctuations at the permil level from that source alone Instrumental fluctuations at the level of some permil may result from small instabilities in the data acquisition electronics, in the photomultiplier gains and in the scintillation yields, that all depend on temperature They may also result at a similar level from small deformations of the telescope support frame when its orientation is being changed There are therefore good arguments to attach an additional point-to-point uncertainty of permil to the rate measurements This has been done and the uncertainties listed in Table have been obtained by adding in quadrature for each measurement the statistical uncertainty, the angular setting uncertainty and a 0.8% uncertainty meant to account for the effects that have just been discussed In addition to the point-to-point uncertainties listed in Table there exist also other sources of uncertainties: an overall scale uncertainty of 2.2% and additional point-to-point uncertainties that are attached to the procedure of electron and hadron subtractions and amount to (1 − cos θ ) × 2.3% Both were discussed in detail in [2] They should be ignored when comparing the present data with the 2002 data [2] as they are common to both sets In the next section, when comparing the present data to earlier data or to model predictions, care will be taken to clearly specify which are the uncertainties that need to be taken into account in each case Evaluation of the cosmic muon flux The evaluation of the muon flux from the measurements of the muon rates presented in the preceding section requires the knowledge of the detector acceptance For convenience, in presenting and discussing the data, we introduce a function J (θ, ϕ), where θ is the zenith angle and ϕ the azimuth, that is the ratio between the muon flux in direction (θ, ϕ) and the value taken by the expression cos2 θ × (1 − b sin2 θ ) introduced above J (θ, ϕ) has the advantage of being nearly constant over the region of the sky covered here, it is expected to be of the order of 70 m−2 sr−1 s−1 , and therefore is best suited to display azimuthal oscillations and to reveal possible disagreements with earlier measurements and/or with model predictions The detector acceptance is calculated from a simple Monte Carlo simulation of the detector geometry that has been shown to be reliable at the 1.5% level [1,2] The geometric acceptance is of the order of 2.22 × 10−2 m2 sr and the efficiency of the selection criteria is between 91% and 92% Muons are generated with an incoming flux of the form cos2 θ × (1 − b sin2 θ ) × m−2 sr−1 s−1 and the simulated detected rate, RMC (θ0 , ϕ0 ), is evaluated for each direction (θ0 , ϕ0 ) of the telescope axis It is then straightforward to transform the muon rate Rµ measured with the telescope axis pointing in direction (θ, ϕ) into an experimentally measured value of J (θ, ϕ) by simply dividing Rµ by RMC (θ, ϕ) The values Jexp (θ, ϕ) obtained in this manner are listed in Table with their associated point-to-point uncertainties Similarly, the flux predictions [5] of the Honda model, that are available in the form of an array of 10×18 solid angle bins of equal sizes, namely 10 bins in cos θ and 18 in ϕ, are easily translated in terms of J (θ, ϕ) from the average value taken by cos2 θ × (1 − b sin2 θ ) RAPID COMMUNICATION P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 in each bin As the average value taken by J (θ, ϕ) in each bin displays only small (less than ±9% on average) variations over the sky coverage of the present experiment, it is easy to calculate by interpolation the predicted value JH (θ, ϕ) taken by J (θ, ϕ) in any direction (θ, ϕ) These are listed in Table The statistical uncertainties attached to JH (θ, ϕ)are very small and the systematic uncertainties are expected to be in the region of a percent [7] A 9parameter parameterization of JH (θ, ϕ) shows that the level of fluctuation that subsists with respect to a smooth function amounts to ±0.25 m−2 sr−1 s−1 , namely ±0.35%, on average Independently from the validity of the model used in the simulation, these numbers give an estimate of the uncertainties having a purely technical origin Comparison of the results with earlier measurements and with the Honda model prediction 4.1 General remarks When comparing the results of the present experiment with earlier measurements and/or with model predictions a number of points need to be kept in mind • The material that must be traversed by a muon to survive in the selected event sample corresponds [2] to a 120 MeV c−1 momentum cut-off The experimental fluxes listed in Table have not been corrected for this effect and the predicted fluxes have been obtained by integrating the predicted muon momentum spectra from 120 MeV c−1 upward The data in Table 1, both Jexp (θ, ϕ) and JH (θ, ϕ), can therefore be compared directly but when comparing them with the results presented in [1,2], that were for zero momentum cutoff, a correction has to be made Rather than correcting the present data to include muons below 120 MeV c−1 momentum we prefer to correct the earlier data, in particular to lower the value quoted for a in [2] by the 1.7% correction that had been applied to it, namely a = 70.8 (instead of 72.0) ± 1.6 m−2 sr−1 s−1 This way all comparisons will be made for a momentum cut-off of 120 MeV c−1 • In [2] a small correction for solar activity was taken into account when comparing the vertical flux measured in [1] with that measured in [2] It was brought to our attention by Honda [7] that the solar spot data [8] that we had been using are less reliable for that purpose than the neutron monitor data available from the Chicago–New Hampshire Collaboration [9] The predictions of the Honda model used here have therefore been calculated for maximum solar activity as there has not yet been any significant increase of the monitored neutron rates since 2001 It is therefore preferable not to apply any solar activity correction to the experimental data and to keep that in mind when comparing measurements performed in different periods In particular the value of the vertical flux measured in [1] that should be compared with the present data is, after inclusion of the 120 MeV c−1 cut-off and using a cos2 law for the extrapolation to vertical incidence, 71.3 ± 2.8 m−2 sr−1 s−1 • The measurements presented in [1,2] were made at different periods of the year than the present measurements (April 2001 for [1], October to December 2002 for [2] and February to May 2003 for the present data) In addition to the solar activity effects RAPID COMMUNICATION 10 P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 mentioned above, one might think that seasonal differences in the average atmospheric pressure might cause detectable variations in the cosmic muon rate However the 0.8% point-to-point uncertainty that was discussed in Section 2.3 should cover such effects • In both experimental data and model predictions the geographical south has been taken as origin of the azimuth scale, with 0◦ pointing to the south, 180◦ to the north, 90◦ to the east and 270◦ to the west In Hanoi the magnetic declination currently amounts to 1.5 ◦ W 4.2 Comparison with earlier data With the above remarks in mind, it is now possible to compare the results of the present measurements with the results presented in [1,2] Without solar activity corrections and for a 120 MeV c−1 momentum cut-off, the values taken by the vertical muon flux in units of m−2 sr−1 s−1 are 71.3 ± 2.8 in April 2001, 70.9 ± 0.6 in October–December 2002 and 70.3 ± 0.4 in February–May 2003, the latter two (2002 and 2003) being subject in addition to a common scale uncertainty of ±2.2% In the comparison between the 2002 and 2003 data only the point-to-point uncertainties listed in Table must be taken into account as the detector and the data collection, reduction and analysis were identical The experimental values are equal within errors, their weighted average being 70.4 ± 0.3 (the 2.2% scale uncertainty being ignored) In 2001, however, the detector, the acceptance, the method of background subtraction were all very different and only some small sources of uncertainties were common to the telescope measurements When comparing the three vertical measurements, they are again equal within errors with a weighted average of 70.6 ±1.4 (this time taking the scale uncertainties in due account) Therefore, at the level of sensitivity of the three measurements, there has been no significant variation of the vertical cosmic muon rate over the two year period between Spring 2001 and Spring 2003 Zenith angle distributions were measured in 2002 and 2003 only, in 2002 with the telescope pointing to the north and in 2003 with the telescope pointing to the west or to the east The 2002 data, JN = J (θ, 180◦), are listed in Table and displayed in Fig 2(a), together with the 2003 east–west averages JEW = 12 [J (θ, 270◦) + J (θ, 90◦ )] The 2002 (JN ) and 2003 (JEW ) data appear to be in good agreement with a χ of 13 for 10 dof (using point-to-point uncertainties of the type listed in Table 1) Note however that they have no reason to be exactly equal as the geographic north is not exactly at the node of the azimuthal oscillation The best fit of the 2003 zenith angle distribution to a form JEW (θ, ϕ) = cte × (1 − b sin2 θ ) gives b = 0.9 ± 1.5%, providing no evidence for a zenith angle dependence significantly different from that measured [2] for JN (corresponding to b = ± 1.1%) In this latter fit electron-hadron subtraction point-topoint uncertainties, (1 − cos θ ) × 2.3%, have also been included (in quadrature) 4.3 Comparison with the predictions of the Honda model The predictions of the Honda model [5] are listed in Table The predicted vertical incidence value is 70.9 m−2 sr−1 s−1 in excellent agreement with the experimental best fit value of 70.6 ± 1.4 m−2 sr−1 s−1 reported above RAPID COMMUNICATION P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 11 Table Zenith angle dependence of the muon flux and east-west asymmetry θ (deg) 10 15 20 25 30 35 40 45 50 55 60 65 70 Jexp N (2002) JH N Jexp EW (2003) JH EW Aexp (%) AH (%) 70.9 ± 0.6 70.5 ± 0.6 69.9 ± 0.7 70.4 ± 0.6 69.9 ± 0.6 70.3 ± 0.7 69.4 ± 0.6 70.5 ± 0.8 71.6 ± 0.9 70.8 ± 0.8 71.2 ± 0.9 72.5 ± 0.9 71.9 ± 1.2 72.6 ± 1.5 68.6 ± 1.6 70.9 70.7 70.5 70.6 70.8 71.1 71.5 71.9 72.4 72.9 73.3 73.7 74.1 74.5 70.3 ± 0.4 70.9 71.0 71.2 71.4 71.5 71.7 71.9 72.1 72.4 72.7 73.0 73.3 73.6 73.9 74.2 0.0 ± 1.5 0.0 2.5 4.9 7.4 9.8 12.0 14.0 15.7 17.0 18.1 18.6 18.9 18.5 17.5 15.8 69.9 ± 0.6 71.1 ± 0.6 69.3 ± 0.6 69.4 ± 0.6 70.8 ± 0.6 69.3 ± 0.7 68.4 ± 0.7 70.2 ± 0.7 71.8 ± 1.1 4.0 ± 1.7 8.2 ± 1.6 7.8 ± 1.7 10.1 ± 1.6 13.1 ± 1.6 12.1 ± 1.7 16.1 ± 1.8 16.4 ± 1.6 18.5 ± 2.2 Experimental and predicted values of J are measured in units of m−2 sr−1 s−1 All listed uncertainties are pointto-point uncertainties of the type listed in Table The predicted zenith angle distributions are listed in Table and displayed in Fig 2(a) for both JN and JEW Both are seen to increase with zenith angle Indeed, fits of the form J (θ, ϕ) = cte × (1 − b sin2 θ ) give b = −7.0% for JN and −4.9% for JEW The one-dimensional version of the Honda model [3] was not predicting such an increase, the value of b found for JN in that case was only 0.5% When comparing them with the experimental data a 2.2% overall scale uncertainty and additional (1 − cos θ ) × 2.3% point-to-point uncertainties need to be taken into account as discussed earlier The best fit to the present data, evaluated by taking these in proper account, is obtained by shifting the experimental values (of both JN and JEW ) globally up by 1.9%, as displayed in Fig 2(a) The χ value of the fit is 43 for 20 degrees of freedom, reflecting the slight but significant steeper decrease of the experimental data with respect to the model prediction The prediction of the one-dimensional version of the Honda model [3], to which the 2002 data had been compared in [2], having a slightly flatter zenith angle dependence, was giving a good fit While significant, the disagreement is quite small: between 0◦ and 60◦ the data decrease by a factor 4.36 and the prediction by a factor 4.19, barely 4% less than observed Fig 2(b) shows a comparison between the measured and predicted values of the difference JN − JEW This quantity is sensitive to the phase shift of the azimuthal oscillation with respect to the geographic north The agreement between experiment and model prediction is reasonable, with a χ of 13 for 10 dof (using Table type uncertainties) The east–west asymmetry, A = [J (θ, 270◦) − J (θ, 90◦)]/JEW is displayed in Fig 2(c) and listed in Table as a function of zenith angle for both Jexp and JH The model prediction is found to be in good agreement with the experimental data, with a χ of 11 for 10 degrees of freedom (again using Table type point-to-point uncertainties exclusively) Finally the azimuthal distributions of both data and model prediction are compared in Fig at zenith angles of 50◦ (Fig 3(a)) and 65◦ (Fig 3(b)) The electron–hadron RAPID COMMUNICATION 12 P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 Fig (a) The east–west average of Jexp (θ, ϕ), JEW (θ ) = 21 [Jexp (θ, 90◦ ) + Jexp (θ, 270◦ )], (full circles) is compared with the north data [2] collected in 2002, JN (θ ) = Jexp (θ, 180◦ ), (open circles) and with the prediction [5] of the Honda model (full line and dashed line respectively) The experimental data have been scaled up by 1.9% as demanded by the result of the best fit (see text) The error bars shown on the experimental points are the point-to-point uncertainties of the type listed in Table added in quadrature with the electron–hadron subtraction uncertainties, (1 − cos θ ) × 2.3% (b) The difference J (full circles) between the zenith angle distribution measured in the north direction in 2002 [2] and the average east–west distribution measured in 2003 (this work) is compared with the prediction [5] of the shower model (full line) The error bars correspond to point-to-point uncertainties of the type listed in Table (c) The measured zenith angle dependence of the east–west asymmetry of the cosmic muon flux, A(θ ) = [J (θ, 270◦ ) − J (θ, 90◦ )]/JEW (θ ) (full circles) is compared to the prediction [5] of the Honda model (full line) The error bars shown on the experimental points correspond to the point-to-point uncertainties listed in Table subtraction is not expected to have much azimuthal dependence and it is proper to ignore it as a point-to-point uncertainty when comparing data at a same zenith angle However it contributes an additional scale uncertainty of 0.8% at 50◦ and 1.3% at 65◦ This was taken in due account by fitting a scale factor common to both data sets of the form 1+λ+µ(1−cos θ ), λ and µ being constrained to within ±2.2% and ±2.3% respectively The result, λ = 3.7% and µ = 2.1%, has been used to scale the experimental data up in Fig The uncertainties shown as error bars are the point-to-point uncertainties listed in Table The χ of the fit is 53 for 35 dof A better fit, with a χ of only 26, is obtained RAPID COMMUNICATION P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 13 Fig The azimuthal dependence of Jexp (θ, ϕ) (full circles) is compared to the prediction [5] JH (θ, ϕ) of the Honda model (full line) for two sets of data (a) at θ = 50◦ , (b) at θ = 65◦ The data have been scaled up by 3.7 + 2.1(1 − cos θ )% as demanded by the result of the best fit (see text) The error bars correspond to the point-to-point uncertainties listed in Table by allowing for an azimuthal phase shift ϕ and for a smaller oscillation amplitude (by a fraction ǫ) The best fit gives ϕ = ± 3◦ and ǫ = 16 ± 4%: while there is no significant azimuthal phase shift between data and model prediction, the predicted amplitude of the azimuthal oscillation is slightly smaller in the data than in the model 4.4 Summary The present experiment concludes a series of measurements of the cosmic muon flux made in Hanoi They measure the sum of the fluxes of positively and negatively charged muons, integrated above a momentum cut-off of 120 MeV c−1 The main results can be summarized as follows The vertical muon flux has not varied within errors between April 2001 and May 2003 and is equal to 70.6 ± 1.4 m−2 sr−1 s−1 in excellent agreement with the prediction [5] of the Honda model, 70.9 m−2 sr−1 s−1 RAPID COMMUNICATION 14 P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 The zenith angle distributions were measured toward the geographic north in 2002 and toward the geographic east and west in 2003 The north and east–west average distributions are found equal within errors Overall, the best fit to the model prediction is obtained by scaling the experimental data up by 1.9% (the global scale uncertainty, common to the 2002 and 2003 data, being ±2.2%) Fits of the north and east–west average experimental flux distributions to a form cos2 θ × (1 − b sin2 θ ) give bN = 10.8 ± 1.1% [2] and bEW = 11.7 ± 1.7% While consistent with the prediction of the one-dimensional Honda model used in Ref [2], bN =11.3%, they have a slightly steeper zenith angle dependence than predicted [5] by the three-dimensional version of the model, bN =3.8% and bEW =5.9% The difference between the north and east–west average distributions is very small in both experimental data and model prediction The east–west asymmetry is measured to increase from at θ = 0◦ (trivially) to nearly 20% at θ = 60◦ , in very good agreement with the prediction [5] of the Honda model as shown in Fig 2(c) Azimuthal distributions have been measured at θ = 50◦ and θ = 65◦ Here again, as can be seen from Fig 3, the agreement with the model prediction is qualitatively excellent However, while data and model are well in phase, the measured amplitude of the oscillation is slightly smaller than predicted (by 16 ± 4%) In summary, the Hanoi measurements, being performed on the geomagnetic equator at a place where the rigidity cut-off is maximal, are a useful test of the validity of the Honda model The present results on the east–west asymmetry, and more generally on the azimuthal dependence of the muon flux, are of particular value as they are sensitive to the momentum spectrum of the muons, the earth magnetic field being used as a spectrometer, as well as to their charge ratio Overall, the agreement between the measured fluxes and their predicted values is excellent Two small but significant differences were however revealed: first, at variance with the one-dimensional version [3] of the Honda model used in Ref [2] that gave a good fit to the zenith angle distribution the three-dimensional version [5] predicts a slightly steeper dependence resulting in a 4% effect at 60◦ Second, the predicted amplitude of the azimuthal oscillation is slightly larger, by 16 ± 4%, than experimentally observed Acknowledgements We are deeply indebted to Professor M Honda for his keen interest in our work, for his scientific advice and for having produced for us files of Monte Carlo predictions tailored to the conditions of the present experiment One of us (D.Q.T) acknowledges financial support from the Rencontres du Vietnam Three others (P.N Diep, P.N Dinh and P.T.T.N.) acknowledge financial support from Odon Vallet fellowships Professors J.W Cronin, Tran Thanh Van and A Watson have played a seminal role in the genesis of VATLY that we are very pleased to acknowledge This work could not have been done without major hardware contributions from various groups and institutes, in particular at CERN and RIKEN We are particularly indebted to RAPID COMMUNICATION P Ngoc Diep et al / Nuclear Physics B 678 (2004) 3–15 15 Professors L Camilleri, G Goggi, L Mapelli, J Panman, D Schlatter, P Schlein and A Yoshida We thank the Pierre Auger Collaboration, and in particular their spokesman Professor A Watson, for their constant interest and support We are grateful to Professors Tran Thanh Van and Nguyen Van Hieu for financial support from the Rencontres du Vietnam and from the Natural Science Council of Vietnam References [1] P.N Dinh, et al., Nucl Phys B 627 (2002) 29 [2] P.N Dinh, et al., Nucl Phys B 661 (2003) [3] M Honda, et al., in: Proceedings of the 2001 Int Cosmic Ray Conf., ICRC 2001, Copernicus Gesellschaft, Hamburg, Vol 3, 2001, p 1162, and references therein [4] Super-Kamiokande Collaboration, Phys Rev Lett 85 (2000) 3999, and references therein [5] M Honda, et al., in: Proceedings of the 2003 Int Cosmic Ray Conf., ICRC 2003, Tsukuba, Japan, 2003, p 1415; M Honda, et al., in preparation [6] P.N Diep, et al., Dependence of the cosmic muon flux on atmospheric pressure and temperature, Vietnam Commun Phys (2003), in press [7] M Honda, et al., private communication [8] J.A Joselyn, et al., Solar cycle 23 project, http://science.msfc.nasa.gov/ssl/pad/solar/predict.htm; E.W Cliver, A.G Ling, Astrophys J Lett L189 (2001) 551 [9] The University of New Hampshire/EOS and Chicago/LASR, Cosmic Physics Instruments in Space, http://ulysses.sr.unh.edu/NeutronMonitor/Misc/neutron2.html  ... are the uncertainties that need to be taken into account in each case Evaluation of the cosmic muon flux The evaluation of the muon flux from the measurements of the muon rates presented in the. .. (full line) The error bars correspond to point-to-point uncertainties of the type listed in Table (c) The measured zenith angle dependence of the east–west asymmetry of the cosmic muon flux, A(θ... Summary The present experiment concludes a series of measurements of the cosmic muon flux made in Hanoi They measure the sum of the fluxes of positively and negatively charged muons, integrated