Charm Production Cross Section

Transcription

1 Charm Production Cross Section Kanglin He, Ming Yang November 4, 26 Contents 1 Introduction 1 2 The 1 resonances around 4.GeV ψ(377) ψ(44) ψ(416) ψ(4415) Angular distribution and correlation Charm Cross Section Below D D Threshold 7 4 Charmed Cross Section Above the D D Threshold The momentum distribution of D mesons The decay fraction of D and Ds Systematic uncertainties on the momentum distribution Full reconstruction of DD events A scan experiment Initial State Radiation Correction 9 1 Introduction Fig. 1.1 shows the R values of e + e hadrons in charm region. Below the open charm DD threshold, the clear spectrum of relatively narrow states, J/ψ and ψ can be identified with the 1S and 2S cc states predicted by the potential models, which incorporate a color Coulomb term at short distances and linear scalar confirming term at large distances. Above the DD threshold, there several resonance peaks, dominantly decay into open-flavor final states, serves as factories for production of charmed mesons. In these strong decays, the initial cc meson decays through production of a light qq quark-antiquark pair (q = u, d, s), followed by separation into two charmed mesons. The mechanism of the strong decay process of these charmonium is still poorly understood. In quark model calculations this decay process is modelled by a simple phenomenological qq pair production amplitude. The qq pair is usually assumed to be produced with vacuum ( ++ ) quantum numbers, and the variants of the decay model make different assumptions regarding the spatial dependence of the pair production amplitude relative to the initial cc pair. A detailed study of the charm cross section above DD threshold, may provide a wealth of information about the strong dynamics with heavy and light quarks. Experimentally, the charm cross section (σ charm ) are measured through: σ charm = N charm (1.1) L where L the integrated luminosity, and N charm the number of produced charmed meson pairs. N charm can be obtained by a tagging technique, measuring the products of a certain tag S = N charm ɛ B 1

2 R J/ψ ψ(2s) Mark-I Mark-I + LGW Mark-II PLUTO DASP Crystal Ball BES ψ 377 ψ 44 ψ 416 ψ 4415 c Figure 1.1: R value above Charm threshold. Figure 1.2: Charm cross section for DD, D D, D D, D s D s, DsD s and Ds ± Ds for the CLEO-c scan run., S the observed tags, ɛ the detection efficiency, B the decay fraction. The accurate measurements of charm production cross section require both larger integrated luminosity and larger (ɛ B). BEPCII and BESIII will perform an analysis with high-statistic charm data sets in future to get the precision charm production cross section above the DD threshold. During late summer and early fall of 25, in order to determine the optimal center-mass-energy point for D s physics study, CLEOc experiment spent two month data taking in the energy range GeV. The scan that was conducted consisted of twelve points, totalling about 6pb 1. At each energy point, three, five and eight decay models were employed for D, D + and D + s, respectively. The different production channels can be distinguished based on the kinematics of reconstructed tags reflected through the selection variables M inv and p D (momentum of tags). The preliminary results(partially evaluated systematic uncertainties, no correction for multi-body contributions, not radiatively corrected)[3] for charm cross section are drawn in Fig CLEOc s results are agreed well with previous measurements[8, 9, 1, 11, 12]. As shown in Fig. 1.2, there is very little DD production at any energy, a sharp peak structure in D D near D D threshold, and a broad peak or plateau in D D. The charm cross section through this region is considerable, comparable to that at ψ(377) ( 6nb). There is a visible but disappointingly small peak in D s D s (.3nb), and more impressive broad peak at about 4.17GeV which offer about 1 nb of D sd s production. The CLEO-c scan data suggest a clear evidence for multi-body production such as e + e D Dπ, since the two-body modes do not account for all charmed-meson production. It s a very interesting topic, can be confirmed at BESIII. 2

3 E cm /GeV ψ(4415) ψ ψ (44) (416) ψ(377) ψ(2s) 4S 2D 3S 1D 2S DD DD* D D*D* s D s * D s D s D 1 s*d s * D D DD 2 * 3 J/ψ 1S charm meson pair production threshold Figure 2.1: The spectra of the cc system above the DD threshold, and of the lowest-lying charmed-meson decay channels. Comparison of the mass with the potential model prediction, ψ(377), ψ(44), ψ(416) and ψ(4415) are assigned to be 1 3 D 1, 3 3 S 1, 2D 3 D 1 and 4 3 S 1 cc states. 2 The 1 resonances around 4.GeV The four known cc states above the DD threshold, ψ(377), ψ(44), ψ(416) and ψ(4415), are of special interest because they are easily produced at e + e collider. The cross section in the region around 4.GeV can be studied by successive onset of the specific channels with the charmed mesons: DD, D D, D s D s, etc. Fig. 2.1 shows the spectra of the cc system above the DD system, and of the lowest-lying charmed-meson decay channels. Tab.2.1 listed the decay widths of 4 cc states around 4. GeV predicted by Ref.[1]. 2.1 ψ(377) The ψ(377) lies below the DD threshold and decays predominately into DD pairs, to make an ideal D factory. The ψ(377) is generally assumed to be the 1 3 D 1 cc state. Some of theory model predict Γ(ψ(377) DD) = 43MeV[1] for a pure 3 D 1 state, which is rather wider than the experimental value of 23.6 ± 2.7 MeV[2]. It can be explained by introducing an additional 2 3 S 1 component. The partial width of a mixed 2S 1D state is given by ψ(377) = cos θ 1 3 D 1 + sin θ 2 3 S 1. (2.1) fitting the experimental ψ(377) width requires a mixing angle of θ = 17.4 ± 2.5. On the basis of isospin conservation and the phase space alone, one expects σ(ψ(377) D ( ) D 3 ) σ(ψ(377) D + D ) = pd = (2.2) p D + in the calculation of Ref.[4], the ratio of D to D + produced at ψ(377) peak is predicted by the value of 1.36, suppressed by the 3 D 1 from factor. If the D and D + cross section near ψ(377) can be measured with sufficient precision, if may be possible to observe the momentum dependent form factor. 3

4 Mode ψ(377) ψ(44) ψ(416) ψ(4415) DD D D D D D s D s DsD s DsD s.7 D 1 D 32 D2D 23 total experiment[2] (23.6 ± 2.7) (52 ± 1) (78 ± 2) (43 ± 15) Table 2.1: open-flavor strong decays widths (MeV) for ψ(377), ψ(44), ψ(416) and ψ(4415). s σ(d D ) σ(d + D ) Mark-III[5] ±.25 ± ±.3 ±.15 BES-II[6] ±.29 ± ±.28 ±.12 CLEO-c[7] ± ± Table 2.2: Charm Cross Section at ψ(377). The charm cross section are well measured by Mark-III[5], BESII[6] and CLEO-c[7]. The results are listed in Tab ψ(44) In the mass region of ψ(44), it is a very interesting case for the study of strong decays. Four open charm modes are energetically allowed, DD, D D, D D and D s D s, although D D has little phase space. There are some experiment evidence for the three non-strange modes and one D s D s mode from BESI and CLEO-c, listed in Tab The σ DD is pretty low,.3nb. The cross sections of D D and D D are approximately equal. The reported relative branching fractions (scaled by p 3 ) show a very strong preference for a D final states, D D D D D D. This motivated suggestions that the ψ(44) might be a D D molecule. The cross section of mode D s D s reported by BESI and CLEO-c are around.3 nb, corresponding to a branching fraction of about 4%, lower than the predicted value( 11%. This branching fraction is of special interest because it determines the event rates available for studies of D S weak decays. The D D mode is especially interesting for strong decay studies, since there three independent decay amplitudes for 1 cc D D, 1 P 1, 5 P 1 and 5 F 1. Since the decay momentum is very low near D D threshold, the amplitude of 5 F 1 could be ignored. The ratio of the nonzero amplitudes is 5 P 1 / 1 P 1 = 2/ ψ(416) Like the ψ(377), there may also be a significant S-wave cc component, since the ψ(416) has a much large e + e width than one would expect for a pure D-wave cc state. There are five open-flavor decay modes available for ψ(416): DD, D D, D D, D s D s and DsD s. The experiment results from Mark-II, Mark-III and CLEO-c are listed in Tab The leading mode is D D with a branching fraction greater than 5%, followed by comparable D D and DsD s modes, and a somewhat weaker DD. The D s D s cross section is very small. The mode of D D is again especially interesting, due to the three decay amplitudes allowed for this final state. For a pure D-wave cc assignment the ratio of the two D D P-wave amplitudes is independent of radial wave function, and is 5 P 1 / 1 P 1 = 1/ 5. The 5 F 1 amplitude is predicted to be the largest, whereas it is zero for an S-wave cc state. A determination of these D D decay amplitude ratios would be an extremely interesting test of decay models. 4

5 ψ(44) ψ(416) Modes BES-I[8, 9] CLEO-c[3] Mark-II[1] Mark-III[11, 12] CLEO-c[3] D D.19 ±.5 DD D + D.13 ± ±.3.23 ±.4 ±.5.2 D D 2.46 ±.6 D D D ± D 2.31 ± ± ±.1 ±.3 2 D D 2.7 ±.4 D D D ± D.87 ± ± ±.2 ±.6 5 D + S D S.32 ±.56 ± D ± S D S.83 ±.17 ±.31.9 Table 2.3: Charm cross section at ψ(44) and ψ(416). 2.4 ψ(4415) Ten open-charm strong decay modes are allowed for the ψ(4415), seven with cn meson states (n = u, d), and three with cs. Experimentally, to date nothing has been reported regarding the exclusive charmed hadronic decay modes of the ψ(4415). 1) The largest exclusive mode is predicted to be the S+P combination D 1 D, where D 1 is the 1 ++ axial mesons near GeV. Since D 1 is rather narrow (Γ 2 3 MeV) and decays dominantly to D π, there should be a strong ψ(4415) signal in DD π final states. 2) The second-largest decay modes is predicted to be another S+P mode, D 2D. The D 2 is also moderately narrow, can also be isolated from the observed final state. The D 2 has significant branching ratio to both D π and Dπ, so the D 2D mode of the ψ(4415) should be observable in both DDπ and DD π. 3) The D D mode should be comparable in strength to D 2D. If the ψ(4415) is indeed an S-wave state to a good approximation, we expect this ratio to be 5 P 1 / 1 P 1 = 2/ 5, and the 5 F 1 amplitude should be zero. 4) It is interesting to note that ψ(4415) decays may provide access to the recently discovered D s(2317) although the channel D S D s(2317) has a threshold of 4429 MeV, 14 MeV above the nominal mass of ψ(4415), the width of the ψ(4415) and the fact that the decay ψ(4415) D S D s(2317) is purely S-wave implies that one may observe significant D s(2317) production just above threshold, near E cm = 4435 MeV. 2.5 Angular distribution and correlation The strong decays of the vector ψ(44) and ψ(416) to D D are especially interesting, since this is their only multiamplitude decay mode. The decay to DD and D D are single amplitude decays, respectively 1 P 1 and 3 P 1, so one learns nothing new about the decay process by studying their angular distributions. The decays to D D however have three allowed amplitudes, 1 P 1, 5 P 1 and 5 F 1, and an experimental determination of the ratios of these amplitudes can be used as an important test of the decay model, specifically of the quantum numbers of the light qq pair produced in the decay. In this section we ll give a brief description of the angular distribution in non-relativistic language. The DD production is purely P-wave with amplitude A DD η p, (2.3) where η is the γ polarization vector and p is the three-momentum of the D in the center-of-mass system. For the unpolarized beam, we have η i η j = δ ij ˆn i ˆn j, (2.4) pol 5

6 the angular distribution is quite simple: A DD 2 1 cos 2 θ = sin 2 θ. (2.5) For D D or D D production, S = 1. To get the correct parity we need L odd, and since J = 1, L 2. Thus L = 1. The decay amplitude of D D mode has the form of A D D the D decays into Dπ or Dγ. Each decay has only a single amplitude η ( p ɛ), (2.6) A D Dπ ɛ q, A D Dγ ɛ (ˆk Ê) ɛ ˆB, (2.7) where ɛ the D polarization vector, q the pion momentum, ˆk the γ direction, and Ê and ˆB represent the photon s electric and magnetic polarization vectors with ˆB i ˆBj = δ ij ˆk i ˆkj (2.8) The distributions are given by pol Ê i Ê j = pol A D D,D Dπ A D D cos 2 θ, cos 2 θ π 1 cos 2 θ Dπ, (2.9) cos2 θ γ A D D,D Dγ For D D production, S=, 1 and 2 are possible. However, since P = ( 1) L and C = ( 1) L+S, L is odd and S is even. if S = 2, then L = 1 or 3. For a purely 3 S 1 state, the amplitude of L = 3 (F-wave) is zero. Or near the D D threshold, F-wave could be ignored since the p D is close to zero. Let A and A 2 denote the production amplitudes for S = and S = 2 states, we have [ 1 = A ( ɛ ɛ)( p η) + A 2 2 ( ɛ p)( ɛ η) ( ɛ η)( ɛ p) 1 ] 3 ( ɛ ɛ)( p η), (2.1) A D D where ɛ and ɛ the polarization vectors of D and D. The amplitudes are normalized in such a way that the total cross section is given by A D D 2 A A 2 2. Then the angular distribution of the production rate has the form A D D 2 1 A A 2 7 A A 2 cos2 θ (2.11) The angular distribution for pions and photons produced in the process e + e D D (Dπ 1 )(Dπ 2 ), (Dπ)(Dγ), and (Dγ 1 )(Dγ 2 ), and the correlations are listed as following[13]: A D D 2 ( ) cos2 θ ππ A cos 2 θ ππ A 2, A D D 2 A D D 2 A D D 2 A D D 2 (1 17 ) cos2 θ γπ A ( 1 cos 2 ) θ γπ A 2, 7 ( ) cos2 θ γγ A ( 1 + cos 2 ) θ γγ A 2, 13 ( 1 21 ) 47 cos2 θ π A A A 2 cos ϕ ( 1 3 cos 2 ) 72 θ π + 47 A 2, ( ) 73 cos2 θ γ A A A 2 cos ϕ ( 3 cos 2 θ γ 1 ) A 2. (2.12) 6

7 where ϕ the relative phase of the amplitude A and A 2. The two principal models assumed by theorists to study these cc decays at present are the 3 P model[14] and the Cornell(timelike vector) model[15]; these give different predictions for the relative D D decay amplitudes, which have not been tested experimentally. At BES3, we can measure these amplitude ratios. This important information will allow theorists to formulate more accurate models of cc strong decays, and should greatly improve our understanding of this dominant QCD strong decay process generally. 3 Charm Cross Section Below D D Threshold Near charm threshold, charmed mesons, D D, D + D and D + S D S, are pairly produced. The double tag method will be applied to obtain a mode independent charm cross section measurement. For specific hadronic decay modes, i and j, the number of produced DD pairs can be expressed as the number of single tags and double tags: N DD = 1 2 S i S j ɛ ij D ij ɛ i ɛ j 1 4 S2 i ɛ ii D ii ɛ 2 i where,n DD the total number of produced DD pairs which are independent of the decay fractions, S i and D ij the number of single tags and double tags, ɛ i and ɛ ij, ɛ ii the detection efficiencies of single and doubles tags. Many systematic uncertainties could be canceled in the double-tag measurement. Let ɛ ij 1, ɛ ii ɛ i ɛ j ɛ 2 1 i in (3.1), and ignore the error of single tags, the calculation could be predigested. The precision of number of produced charm meson pairs is estimated by i j i = j (3.1) N N 1 Dij = 1 D (3.2) where D presents the total number of double tags. At BESIII, about 4, and 2, D and D + double tags are expected to be reconstructed in 15fb s = GeV. The statistic error of number of produced DD pairs would be ignored. The dominant systematic uncertainties are from the luminosity (L ) measurement which are expected to be at 1% level. If we take 3fb 1 data at 4.3GeV or 4.17GeV, about 75 and 2,2 D S double tags will be acquired, corresponding to a statistic error of 2.% or 1.2%. Both statistical and systematical can contribute to the precision of D S cross section. 4 Charmed Cross Section Above the D D Threshold Both ψ(44) and ψ(416) are above the D threshold, mainly decay into D final states with large fractions. The low Q value of the hadronic decay modes of the D provides a clean method to identify charm. This method is used to determine charmed meson cross section above the D production threshold. The momentum distributions are monochromatic for DD production and the directly D meson rise from DD procedure. It s different for D meson decay from D meson. The momentum distribution of tagged D approximates monochromatic while D meson decaying to π D or π + D due to the low decay Q value, but it becomes board while D decaying to γd. Figure 4.1 shows the momentum distribution of D meson in s=4.3gev. D meson from DD, D D(DD ) and D D can be identified cleanly. 4.1 The momentum distribution of D mesons The p D distribution for D s produced in the decay of D s depends on the momentum of the parent D in the lab frame and the angular distribution of the D in the D rest frame. In the case of D D, the angular distribution of the D in the D frame can be uniquely predicted. In the case of D D, it s much complicated. Initial state radiation further distorts the shape of the p D distribution by reducing the effective center-of-mass energy. 7

8 Number of events /(5MeV/c) Daughter D (π D ) Daughter D (π + D ) Total All Bachelor D 1 Daughter D (γd ) P D /(MeV/c) Number of events /(5MeV/c) Total all Bachelor D + Daughter D + (π D + ) 2 Daughter D + (γd + ) P D + /(MeV/c) Figure 4.1: p D distribution at s = 4.3GeV. The shape of p D can be simulated in Monte Carlo event generator [16, 17]. D + D D + π + D D + D D ± π D ± D D D π D D + D D ± γd ± D D D γd p D + D ± D D ± π D ± D ± D D ± π ± D D ± D Direct D p D D D D ± D D ± γd ± D π D D D + D Direct D ± D D γd D D Direct D D D Direct D (4.1) The charm cross section and decay branching ratio of D can be together determined by fitting to the p D distribution. 4.2 The decay fraction of D and D s 4.3 Systematic uncertainties on the momentum distribution The p D distribution are distorted by the ISR effects and the lineshape of ψ(377), ψ(44) and ψ(416). The systematic uncertainties on the simulated ISR events are estimated by varying the resonant parameters of ψ s in generator. As shown in Table 2.1, the errors of the resonant parameters for ψ(44) and ψ(416) are quite large by the current experimental measurements. The partial decay width of ψ s to certain charm meson pairs is also energy dependent, by a factor of ( ) 2L+1 q Γ i = Γ m i (4.2) m q where q and q is the decay momentum in center-mass of m and m, the subscript indicate the q and m take the peak value of ψ s; L is the number of angular momentum. 8

9 The peak mass of ψ s are very close. With the large decay widths, the interference between ψ s and continuum contribution will distort the lineshapes. A fine scan experiment is suggested to investigate the interference phase of all possible physics processes. 4.4 Full reconstruction of DD events Above D production thresgold, we assumed that events containing charmed D mesons arise from DD and D D productions (the DD procedure was the simplest case), so that the detection of a single D in an event implies that the recoiling system is a monochromatic D or a D decay from D mesons. For the D tag, it may comes from neutral charmed D pair productions or charged charmed D pair productions. For the D + tag, it comes solely from the charged charmed D pair productions. We assumed that the neutral D mesons total decay to D mesons with a 1% decay fractions because of the kinematics limitation, the charged D mesons are decayed to D meson with a decay fractions Br(D + π + D ) and D + mesons with a decay fractions 1 Br(D + π + D ). Table 4.1 listed the observed possibility of single-tags and double-tags via different charmed D mesons productions. The data are searched for events containing either one or two reconstructed D mesons. The momentum distributions of reconstructed D tags provided an additional information to identify the DD,DD,D D productions (see Figure.1). By comparing the observed number of a single D is reconstructed with the number of partial reconstructed DD events for different charmed D meson productions, we could employed a χ 2 minimization fit. To determine the individual branching ratios(b i ) and the number of produced DD, D D pairs (N = σl), the observed number of single-tags(s i ) and double-tags(d ij ) can be expected as (see Table 4.1) and Si = 2N ɛ i B i + N + ɛ i B i B+ S + i = N + ɛ i B i + N + ɛ i B i (1 B+) Dij = δ ij N ɛ ij B i B j D ++ ij = δ ij N + ɛ ij B i B j (1 B+) D + ij = N + ɛ ij B i B j B+ Si = 2N ɛ i B i + 2N + ɛ i B i B+ S + i = 2N + ɛ i B i (1 B+) Dij = δ ij (N ɛ ij B i B j + N + ɛ ij B i B j (B + )2 ) D ++ ij = δ ij N + ɛ ij B i B j (1 B+) 2 D + ij = 2N + ɛ ij B i B j B+(1 B+) for DD production (4.3) for D D production (4.4) with N = σ D D L and N + = σ D± D L in (4.3) N = σ D D L and N + = σ D ± D L in (4.4), and where B i,j is the individual branching fractions in {i, j} th D decay mode, ɛ i is the efficiency for reconstructing a single-tag in the i th D decay mode, ɛ ij is the partial reconstruction efficiency for DD(from DD and D D ) decay mode i and j, B+ = Br(D + π + D ). The efficiencies are determined by a detail Monte Carlo simulation of DD, D D production and decay, including the detector response. In final, we form a χ 2 expression: χ 2 = i (Smeasure i Spredict i )2 + ij σ 2 S i measure (D ij measure D ij predict )2 σ 2 D ij measure where, the index {measure,predict} represent for the number of tags obtained from {measurement, prediction}, σ are the errors of measurement. 4.5 A scan experiment 5 Initial State Radiation Correction The production cross section for charm are obtained by correcting the observed cross section for the effects of initial stat radiation (ISR). The ISR correction is dependent on the cross section for all energies less than (4.5) 9

10 Single Tags Double Tags Modes D D + D vs D D + vs D D vs D 2ɛ i B i δ ij ɛ ij B i B j DD ± ɛ i B i B + ɛ i B i (1 + B +) + δ ij ɛ ij B i B j B + + ɛ ij B i B j B + 2ɛ i B i δ ij ɛ ij B i B j D D ± 2ɛ i B i B + 2ɛ i B i B + + δ ij ɛ ij B i B j (B + )2 δ ij ɛ ij B i B j (B +) + 2 2ɛ ij B i B j B + B+ + 2ɛ i B i δ ij ɛ ij B i B j DD ± 2ɛ i B i δ ij ɛ ij B i B j Table 4.1: The observation possibility of single-tags and double-tags above charm threshold. Here, B i, B j are the decay fraction of D (i th, j th ) tagged channels, ɛ i is the acceptance of single-tags, { ɛ ij is the acceptance of double-tags, B+ =Br(D + π + D ), B + + = 1 B+, 1(i=j) the symbol δ ij = 2(i j). The acceptance ɛ was weighted by detection efficiencies for D meson arise from different decay mode of D mesons ρ ω φ J/ψ ψ ψ(43) ψ(377) ψ(416) Figure 5.1: The Born cross section (R-unit) of e + e hadrons from.3gev to 4.3GeV in KK. the nominal energy. Kuraev and Fadin[18, 19, 2] give the observed cross section σ as an integral over the idealized no-radiation cross section σ with a kernel function σ(s) = σ(s(1 x))f KF (x, s), where s = W 2 and x = (W 2 Weff 2 )/W 2, W is the nominal energy of c.m.s., W eff is the effective energy after ISR. The idealized Born cross section σ in charm region is drawn in Figure 5.1, and the kernel function is given by [ F KF (x, s) = tx t t + α ( π 2 π 3 1 ) ( 9 + t t2 1x 1 + 3(1 x)2 [4 (2 x) ln 8 x ) 1. Kuraev and Fadin claim.1% accuracy for F KF (x, s)[2]. where t = 2 α (ln W 2 π m 2 e The Breit-Wigner resonance amplitude and cross section are given by )] ( 32 π2 t 1 x ) 12 ] 2 (5.1) ln(1 x) 6 + x κ A(W ) = W M + iγ/2 σ(w ) = A 2 κ = (W M) 2 + Γ 2 /4 (5.2) 1

11 where κ = 3πΓ ee Γ/M 2 (5.3) Note that the amplitude A is complex, with a phase φ res = tan 1 [(Γ/2)/(W M)] that start near at low W, pass through 9 at W = M, and approaches 18 for large W. If the interference between ψ s exist with the relative phase α s, we can write the total amplitude and the cross section as: A(W ) = i A i exp(iα i ), σ(w ) = A 2 (5.4) Synchrotron radiation and the replacement of the radiated energy by the RF accelerating cavities leave each beam with a spread in energies, resulting in an essentially Gaussian distribution in center-of-mass collision energy W centered on the nominal W G(W, W ) = 1 exp [ (W W ) 2 ] 2π 2 2 (5.5) where is the beam energy spread. The convolution of Breit-Wigner, radiative tail, and the energy Gaussian must be done numerically. For the final state we have to put into the definition of Γ ee all effects that contribute to 1 e + e, including all 1 e + e γ decays. The value of Γ ee would depend on the minimum detectable photon energy, and the various partial decay widths would not add up to the total Γ. In 1 e + e (γ) decay the divergences in the soft limit of photon emission graphs and in the vertex corrections cancel[21]. But in e + e 1 production there is no photon in the initial state, so there is no cancelation. The correct procedure is use the final state definition of Γ ee = Γ(1 e + e (γ)), and to correct the lowest order prediction for the e + e 1 process to account for whatever radiative effects are not included already in the definition of Γ ee Γ ee = Γ ee(1 + δ vac ) (5.6) where Γ ee is the experimental width, Γ ee is the lowest order width. The vacuum polarization factor (1+δ vac ) includes both leptonic and hadronic term. It varies from charm threshold to 4.14GeV by less than ±2%. It is treated as a constant with the value of References [1] T. Barnes, S. Godfrey and E. S. Swanson, arxiv:hep-ph/552 [2] PDG 6 [3] R. Poling, arxiv:hep-ex/6616 [4] E. Eichten, et al. Phys. Rev. D 21, 23 (198) [5] J. Adler et al. Phys. Rev. Lett. 6, 89 (1988) [6] K. L. He, J. Y. Zhang, W. G. Li, Bes analysis Note. [7] Q. He et al. Phys. Rev. Lett. 95, (25) [8] J. Z. Bai et al. Phys. Rev. D 52, 3781 (1995) (1 + δ vac ) = 1.47 ±.24 (5.7) [9] H. W. Zhao, PhD. thesis(in Chinese);K. L. He, PhD. thesis (in chinese) [1] M. W. Coles et al. Phys. Rev. D 26, 219 (1982) [11] J. Adler et al. Phys. Lett. B 28, 152 (1988) [12] G. Blaylock et al. Phys. Rev. Lett. 58, 2171 (1987) 11

12 [13] N. Cahn, B. Kayser,Phys. Rev. D 22, 2752 (198) [14] T. Barnes, S. Godfrey and E. S. Swanson, Phys. Rev. D 72, 5426 (25) [15] E. J. Eichten, K. Lane and C. Quigg, Phys. Rev. D 73, 1414 (26), Phys. Rev. D 73, 7993 (26) [16] arxiv:hep-ph/ [17] one paper. [18] E. A. Kuraev and V. S. Fadin, Sov. J. Nucl. Phys. 41, 466(1985). [19] G. Altareli and G. Martinelli, CERN Yellow Report 86-2(1986)47; O. Nicrosini and L. Trentadue, Phys. Lett. B 196, 551 (1987). [2] F. A. Berends, G. Burgers and W. L. van Neeren, Nucl. Phys. B 297, 429 (1988); Nucl. Phys. B 34, 921 (1986). [21] T. Kinoshita, J. Math. Phys. 3, 65(1962); T. D. Lee and M. Nauenberg, Phys. Rev. B133, 1549(1964). 12