Hadronic cross section of e + e − annihilation at bottomonium energy region
aa r X i v : . [ h e p - ph ] J un Hadronic cross section of e + e − annihilation at bottomonium energy region Xiang-Kun Dong,
2, 3, ∗ Xiao-Hu Mo,
1, 3, † Ping Wang, ‡ and Chang-Zheng Yuan
1, 3, § Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China (Dated: June 24, 2020)
Abstract
The Born cross section and dressed cross section of e + e − → b ¯ b and the total hadronic cross section in e + e − annihilation in the bottomonium energy region are calculated based on the R b values measured bythe BaBar and Belle experiments. The data are used to calculate the vacuum polarization factors in thebottomonium energy region, and to determine the resonant parameters of the vector bottomonium(-like)states, Y (10750) , Υ(5 S ) , and Υ(6 S ) . PACS numbers: 13.66.Bc, 13.25.Gv, 14.40.Rt ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] . INTRODUCTION The cross section of e + e − annihilation into hadrons is essential information for Quantum Elec-trodynamics (QED) as it is related to the vacuum polarization (VP) of the photon propagator.The measurement of these cross sections is one of the important topics in various e + e − collidersfrom low to high energy, and the precision of the measurements has been successively improvedsince the running of the first generation of e + e − colliders [1]. The data have been used in manycalculations involving the photon propagator, especially in the high precision calculations of theanomalous magnetic moment of the µ , a µ , and the running of the fine structure function, α ( s ) ,where s is the center-of-mass (CM) energy squared [2–4].The cross section of e + e − annihilation into hadrons is often reported in terms of R value,defined as R = σ B ( e + e − → hadrons) σ B ( e + e − → µ + µ − ) , (1)where σ B ( e + e − → µ + µ − ) = πα (0)3 s , is the Born cross section of e + e − → µ + µ − . The experimen-tal measurements of the R values are compiled in Ref. [1]. There are many data at low energies( √ s < GeV) with precision at 1% level; while the measurements are sparse and less preciseat higher energies, for example, the charmonium ( . < √ s < . GeV) and the bottomonium( . < √ s < . GeV) energy regions. One of the reasons of less measurements at high energyis the smaller contribution to the VP, and another reason is the fact that fewer experiments weredesigned in these energy regions.The cross sections of e + e − → b ¯ b were measured in much higher precision by BaBar [5] andBelle [6] experiments in the bottomonium energy region, i.e., √ s = 10 . to . GeV, than byCUSB [7] and CLEO [8] experiments more than 30 years ago. However, neither BaBar nor Belle(let alone CUSB and CLEO) did radiative corrections to the measured cross sections, so the datacannot be used directly for many calculations where the Born cross sections are needed as input.In this paper, we describe how to get the Born cross section based on the published data fromthe BaBar and Belle experiments with some reasonable assumptions. We report the Born crosssections from these experiments and discuss the usage of the data samples in the calculation of theVP factors especially in the bottomonium energy region, and the fit to the dressed cross sectionsto extract the resonant parameters of the vector bottomonium states. We also discuss a possibledetermination of the VP directly by measuring e + e − → µ + µ − cross sections with high luminositydata at the Belle or Belle II experiment, and a strategy to search for the production of invisibleparticles in e + e − annihilation. II. RADIATIVE CORRECTION
The experimentally observed cross section ( σ obs ) is related to the Born cross section via σ obs ( s ) = x m Z F ( x, s ) σ B ( s (1 − x )) | − Π( s (1 − x )) | d x, (2)where σ B is the Born cross section, F ( x, s ) has been calculated in Refs. [9–11] and | − Π( s ) | is theVP factor; the upper limit of the integration x m = 1 − s m /s , where √ s m is the experimentally2equired minimum invariant mass of the final state f after losing energy to multi-photon emission.In this paper, √ s m corresponds to the B ¯ B mass threshold, which is . GeV.The radiator F ( x, s ) is usually expressed as [9] F ( x, s ) = x β − β · (1 + δ ′ ) − β (1 − x )+ 18 β (cid:20) − x ) ln 1 x − (1 + 3(1 − x ) ) x ln(1 − x ) − x (cid:21) , (3)with δ ′ = απ ( π −
12 ) + 34 β + β ( 932 − π
12 ) , (4)and β = 2 απ (cid:18) ln sm e − (cid:19) . (5)Here the conversion of soft photons into real e + e − pairs is included.The Born cross section is thus calculated from σ B ( s ) = σ obs ( s )(1 + δ ( s )) · | − Π( s ) | , (6)where (1 + δ ( s )) is the initial state radiation (ISR) correction factor.It is obvious that both (1 + δ ( s )) and | − Π( s ) | depend on the Born cross section from thresholdup to the CM energy under study, while the Born cross section is the quantity we want to measure.These two factors can only be obtained using the measured quantities with an iteration procedure.The pure ISR correction factor (1 + δ ( s )) depends only on the line shape of e + e − → b ¯ b crosssection, while | − Π( s ) | depends also on the R values in the full energy range, we use a two-stepprocedure to get the Born cross sections. A. ISR correction factor
The ISR correction factor is obtained with an iterative procedure, following Ref. [12], via σ obs i +1 ( s ) = Z x m F ( x, s ) σ dre i ( s (1 − x )) d x, (7)
11 + δ i +1 ( s ) = σ dre i ( s ) /σ obs i +1 ( s ) , (8) σ dre i +1 ( s ) = 11 + δ i +1 ( s ) σ obs ( s ) (9)where σ dre ( s ) = σ B ( s ) | − Π( s ) | is the dressed cross section. At the zeroth step of the iteration, theobserved cross sections are inserted into the integral, playing the role of the dressed cross sections,i.e. σ dre0 ( s ) = σ obs ( s ) . The iteration is continued until the difference between the two consecutiveresults is smaller than a given upper limit. The result from the last iteration, denoted by (1+ δ f ( s )) ,is regarded as the final ISR correction factor. 3 . Vacuum polarization factor A similar procedure is used to calculate the VP factor in the bottomonium energy region. Inthis calculation, however, the total hadronic cross section is used rather than that of e + e − → b ¯ b only. Moreover, instead of depending on the hadronic cross sections in the bottomonium energyregion, the VP factor depends on the R values in the full energy region. In addition, there is alsocontribution from leptons. The VP factor includes two terms [13] Π( s ) ≡ X j = e, µ, τ Π l ( s, m j ) + Π h ( s ) . (10)The first term is the contribution from the leptonic loops with Π l ( s, m ) = Π R + i Π I (11)for lepton with mass m . For ≤ s < m , we define a = (4 m /s − / , Π R = − απ (cid:20)
89 + a − (cid:18)
12 + a (cid:19) · a · cot − ( a ) (cid:21) , Π I = 0 , (12)while for s ≥ m , we define a = (1 − m /s ) / and b = (1 − a ) / (1 + a ) , Π R = − απ (cid:20) − a (cid:18) − a (cid:19) · a · ln b (cid:21) , Π I = − aα (cid:18) m s (cid:19) . (13)The second term in Eq. (10) is the contribution from the hadronic loops. This quantity Π h ( s ) is related to the total cross section σ ( s ) of e + e − → hadrons in the one-photon exchange approxi-mation through a dispersion relation Π h ( s ) = s π α Z ∞ m π σ ( s ′ ) s − s ′ + iǫ ds ′ . (14)Using the identity x + iǫ = P x − iπδ ( x ) , we have Π h ( s ) = − s π α P Z ∞ m π σ ( s ′ ) s ′ − s ds ′ − i s πα σ ( s ) . (15)We follow the procedure in Ref. [14] to calculate the first term in the above equation. First,the integration is performed analytically for narrow resonances J/ψ , ψ (3686) , Υ(1 S ) , Υ(2 S ) , and Υ(3 S ) . Second, for the high energy part, it is assumed that R ( s ) = R ( s ) is a constant above acertain value s . And third, the integral between threshold and s is carried out numerically afterseparation of the principle value part. Thus we have ℜ Π h ( s ) = 3 sα X j Γ je + e − M j s − M j ( s − M j ) + M j Γ j + α π R ( s ) ln (cid:12)(cid:12)(cid:12)(cid:12) s − s s (cid:12)(cid:12)(cid:12)(cid:12) − s π α Z s m π σ nr ( s ′ ) − σ nr ( s ) s ′ − s d s ′ − sσ nr ( s )4 π α ln (cid:12)(cid:12)(cid:12)(cid:12) s − s m π − s (cid:12)(cid:12)(cid:12)(cid:12) , (16)4here Γ j , Γ je + e − , and M j denote total width, partial width to e + e − pair, and mass of the resonance j , respectively. Here, σ nr ( s ) is the σ ( s ) in Eq. (15) with the contributions from narrow resonancessubtracted.We use experimental measurements or theoretical calculations of R values in different energyregions in the calculation of the VP factors:1. For m π < √ s < . GeV, we consider e + e − → π + π − only, with the π form factorobtained through [15] F π ( s ) = 1 + 16 h r i π s + c s + c s , (17)where h r i π = 0 . , c = 6 . , and c = − . .2. For . < √ s < . GeV, we used R values from PDG compilation [1, 16].3. For . < √ s < . GeV, we use R values from the BES collaboration [17, 18].4. For . < √ s < . GeV, we use the R b values provided by the Belle and BaBarcollaborations [5, 6] with proper handling of the ISR correction and VP correction describedbelow.5. For all the other energy regions, we use R values from pQCD calculation [16, 19] R QCD ( s ) = R EW ( s )[1 + δ QCD ( s )] , (18)where R EW ( s ) = 3Σ q e q is the purely electroweak contribution neglecting finite-quark-masscorrections with e q the electric charges of the quarks; the QCD correction factor is given by δ QCD ( s ) = X i =1 c i (cid:20) α s ( s ) π (cid:21) i , (19)with parameters defined in Refs. [16, 19].Replacing pQCD calculations with recent KEDR measurements [20, 21] for √ s between 2 and3.7 GeV gives very similar results in the bottomonium energy region of interest.In the bottomonium energy region, the dressed cross section of e + e − → b ¯ b is denoted by σ dre b ( s ) = σ B b ( s ) | − Π( s ) | = (1 + δ f ( s )) σ obs ( e + e − → b ¯ b ) where σ obs ( e + e − → b ¯ b ) is the observedcross section provided by the Belle and BaBar collaborations [5, 6], and the Born cross sectionof e + e − → u, d, s, c -quarks from the pQCD calculation is denoted by σ B udsc ( s ) . Then σ B0 ( s ) = σ B udsc ( s ) + σ dre b ( s ) is taken as zeroth order approximation of the Born cross section of e + e − → hadrons . Together with the Born cross sections in other energy regions we obtain the first orderapproximation of the VP factor, | − Π ( s ) | , via Eqs. (15) and (16). Then we use σ B i ( s ) = σ B udsc ( s ) + σ dre b ( s ) / | − Π i ( s ) | , the i th order approximation of σ B ( s ) , to calculate | − Π ( i +1) ( s ) | . We iterate thisprocedure until | − Π i ( s ) | is stable and take it as the final VP factor | − Π f ( s ) | .5 . Born cross section The final Born cross section of e + e − → b ¯ b can then be calculated with Eq. (6) with the ISRcorrection factor and VP factor calculated above, i.e., σ B b ( s ) = σ obs ( s )(1 + δ f ( s )) | − Π( s ) | f . (20) III. THE DATA
Both BaBar [5] and Belle [6] experiments measured R b in the bottomonium energy region: R b ≡ σ ( e + e − → b ¯ b ) σ B ( e + e − → µ + µ − ) , where the denominator is the Born cross section of e + e − → µ + µ − . In both experiments, neitherISR correction, nor the VP correction was considered, so the reported R b corresponds to the ob-served cross section. In both experiments, the contribution of the narrow Υ states from the initialstate radiation, i.e., Υ(1 S ) , Υ(2 S ) , and Υ(3 S ) states can be removed from the data supplied in thepapers.The BaBar measurement [5] was based on data collected between March 28 and April 7, 2008at CM energies from 10.54 to 11.20 GeV. First, an energy scan over the whole range in 5 MeVsteps, collecting approximately 25 pb − per step for a total of about 3.3 fb − , was performed.This was then followed by a 600 pb − scan in the range of CM energy from 10.96 to 11.10 GeV,in 8 steps with non-regular energy spacing, performed in order to investigate the Υ(6 S ) region.Altogether, there are 136 energy points [5]. In the BaBar paper, the ISR produced narrow Υ states,i.e., Υ(1 S ) , Υ(2 S ) , and Υ(3 S ) states were included in R b , but in the data file supplied, theircontribution is listed and can be removed from the data.The Belle measurement [6] was done with the scan data samples above 10.63 GeV at total78 data points. The data consist of one data point of 1.747 fb − at the peak √ s = 10 . GeV;approximately 1 fb − at each of the 16 energy points between 10.63 and 11.02 GeV; and 50 pb − at each of 61 points taken in 5 MeV steps between 10.75 and 11.05 GeV. The non-resonant q ¯ q continuum ( q ∈ { u, d, s, c } ) background is obtained using a 1.03 fb − data sample taken at √ s =10 . GeV. Belle experiment supplied a data file of R b with the ISR produced Υ(1 S ) , Υ(2 S ) , and Υ(3 S ) states removed (defined as R ′ b in Belle paper [6]).The BaBar and Belle measurements [5, 6] are shown in Fig. 1. Notice that the definitions of R b are different in these papers. After removing the ISR contribution of the narrow Υ states fromBaBar results, the R b values and the comparison between the two experiments are shown in Fig. 2.In the following analysis, R b refers to the results after removing the ISR contribution of the narrow Υ states, R dre b refers to the dressed cross section after the ISR correction is applied, and R B b refersto the Born cross section after the ISR and VP corrections are applied.We can see from Fig. 2 that Belle results are systematically larger than BaBar measurements.To get the size of the systematic difference, we calculate the ratio between the Belle and BaBarmeasurements in the energy region covered by both experiments. Figure 2 shows the ratio of the R b between Belle and BaBar measurements, the ratios are fitted with a constant with a good fit6 [GeV]s b R BB B B* *B B* s B s B * s B s B * s B* s B (GeV)s b R FIG. 1: R b data from BaBar [5] (left) and Belle [6] (right) experiments. Error bars are statistical only. Thecurves are the fit described in the original paper. √ s (GeV) R b BelleBaBar √ s (GeV) R b ( B e ll e / B a B a r) FIG. 2: Comparison of R b data from BaBar (open cycles) and Belle (Red dots) [top panel] experiments andthe ratio of R b between Belle and BaBar measurements [bottom panel]. Error bars are combined statisticaland systematic errors, and the line is a fit to the ratio of Belle and BaBar measurements. χ / ndf = 56 / , where ndf is the number of degrees of freedom. This indicates that theBelle and BaBar measurements differ by a factor of f = 1 . ± . , (21)which is more than σ from one if they are the same. A. Combination of Belle and BaBar data
The BaBar experiment measured the R b above 11.1 GeV which is very flat. This indicatesthat the bottomonium resonance region has been passed and the flat continuum region has beenreached. At CM energy well above the open-bottom threshold, R values (the total cross sectionof e + e − annihilation) and R b can be calculated with pQCD with five different flavors of quarks.And this can be compared with the BaBar measurement. If we assume the difference in R b be-tween Belle and BaBar can be extrapolated to energy region above 11.1 GeV, by comparing theexpected Belle measurements in this energy region and the pQCD expectation, we can check thenormalization of the Belle data.To compare R b with pQCD calculation, ISR correction and VP correction should be appliedto the Belle and BaBar measurements since pQCD calculates the Born cross sections. Usingthe ISR correction factors (point by point correction, average correction factor δ ≈ . above 11.1 GeV) and VP factors ( | − Π | ≈ . ) calculated below, a fit to the R B b from BaBarexperiment for CM energies between 11.10 and 11.21 GeV yields R B b = 0 . ± . with theerror dominated by the common systematic error.Assuming Eq. (21) applies to R B b at CM energy above 11.1 GeV for Belle measurement, weextrapolate the Belle measurement to this energy region so that we would expect R B b = (0 . ± . × (1 . ± . . ± . . (22)Calculating R B b and the total continuum R values from udsc -quarks in pQCD according toEq. (18), we find that R B b is almost a constant for CM energy between 11.10 GeV and 11.21 GeV,which is 0.351 with an uncertainty negligible compared with the experimental measurement; and R Budsc can be well parameterized as a linear function of CM energy ( √ s in GeV) between 10 and12 GeV, i.e., R Budsc = 3 . − . × − √ s. (23)Figure 3 shows the comparison between the BaBar measurements, Belle expected, and thepQCD calculated R B b , we can find that Belle data agree with pQCD reasonably well (within about σ , the common error of the Belle measurements at high energy is about ± . , similar to theBaBar measurements) while the BaBar measurements are about σ lower than pQCD calculation.As a consequence, we assume BaBar measurement suffers from a normalization bias, and the Bellemeasurement is normalized properly. In the analysis below, we increase the BaBar measurementsby the factor f in Eq. (21) and combine them with the Belle measurements to treat them as a singledata set. The normalized and combined data are shown in Fig. 4.In the remainder of this work, the threshold of open-bottom production is set to be 10.5585 GeV,larger than the first two energy points in BaBar experiment. Therefore, these two data were omittedin our analysis. 8 .20.30.40.50.6 11 11.05 11.1 11.15 11.2 √ s (GeV) R b B Belle expectedBaBar correctedpQCD calculated
FIG. 3: The R B b data from BaBar after ISR and VP corrections (open cycles) and the fit with a constantfunction (blue dashed line, R B b = 0 . ) and the expected Belle results (Red dash-dotted line, R B b =0 . × .
066 = 0 . ), and the pQCD calculation (pink line, R B b = 0 . ). Error bars are combinedstatistical and systematic errors. √ s (GeV) R b BelleBaBar (normalized) √ s (GeV) R b FIG. 4: Normalized R b data from BaBar (open cycles) and Belle (Red dots), which will be treated as asingle data set. Error bars are combined statistical and systematic errors. B. Parametrization of R b To calculate the ISR correction factors, the measured R b will be used as input. To avoid thepoint-to-point statistical fluctuation, one may parameterize the line shape with a smooth curve.There is no known function describing the line shape a priori, so one may parameterize the lineshape with any possible combination of smooth curves.We use the “robust locally weighted regression” or “ LOWESS ” method to smooth the exper-imental measurements. The principal routine
LOWESS computes the smoothed values using the9ethod described in Ref. [22]. This method works very well only for slowly varying data, whichmakes the procedure at the
Υ(4 S ) region work improperly. As a consequence, we use the datapoints directly for √ s < . GeV and use the smoothed data for the other data points. Figure 5shows the smoothed R b , which looks very reasonable. √ s (GeV) R b MeasuredSmoothed √ s (GeV) R b FIG. 5: Belle and BaBar combined R b data (red circles with error bars) and the results after smoothing (bluedots). Error bars are combined statistical and systematic errors. In the following analysis, we use a straight line to connect two neighboring points. As thesepoints are after smoothing and the step is not big, there is no big jump between neighboring points,so we do not expect significant difference between a straight line and a smooth curve.
IV. CALCULATION PROCEDUREA. Calculation of ISR correction factors
We follow the procedure defined in Eqs. (7), (8), and (9) to calculate the ISR correction factors.In doing this for experimental data, we assumed the detection efficiencies for b ¯ b events withoutISR and those with different energy of ISR photons have been estimated reliably within the quotedsystematic uncertainties at both BaBar [5] and Belle experiment [6]. The iteration is continueduntil the difference between two consecutive results is less than 1% of the statistical error of theobserved R b .In the energy region where the cross section varies smoothly, the ISR correction factors becomestable after a few iterations while in the Υ(4 S ) energy region, due to the rapid change of the crosssection in narrow energy region, the ISR correction factors only converge to within 1% after morethan ten iterations. We iterate 20 times, and the maximum difference is less than 0.5% within thefull energy region. Figure 6 shows the final ISR factors as well as the corrected R dre b values.10
123 10.6 10.7 10.8 10.9 11 11.1 11.2 R b R obsb R dreb R b R obsb R dreb √ s (GeV) / ( + d ) ISR factor
FIG. 6: Normalized R b (red in the top two panels) and the ISR corrected R b (or R dre b , blue in the top twopanels), errors are not shown. The bottom panel shows the ISR correction factor. B. Calculation of vacuum polarization factors
Taking R dre b , the ISR corrected R b obtained in previous subsection, as the approximation of R B b and adding the pQCD calculation of the udsc -quark contribution to R values (refer to Eq. (23)),we calculate the VP factors in the bottomonium energy region. After obtaining the VP factor | − Π | , we use R dre b / | − Π | as input to calculate | − Π | again and we iterate this process. It turnsout that after three iterations, the VP factor | − Π | becomes stable so we take the values from thisround as the final results, and we obtain R B b with Eq. (20). Figure 7 shows the VP factors fromthis calculation. C. Estimation of errors
In the previous two subsections we obtain the ISR correction factor and VP factor and in turnthe Born R B b via Eq. (20). During the calculation, however, only the central values of the observed R b are used. Instead of just scaling the errors of the original measurements by the obtained ISRcorrection factors and the VP factors, we perform a toy Monte Carlo sampling to investigate howthe errors (both statistical and systematic) of the original measurements impact the obtained R B b .11 .0651.071.0751.0810.5 10.6 10.7 10.8 10.9 11 11.1 11.2 iteration 1 iteration 2 √ s (GeV) / | - P | iteration 3 -0.6-0.300.30.60.910.5 10.6 10.7 10.8 10.9 11 11.1 11.2 iteration 2-1 √ s (GeV) D VP / VP * iteration 3-2 FIG. 7: The VP factors from 3 iterations (top) and the difference between two iterations (bottom). Noticethat the difference between the second and the third iterations is very small and the curves for the VP factorsare almost indistinguishable.
At each energy point where the Belle or BaBar measurement [5, 6] was performed, we perform10,000 samplings of the observed R b according to a Gaussian distribution for which the mean valueand standard deviation are the central value and statistical error of the observed R b , respectively.These samples will be used to estimate the statistical errors of the deduced quantities. In addition,the uncommon systematic errors are also added to the samples in the same way. For the commonsystematic error, the same error is added to each sample at all energy points in Belle or BaBarexperiment. These samples with both statistical and systematic errors considered will yield thetotal errors of the deduced quantities. In the energy regions . < √ s < . GeV and . < √ s < . GeV, the data from the PDG compilation [1, 16] and BES collaboration [17, 18] areassumed to be completely correlated when we perform the sampling.With each sample as input, we repeat the calculation described in the previous two subsectionsto obtain the ISR correction factor, VP factor, and R B b for this sample. Finally a distribution of R B b at each energy point is observed, so are the ISR correction factor and the VP factor. Wefind that these distributions also satisfy Gaussian distribution well so the fitted mean and standarddeviation are taken as the central value and error of the corresponding quantities, respectively.The covariances of the distributions of Born and dressed cross sections at different energy pointsare also available in the supplemental material [23]. These covariances are useful in calculationswhere R B b or R dre b are inputs such as extracting the resonant parameters of the Y (10750) , Υ(5 S ) ,and Υ(6 S ) by fitting R dre b . 12 √ s (GeV) R b MeasuredBorn √ s (GeV) R b FIG. 8: Comparison of measured R b (open cycles) and Born R B b (solid dots). The error bars are thecombined statistical and systematic errors. The dotted vertical lines are the thresholds for bottom mesonproductions. Recall that we smoothed the observed R values using the “Lowess” method before we calcu-lated the dressed and Born ones. In principle, one can use different methods to smooth the data,which will result in uncertainty of the final results. We test another smoothing method, “Smooth-ing spline” [24], and find that such uncertainty is negligible when compared with the originalerrors. V. FINAL RESULTS ON R B b After all the above operations, we obtain the R B b as well as its total uncertainty from the com-bined BaBar and Belle measurements as shown in Fig. 8 and the Table in the Appendix. We findthat the R B b values are very different from the R b reported from the original publications [5, 6] andthe differences are energy dependent. Common features are that the peaks are even higher and thevalleys become deeper, the two dips at the B ¯ B ∗ + c.c. and B ∗ ¯ B ∗ thresholds are more significant,the peaks corresponding to the Υ(5 S ) and Υ(6 S ) increase significantly, and there is a prominentdip at 10.75 GeV.The total R value corresponding to the production of udscb quarks can be obtained directly byadding the R B b to the udsc -quark contribution calculated from pQCD, as indicated in Eq. (23). VI. SUMMARY AND DISCUSSIONS
From the BaBar and Belle measurements of the observed e + e − → b ¯ b cross sections, we doISR correction to obtain the dressed e + e − → b ¯ b cross sections from 10.56 to 11.21 GeV. These13ressed cross sections are the right ones to be used to determine the resonant parameters of thevector bottomonium states. Together with the R values measured at other energy points and the R values calculated with pQCD, we calculate the VP factors. By applying VP correction, weobtain the Born cross section of e + e − → b ¯ b from threshold to 11.21 GeV. These cross sectionscan be used for all the calculations related to the photon propagator, such as a µ , the µ anomalousmagnetic moment, and α ( s ) , the running coupling constant of QED [2, 3].In the following parts of this section, we discuss the usage of the data obtained in this study. A. Vacuum polarization
The VP factors have been calculated by many groups [14, 25–28], with the experimental dataand various theoretical inputs when the data are not available or less precise. Different techniqueson how to handle the discrete data points and how to correct possible bias in data were developed.All these different treatments yield very similar results on hadronic contribution to a µ and onthe running of the α at M Z , which indicates that the methods are all essentially applicable withcurrent precision of data.Previous calculations of the VP factors in the bottomonium energy region used either the reso-nant parameters of the Υ(4 S ) , Υ(5 S ) , and Υ(6 S ) reported by previous experiments [7, 8] whichare very crude [26, 27] or the experimental data from previous experiments [7, 8] which gave theobserved cross sections [28]. We recalculate the VP factors by using R B b obtained in this analysisbased on high precision data from BaBar and Belle experiments [5, 6], with the ISR correction andVP factors properly considered. Although these new data have little effect on the VP factors farfrom the bottomonium energy region, they do change the VP factors in the bottomonium energyregion as is shown in Fig. 9. The difference between this and the previous calculations [26, 27] isvisible at some energies although all the calculations agree within errors. B. Bottomonium spectroscopy
There are very clear structures in R B b distribution shown in Fig. 8. From low to high energy,we identify the Υ(4 S ) at 10.58 GeV, dips due to B ¯ B ∗ + c.c. and B ∗ ¯ B ∗ thresholds at 10.61 and10.65 GeV, respectively, a dip at 10.75 GeV that may correspond to the Y (10750) [29], and the Υ(5 S ) and Υ(6 S ) at 10.89 and 11.02 GeV, respectively.The observed R b values were used to extract the resonant parameters of the Υ(5 S ) and Υ(6 S ) in the BaBar [5] and Belle [6] publications. As the ISR correction effect is significant and isenergy dependent, this suggests that the fit results are not reliable. To avoid the dip at around10.75 GeV, both BaBar and Belle fitted data above 10.80 GeV only. A recent study of e + e − → π + π − Υ revealed a new state, the Y (10750) , with a mass of (10752 . ± . +0 . − . ) MeV/ c and width (35 . +17 . − . . − . ) MeV [29], at exactly the position of the dip in R B b . This indicates that the dipis very likely to be produced by the interference between a Breit-Wigner function and a smoothbackground component.We do a least-square fit to the dressed e + e − → b ¯ b cross sections ( σ dre = σ B | − Π | ) above10.68 GeV with the coherent sum of a continuum amplitude (proportional to / √ s ) and threeBreit-Wigner functions with constant widths representing the structures at 10.75, 10.89, and14 .0651.071.0751.081.08510.4 10.6 10.8 11 11.2 √ s (GeV) / | - P | Jegerlehner 2019Ignatov 2016This work
FIG. 9: VP factors in the bottomonium energy region and the comparison with previous calculations [26,27]. The solid lines are the central values and the error bars or bands show the uncertainties.
BW = e iφ √ π Γ e + e − Γ s − m + im Γ , where m , Γ , Γ e + e − , and φ are the mass, total width, electronic partial width of the resonance, andthe relative phase between the resonance and the real continuum amplitude, respectively, and theyare all free parameters in the fits.Eight sets of solutions are found from the fit [30], with identical total fit curve, identical fitquality ( χ = 274 with 188 data points and 13 free parameters), and identical masses and widthsfor the same resonance, but with significantly different Γ e + e − and φ .Figure 10 shows one of the solutions of the fit, and Table I lists the resonant parameters from8 solutions of the fit. The masses and widths of the resonances agree with those from Ref. [29]but with improved precision because of the much better measurements used in this work com-pared with those in exclusive e + e − → π + π − Υ analyses. The Γ e + e − values determined from thisstudy allow us to extract the branching fractions of π + π − Υ of these resonances by combining theinformation reported in Ref. [29], and to understand the nature of these vector states [31–35].In this analysis, we assumed that all the resonances are Breit-Wigner functions with constantwidths and the continuum term is a smooth curve in the full energy region and they interferewith each other completely. In fact, the R B b or the total cross section has contributions fromdifferent modes, including open bottom and hidden bottom final states, the parametrization of theline shape should be very complicated due to the coupled-channel effect [36] and the presence15 √ s (GeV) s d r e ( e + e - → bb - ) ( pb ) FIG. 10: Fit to the dressed cross sections with coherent sum of a continuum amplitude and three Breit-Wigner functions. The solid curve is the total fit, and the dashed ones for each of the four components fromSol. 1 in Table I. The magnitudes of these components are different in different solutions.TABLE I: Resonant parameters from the fit to dressed cross sections. There are 8 solutions with identicalfit quality, and the masses and widths of the resonances are identical in all the solutions. The uncertaintiesare combined statistical and systematic uncertainties in experimental measurements.Solution Parameter Y (10750) Υ(5 S ) Υ(6 S ) c ) ± ± ± Width (MeV) . ± . . ± . . ± . Γ e + e − (eV) . ± . . ± . . ± . φ (degree) ± ± ± Γ e + e − (eV) . ± . . ± . ± φ (degree) ± ± ± Γ e + e − (eV) . ± . ±
14 11 . ± . φ (degree) ± ± ± Γ e + e − (eV) . ± . ±
19 363 ± φ (degree) ± ± ± Γ e + e − (eV) ±
24 23 . ± . . ± . φ (degree) ± ± ± Γ e + e − (eV) ±
27 27 . ± . ± φ (degree) ± ± ± Γ e + e − (eV) ±
32 534 ±
18 11 . ± . φ (degree) ± ± ± Γ e + e − (eV) ±
34 622 ±
25 370 ± φ (degree) ± ± ±
16f many open bottom thresholds, B ¯ B , B ¯ B ∗ + c.c. , B ∗ ¯ B ∗ , B s ¯ B s , B s ¯ B ∗ s + c.c. , B ∗ s ¯ B ∗ s , B ¯ B + c.c. , B ∗ ¯ B + c.c. , B ∗ ¯ B + c.c. , B ¯ B + c.c. , and so on, and even πZ b (10610) , πZ b (10650) . Thesituation becomes somewhat simpler in a single final state like π + π − Υ( nS ) ( n = 1 , , ) [6]and π + π − h b ( mP ) ( m = 1 , ) [37] although the intermediate structure in the three-body finalstate is also complicated. The results from these fits may change dramatically by including moreinformation on each exclusive mode.We also attempt to add one more Breit-Wigner function to fit the cross sections, the fit qualityimproves slightly with a state at m = (10848 ± MeV/ c with a width of (28 ± MeV, a stateat m = (10831 ± MeV/ c with a width of (14 ± MeV, or a state at m = (11065 ± MeV/ c with a width of (73 ± MeV. In all these cases, the significance of the additional state is lessthan σ .The data obtained in this analysis can be used to extract resonant parameters of these states if abetter parametrization of the cross sections is developed. C. Search for the production of invisible particles
The experimentally observed e + e − → µ + µ − cross section, with radiative correction is ex-pressed as σ ( s )( e + e − → µ + µ − ) = Z x m πα s (1 − x ) F ( x, s ) | − Π( s (1 − x )) | dx, (24)where F ( x, s ) is expressed in Eq. (3), x m = 1 − s m /s , with √ s m the required minimum invariantmass of the µ pair in the event selection. Here the mass of µ is neglected for charm and beautyfactories. In Eq. (24), the cross section is calculated by QED without any ambiguity except thevacuum polarization Π( s ) which is expressed by Eqs. (10–15), with the hadronic contributiondepending on the experimental measured data as input.If the e + e − → µ + µ − cross section is measured to a high precision, then Π( s ) can be obtained.Thus Π( s ) is measured from the experiment directly [38]. It can then be compared with thatcalculated by Eqs. (10–15). This provides a test of QED at high luminosity flavor factories. InEq. (10), the leptonic term Π l ( s, m ) is expressed in terms of the QED fine structure constant α (0) and lepton masses, which are all known to very high accuracy, while the hadronic term Π h ( s ) mustbe evaluated with Eq. (15) with the input of experimental measured hadronic cross sections. It isseen in Eq. (15) that Π h ( s ) is most sensitive to the hadronic cross section at energies close to s ,which can be measured with the same experiment. Therefore such a test of QED can be performedwith two sets of data, e + e − → µ + µ − and e + e − → hadrons , collected within the same experiment.Any discrepancy would mean that there are missing hadronic final states, or even a final state thatescaped detection, or is invisible by current detection technology. This provides a test of QED andsearch for new physics. We propose for it to be included in physics goals in future high-luminosityfrontier physics. Acknowledgments
This work is supported in part by National Natural Science Foundation of China (NSFC) un-der contract Nos. 11521505, 11475187, and 11375206; Key Research Program of Frontier Sci-ences, CAS, Grant No. QYZDJ-SSW-SLH011; the CAS Center for Excellence in Particle Physics17CCEPP); and the Munich Institute for Astro- and Particle Physics (MIAPP) which is fundedby the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’sExcellence Strategy-EXC-2094-390783311. [1] V. V. Ezhela, S. B. Lugovsky and O. V. Zenin, hep-ph/0312114.[2] M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur. Phys. J. C , no. 3, 241 (2020).[3] A. Keshavarzi, D. Nomura and T. Teubner, Phys. Rev. D , no. 1, 014029 (2020).[4] F. Jegerlehner, EPJ Web Conf. , 00022 (2018).[5] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. , 012001 (2009).[6] D. Santel et al. [Belle Collaboration], Phys. Rev. D , no. 1, 011101 (2016).[7] D. M. J. Lovelock et al. , Phys. Rev. Lett. , 377 (1985).[8] D. Besson et al. [CLEO Collaboration], Phys. Rev. Lett. , 381 (1985).[9] E. A. Kuraev and V. S. Fadin, Sov. J. Nucl. Phys. , 466 (1985) [Yad. Fiz. , 733 (1985).][10] F. A. Berends, “Z Line Shape”, CERN (1989), edited by G. Altarelli, R. Kleiss and C. Verzeg-nassi.[11] G. Montagna, O. Nicrosini, F. Piccinini and L. Trentadue, Nucl. Phys. B , 161 (1995).[12] X. K. Dong, L. L. Wang and C. Z. Yuan, Chin. Phys. C , no. 4, 043002 (2018).[13] W. Greiner, J. Reinhardt. Quantum Electrodynamics (3rd Ed.), Springer-Verlag, Berlin, 1994.[14] F. A. Berends and G. J. Komen, Phys. Lett. , 432 (1976).[15] M. Davier, S. Eidelman, A. Hocker and Z. Zhang, Eur. Phys. J. C , 497 (2003).[16] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D , no. 3, 030001 (2018).[17] J. Z. Bai et al. [BES Collaboration], Phys. Rev. Lett. , 594 (2000).[18] J. Z. Bai et al. [BES Collaboration], Phys. Rev. Lett. , 101802 (2002).[19] G. Rodrigo, A. Pich and A. Santamaria, Phys. Lett. B , 367 (1998).[20] V. V. Anashin et al. [KEDR Collaboration], Phys. Lett. B , 42 (2019).[21] V. V. Anashin et al. , Phys. Lett. B , 174 (2017).[22] William S. Cleveland (Wadsworth, 555 Morego Street, Monterey, California 93940), “The Elements ofGraphing Data”. We use the program from wikipedia (https://en.wikipedia.org/wiki/Local regression)supplied by the author.[23] See supplemental material at http://cpc.ihep.ac.cn/article/doi/10.1088/1674-1137/44/8/083001 for thefinal results shown in the Appendix and the covariances of the R B b and R dre b et al. [Working Group on Radiative Corrections and Monte Carlo Generators for Low Ener-gies], Eur. Phys. J. C , 093003 (2004); K. Hagiwara,A. D. Martin, D. Nomura and T. Teubner, Phys. Lett. B , 173 (2007).[29] R. Mizuk et al. [Belle Collaboration], JHEP , 220 (2019).[30] K. Zhu, X. H. Mo, C. Z. Yuan and P. Wang, Int. J. Mod. Phys. A , 4511 (2011).[31] See, for example, B. Chen, A. Zhang and J. He, Phys. Rev. D , no. 1, 014020 (2020), and references herein.[32] A. Ali, L. Maiani, A. Y. Parkhomenko and W. Wang, Phys. Lett. B , 135217 (2020).[33] Q. Li, M. S. Liu, Q. F. L, L. C. Gui and X. H. Zhong, Eur. Phys. J. C , no. 1, 59 (2020).[34] Z. G. Wang, Chin. Phys. C , no. 12, 123102 (2019).[35] W. H. Liang, N. Ikeno and E. Oset, Phys. Lett. B , 135340 (2020).[36] N. A. Tornqvist, Phys. Rev. Lett. , 878 (1984).[37] A. Abdesselam et al. [Belle Collaboration], Phys. Rev. Lett. , no. 14, 142001 (2016).[38] A. Anastasi et al. [KLOE-2 Collaboration], Phys. Lett. B , 485 (2017). ppendix A: Final results of Born cross section R B b and dressed cross section R dre b , together withthe ISR correction factors and vacuum polarization factors. The first error of R B b and R dre b is statisti-cal and the second one is systematic. The errors of ISR factor and VP factor are combined statisticaland systematic errors. √ s (GeV) R B b R dre b / (1 + δ ) 1 / | − Π | ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± s (GeV) R B b R dre b / (1 + δ ) 1 / | − Π | ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± s (GeV) R B b R dre b / (1 + δ ) 1 / | − Π | ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± s (GeV) R B b R dre b / (1 + δ ) 1 / | − Π | ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± s (GeV) R B b R dre b / (1 + δ ) 1 / | − Π | ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.0008