- Email: [email protected]
Measurement of hadron pair production cross sections at BESIII
Available online at www.sciencedirect.com
Nuclear and Particle Physics Proceedings 270–272 (2016) 62–66 www.elsevier.com/locate/nppp
Measurement of hadron pair production cross sections at BESIII Haiming Hu on behalf of the BESIII Collaboration∗ Institute of High Energy Physics, CAS, Beijing, China.
Abstract In this paper, two results using data samples collected with the BESIII detector at BEPCII collider are presented. (1) the Born cross section of e+ e− → p p¯ at 12 center-of-mass energies from 2232.4 to 3671.0 MeV are measured; the effective electromagnetic form factor is deduced under the assumption |G E | = |G M |; the ratio |G E /G M | and |G M | are extracted by fitting the polar angle distribution; the results show that the |G E /G M | ratios are close to unity, which indicate the data are consistent with the assumption |G E | = |G M | within uncertainties. (2) The cross section of e+ e− → π+ π− in the effective center-of-mass energy region between 600 and 900 MeV is measured by the method of the initial state radiation using the data set with an integrated luminosity of 2.93fb−1 taken at center-of-mass energy of 3773 MeV; the pion form factor |Fπ |2 is extracted; and the contribution to (g−2)μ of the measured cross section to the leading order hadronic vacuum polarization is calculated, which shows aππ,LO (600−900 MeV) = (370.0±2.5stat ±3.3sys )·10−10 . μ 1. Introduction In the Standard Model (SM), hadrons (mesons and baryons) are not elementary particles, they are composed of qq¯ and qqq respectively, and the internal structures and dynamics properties of hadrons are described by the corresponding form factors (FFs). Every exclusive reaction channel in e+ e− annihilation has its own FFs. Among of them, the measurements of FFs to e+ e− → p p¯ and π+ π− have special significance. For example, proton and pion are two simplest and typical baryon and meson, and proton is also the basic building block of the matter, and the precision measurement of the FFs of π+ π− is the most important input in the theoretical calculation to the (g − 2)μ discrepancy between the SM prediction and measurement, and so on. Due to the non-perturbative characteristics of the Quantum Chromodynamics (QCD), FFs can not be calculated from the QCD directly, one has to extract the fundamental information of the FFs by fit the measured differential or total cross section in experiments. ∗ Speaker
Email address: [email protected] (Haiming Hu on behalf of the BESIII Collaboration)
http://dx.doi.org/10.1016/j.nuclphysbps.2016.02.014 2405-6014/© 2016 Elsevier B.V. All rights reserved.
This paper is going to introduce these two measurements at Beijing Spectrometers III (BESIII) briefly, the more details about the experiments may refer to the two original papers [1] and [2] respectively, and the related theoretical formula may refer to the references cited therein.
2. The BESIII Experiment and data sets BEPCII is a double-ring e+ e− collider running at center-of-mass energies between 2.0 and 4.6 GeV and reached a peak luminosity of 0.85 × 1033 cm−2 s−1 at center-of-mass energy of 3770 MeV. The BESIII detector is located at the BEPCII [3], which is designed to fulfill the requirement of the tau-charm physics [4]. Using the data samples collected with the BESIII at 12 center-of-mass energies between 2232.4 and 3671.0 MeV, the Born cross section and FFs of e+ e− → p p¯ are measured, and using the data sample taken at center-ofmass energy 3773.0 MeV with an integrated luminosity of 2.93 fb−1 , the Born (bare) cross section and FFS of e+ e− → π+ π− between 600 and 900 MeV are measured with the initial state radiation (ISR) return method.
H. Hu / Nuclear and Particle Physics Proceedings 270–272 (2016) 62–66
3. Measurement about e+ e− → p p¯ 3.1. The related formula In the e+ e− annihilation processes, the lowest order Feynman diagram for e+ e− → p p¯ is shown in Fig. 1. P’μ=(E’p, p’)
k’μ=(E’e, k’) e+
p
γ*
jμ=
p Pμ=(Ep, p)
Figure 1: The Feynman diagram of
e+ e−
The leptonic current jμ is known in QED, the vertex Γμ (p, p ) in the hadronic current (1)
can be parameterized in term of two time-like FFs, F1 and F2 [5], Γμ (p , p) = γμ F1 (q2 ) +
iσμν qν κ p F2 (q2 ), 2m p
(2)
where q2 > 0. The Eq. (2) can be rearranged by defining the Sachs electric (G E ) and magnetic (G M ) FFs [6]: G E (q2 ) = F1 (q2 ) +
q2 κ p F2 (q2 ), 4m2p
G M (q2 ) = F1 (q2 ) + κ p F2 (q2 ).
(3) (4)
The differential cross section and Born cross section can be written as the function of |G E | and |G M | [7], 2 dσBorn (s) = α 4sβC [|G M (s)|2 (1 + cos2 θ p ) dΩ +
4m2p s
|G E (s)|2 sin2 θ p ],
σBorn (s) =
2m2p 4πα2 β C[|G M |2 + |G E |2 ]. 3s s
(5)
(6)
The meaning of the symbols in above equations are explained in Ref. [1]. In the previous experiments, one often introduce the effective form factor |G|, and assumes |G| = |G E | = |G M |, then |G| can be deduced by |G|2 =
3s σBorn . 4πα2 βC(1 + 2m2p /s)
F(cos θ p ) = Nnorm [1+cos2 θ p +
4m2p
R2 sin2 θ p ], (8) s where Nnorm is the overall normalization factor [1].
σBorn =
→ p p¯ at the lowest order.
Jμ =< p |Jμ (0)|p >= e¯u(p )Γμ (p , p)u(p)
polar angle θ p on the basis of the differential cross section Eq.(5). Some times, one is interested in the ratio R = |G E /G M |, under this case, the Eq.(5) can be rewritten as:
3.2. The data analysis and results Experimentally, the Born cross section of e+ e− → p p¯ is measured by following expression
Jμ=
q2>0
ekμ=(Ee, k)
63
(7)
In principle, one may obtain both |G E | and |G M | respectively by fit the measured distribution of the proton
Nobs − Nbkg , L · ε · (1 + δ)
(9)
where Nobs is the observed number of candidate events, Nbkg the number of background events estimated by MC simulations, L the integrated luminosity of the data sample, ε the detection efficiency determined with MC generator, and (1 + δ) the radiative correction factor. Comparisons of σBorn and |G| measured with BESIII data to the previous experimental measurements are shown in Fig. 2. Compared to the BaBar results [19], the precision of the Born cross section measured √ with BESIII is improved by 30% for data sets with s ≤ 3080.0 MeV, and the precision of the FFs is improved. Both R and Nnorm (G M (s)) can be extracted by fitting the efficiency corrected cos θ p distributions. The polar angular distributions in cos θ p are shown in Fig. 3 for √ s = 2232.4 MeV and 2400.0 MeV, as well as √ for a combined data sample with sub-data samples at s = 3050.0 MeV, 3060.0 MeV and 3080.0 MeV. The distributions are fitted with the Eq. (8), and the fit results are shown in Fig. 3. The ratios R = |G E /G M | are shown in Fig. 4, and the results from the previous experiments are also presented on the same plot for comparison. 4. Measurement about e+ e− → π+ π− The Born (bare) cross section√ of e+ e− → π+ π− at effective center-of-mass energy s can be expressed in term of the form factor Fπ (s ) [8, 9], πα2 β3π |Fπ (s )|2 . (10) 3s In experiment, the measured cross section contains the vacuum polarization effect, but the final state radiation (FSR) correction considered, which is called as the . Bare and dressed cross sections dressed one, σdressed ππ are related by following relation [10]: 2 σdressed α(0) bare dressed σ = = σππ · , (11) α(s) |1 − Π(s)|2 σ0ππ (s ) =
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Figure 2: Comparison of (a) the Born cross section and (b) the effective FF |G| between this measurement and previous experiments, shown on a logarithmic scale for invariant p p¯ masses from 2.20 to 3.70 GeV/c2 .
60
80 60 40 20
70
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cosθp
0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8
cosθp
Figure 3: Efficiency corrected distributions of cos θ p and fit results for data at c.m. energies (a) 2232.4, (b) 2400.0 MeV and (c) a combined sample with c.m. energy at 3050.0, 3060.0 and 3080.0 MeV. The dots with error bars represent data. The solid line (black) represents the overall fit result. The dot-dashed line (in red) shows the contribution of the magnetic FF and the dashed line (in blue) of the electric FF.
where Π(s) is the vacuum polarization factor [11], the explanations can be found in Ref.[12]. |Fπ |2 can be ex= σ(e+ e− → π+ π− ), tracted using the measured σdressed ππ which contains the vacuum polarization, but corrected for FSR effects. 4.1. The methods The Born cross section of e+ e− → π+ π− is measured by two independent normalization schemes with BESIII data [2]. 4.1.1. The first method The following formula is used to measure the cross section and extract the FFs: σbare ππ(γFSR ) =
Nππγ · (1 + δππ FS R )
ππγ L · global · H(s) · δvac
,
(12)
where Nππγ is the number of signal events in data, L the luminosity of the data set, and H the radiator function, ππγ the global efficiency of the signal determined by global MC, δvac the vacuum polarization correction, δππ FSR the
final state radiation correction factor. The tree-level diagrams for ISR and FSR contribution to e+ e− → π+ π− γ can be found in Ref. [13]. 4.1.2. The second method As a cross check, the following formula is used to measure the cross section and FFs: μμγ μμ Nππγ global 1 + δFSR bare bare σμμ . (13) σππ(γFSR ) = ππγ Nμμγ global 1 + δππ FSR In this way, the common factors L, H and δvac for e+ e− → π+ π− γIS R and e+ e− → μ+ μ− γIS R can be canμμγ is the global efficiency of dimuon celed. Where, global μμ selection, δFSR the FSR correction factor to μ+ μ− final state, σbare μμ the bare cross section of dimuon given by 4πα2 βμ (3 − βμ ) σbare , (14) = μμ 3s 2 √ where s is the effective center-of-mass energy of 2
μ+ μ− system in ISR return, βμ = 1 − 4m2μ /s the muon speed, and mμ the mass of muon.
final state correction is considered. The result for |Fπ |2 is shown in Fig. 7, which includes vacuum polarizations, but final state radiation effects are excluded, and the exactly the same fit formula and fit procedure are applied as described in Ref. [14]. Fig. 8 shows the difference between fit and data. The comparisons between the BESIII fit and the previous measurements, BaBar [14], KLOE [15, 16, 17], are illustrated in Fig. 9.
BESIII BaBar PS170
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1600
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1200 1000 800 600 400 200
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Figure 6: The measured bare e+ e− → π+ π− (γFSR ) cross section. Only the statistical error are shown.
600 400 200 0.6
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0.65
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Figure 5: Comparison between the results using luminosity (black) and normalization by σμμ (blue). The lower panel shows the ratio of these results together with a liner fit (blue line) to quantify their difference.
45
BESIII fit
40
BESIII
35 |Fπ|2
luminosity / R ratio
+
σbare(e e-→ π+π-(γ
FSR
)) [nb]
Figure 4: The measured ratio of electric to magnetic FFs |G E /G M | at different c.m. energy from BESIII (filled circles), BaBar at SLAC (open crosses) and PS170 at LEAR/CERN (open circles).
)) [nb]
2.0
65
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2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
σbare(e+e- → π+π- (γ
|GE/GM|
H. Hu / Nuclear and Particle Physics Proceedings 270–272 (2016) 62–66
30 25 20 15
4.2. Data analysis Events of the type e+ e− → π+ π− γISR in the invariant mass range 600 < mππ < 900 MeV are selected from √ the data sample taken at center-of-mass energy s = 3773 MeV and with integrated luminosity of 2.93 fb−1 [2]. The integrated luminosity of the data set is measured with the larger angle Bhabha scattering events, L = (2931.8 ± 0.2stat ± 13.8sys )pb−1 with a relative uncertainty of 0.5%. The track-based muonpion separation based on the Artificial Neural Network (ANN), which is trained using π+ π− γ and μ+ μ− γ MC samples, is used to suppress dominant background e+ e− → μ+ μ− γISR . The possible remaining backgrounds are subtracted using MC method. 4.3. Results The results for σbare (e+ e− → π+ π− (γFSR )) as a func√ tion of s = mππ is illustrated in Fig. 6. The cross section is corrected for vacuum polarization effect, and the
10 0.6
0.65
0.7
0.75
s' [GeV]
0.8
0.85
0.9
Figure 7: The measured squared pion form factor |Fπ |2 , only statistical errors are shown, the red line represents a fit using the GounarisSakurai parametrization.
4.4. Contribution to (g − 2)μ The contribution of BESIII cross section measurement to the hadronic contribution of (g−2)μ is computed according to the following integral: 1 (0.6−0.9GeV) = 3 aππ,LO μ 4π
(0.9GeV) 2
ds K(s )σbare ππ(γ) (s ), (15)
(0.6GeV)2
where K(s ) is the kernel function [18]. As summarized in Fig. 10, the results from KLOE 08 [15], 10 [16], 12 [17], BaBar [14] also showed for a comparison.
H. Hu / Nuclear and Particle Physics Proceedings 270–272 (2016) 62–66
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0.15
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0
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BESIII 0.05
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0.65
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s' [GeV]
0.8
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0.9
Figure 8: Relative difference of the form factor squared from BaBar and the BESIII fit. Statistic and systematic uncertainties are included in the data points. The with of the BESIII band shows the systematic uncertainty only.
BESIII fit KLOE 08
|Fπ|2 / BESIII fit - 1
KLOE 10 KLOE 12
0.05
365
370
375
aπμπ,LO(600
380
-10
- 900 MeV) [10 ]
385
390
395
Figure 10: The contribution of BESIII cross section measurement to the hadronic contribution of (g − 2)μ , the band shows the 1σ range of the BESIII result.
t ρ(770) mass region between 600 MeV and 900 MeV, and the form factor |Fπ | is fitted. The two-pion contri(600 − 900 bution to (g − 2)μ is determined to be aππ,LO μ MeV) = (370.0 ± 2.5stat ± 3.3sys ) · 10−10 . It is found that a deviation of more than 3σ between the SM prediction of (g − 2)μ and its direct measurement is confirmed.
0.15 0.1
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0
-0.05
References
-0.1 -0.15 0.6
0.65
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s' [GeV]
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Figure 9: Relative difference of the form factor squared from KLOE and the BESIII fit. Statistic and systematic uncertainties are included in the data points. The with of the BESIII band shows the systematic uncertainty only.
5. Summary The Born cross section of e+ e− → p p¯ is measured, and the effective form factor |G| is extracted under the assumption |G E | = |G M | using data between 2232.4 MeV and 3671.0 √ MeV. The precision of the Born cross section with s ≤ 3.08 GeV is between 6.0% and 18.9% which is much improved comparing with the previous results from BaBar experiment [19], and the precision is comparable with those of previous results at √ s > 3.08 GeV. The |G√E /G M | ratios and |G M | are extracted at the energies s = 2232.4 MeV and 2400.0 MeV, and a combined data sample with energies of 3050.0 MeV, 3060.0 MeV and 3080.0 MeV. The measured |G E /G M | ratios are close to unity which are consistent with the BaBar experiment in the same q2 region. Using the method of ISR return, a new measurement of the cross section σbare (e+ e− → π+ π− (γFSR )) is performed with an accuracy of 0.9% in the dominan-
[1] M. Ablikim et al. [BESIII Colaboration], Phys. Rev. D91, 112004 (2015). [2] M. Ablikim et al. [BESIII Colaboration], arXiv:1507.08188 [hep-ex]. [3] M. Ablikim et al. [BESIII Colaboration], Nucl. Instrum. Meth. A614, 345 (2010). [4] K. Chao and Y. Wang, Physics at BESIII, Modn. Phys. A Supplement, V24, (2009), Scientific World. [5] L. N. Hand, Rev. Mod. Phys. 35, 335 (1963). [6] F. J. Ernst et al. Phys. Rev. 119, 1105 (1960). [7] N. Cabibbo and R. Gatto, Phys. Rev. 124 1577 (1961). [8] B.L. Ioffe, V.A. Khooze, and L.N. Lipatov, Hard processes, Volume 1, Phenomenology Quark-Parton Model, (1984), North Holland. [9] F. E. Close, An Introduction to Quarks and Partons, (1979), Academic Press. [10] S. Actis et al. arXiv:0912.0749 [hep-ph]. [11] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum field Theory, (1995), Westview Press. [12] A. Hofer, ¨ et al. Eur. Phys. J. C24, 51, (2002). [13] S. Pacetti, Nucl. Phys. B(Proc. Suppl.)162, 172(2006). [14] J.P. Lees et al. [BABAR Collaboration], Phys. Rev. D86, 032013 (2013). [15] F. Ambrosino et al. [KLEO Collaboration], Phys. Lett. B670, 285 (2009). [16] F. Ambrosino et al. [KLEO Collaboration], Phys. Lett. B700, 102 (2011). [17] D. Babusci et al. [KLEO Collaboration], Phys. Lett. B720, 336 (2013). [18] S. Eidelman, F. Jegerlehner, Z. Phys. C67, 585 (1995). [19] J. P. Lees et al. (BaBar Collaboration), Phys. Rev. D 87, 092005 (2013).
Nuclear and Particle Physics Proceedings 270–272 (2016) 62–66 www.elsevier.com/locate/nppp
Measurement of hadron pair production cross sections at BESIII Haiming Hu on behalf of the BESIII Collaboration∗ Institute of High Energy Physics, CAS, Beijing, China.
Abstract In this paper, two results using data samples collected with the BESIII detector at BEPCII collider are presented. (1) the Born cross section of e+ e− → p p¯ at 12 center-of-mass energies from 2232.4 to 3671.0 MeV are measured; the effective electromagnetic form factor is deduced under the assumption |G E | = |G M |; the ratio |G E /G M | and |G M | are extracted by fitting the polar angle distribution; the results show that the |G E /G M | ratios are close to unity, which indicate the data are consistent with the assumption |G E | = |G M | within uncertainties. (2) The cross section of e+ e− → π+ π− in the effective center-of-mass energy region between 600 and 900 MeV is measured by the method of the initial state radiation using the data set with an integrated luminosity of 2.93fb−1 taken at center-of-mass energy of 3773 MeV; the pion form factor |Fπ |2 is extracted; and the contribution to (g−2)μ of the measured cross section to the leading order hadronic vacuum polarization is calculated, which shows aππ,LO (600−900 MeV) = (370.0±2.5stat ±3.3sys )·10−10 . μ 1. Introduction In the Standard Model (SM), hadrons (mesons and baryons) are not elementary particles, they are composed of qq¯ and qqq respectively, and the internal structures and dynamics properties of hadrons are described by the corresponding form factors (FFs). Every exclusive reaction channel in e+ e− annihilation has its own FFs. Among of them, the measurements of FFs to e+ e− → p p¯ and π+ π− have special significance. For example, proton and pion are two simplest and typical baryon and meson, and proton is also the basic building block of the matter, and the precision measurement of the FFs of π+ π− is the most important input in the theoretical calculation to the (g − 2)μ discrepancy between the SM prediction and measurement, and so on. Due to the non-perturbative characteristics of the Quantum Chromodynamics (QCD), FFs can not be calculated from the QCD directly, one has to extract the fundamental information of the FFs by fit the measured differential or total cross section in experiments. ∗ Speaker
Email address: [email protected] (Haiming Hu on behalf of the BESIII Collaboration)
http://dx.doi.org/10.1016/j.nuclphysbps.2016.02.014 2405-6014/© 2016 Elsevier B.V. All rights reserved.
This paper is going to introduce these two measurements at Beijing Spectrometers III (BESIII) briefly, the more details about the experiments may refer to the two original papers [1] and [2] respectively, and the related theoretical formula may refer to the references cited therein.
2. The BESIII Experiment and data sets BEPCII is a double-ring e+ e− collider running at center-of-mass energies between 2.0 and 4.6 GeV and reached a peak luminosity of 0.85 × 1033 cm−2 s−1 at center-of-mass energy of 3770 MeV. The BESIII detector is located at the BEPCII [3], which is designed to fulfill the requirement of the tau-charm physics [4]. Using the data samples collected with the BESIII at 12 center-of-mass energies between 2232.4 and 3671.0 MeV, the Born cross section and FFs of e+ e− → p p¯ are measured, and using the data sample taken at center-ofmass energy 3773.0 MeV with an integrated luminosity of 2.93 fb−1 , the Born (bare) cross section and FFS of e+ e− → π+ π− between 600 and 900 MeV are measured with the initial state radiation (ISR) return method.
H. Hu / Nuclear and Particle Physics Proceedings 270–272 (2016) 62–66
3. Measurement about e+ e− → p p¯ 3.1. The related formula In the e+ e− annihilation processes, the lowest order Feynman diagram for e+ e− → p p¯ is shown in Fig. 1. P’μ=(E’p, p’)
k’μ=(E’e, k’) e+
p
γ*
jμ=
p Pμ=(Ep, p)
Figure 1: The Feynman diagram of
e+ e−
The leptonic current jμ is known in QED, the vertex Γμ (p, p ) in the hadronic current (1)
can be parameterized in term of two time-like FFs, F1 and F2 [5], Γμ (p , p) = γμ F1 (q2 ) +
iσμν qν κ p F2 (q2 ), 2m p
(2)
where q2 > 0. The Eq. (2) can be rearranged by defining the Sachs electric (G E ) and magnetic (G M ) FFs [6]: G E (q2 ) = F1 (q2 ) +
q2 κ p F2 (q2 ), 4m2p
G M (q2 ) = F1 (q2 ) + κ p F2 (q2 ).
(3) (4)
The differential cross section and Born cross section can be written as the function of |G E | and |G M | [7], 2 dσBorn (s) = α 4sβC [|G M (s)|2 (1 + cos2 θ p ) dΩ +
4m2p s
|G E (s)|2 sin2 θ p ],
σBorn (s) =
2m2p 4πα2 β C[|G M |2 + |G E |2 ]. 3s s
(5)
(6)
The meaning of the symbols in above equations are explained in Ref. [1]. In the previous experiments, one often introduce the effective form factor |G|, and assumes |G| = |G E | = |G M |, then |G| can be deduced by |G|2 =
3s σBorn . 4πα2 βC(1 + 2m2p /s)
F(cos θ p ) = Nnorm [1+cos2 θ p +
4m2p
R2 sin2 θ p ], (8) s where Nnorm is the overall normalization factor [1].
σBorn =
→ p p¯ at the lowest order.
Jμ =< p |Jμ (0)|p >= e¯u(p )Γμ (p , p)u(p)
polar angle θ p on the basis of the differential cross section Eq.(5). Some times, one is interested in the ratio R = |G E /G M |, under this case, the Eq.(5) can be rewritten as:
3.2. The data analysis and results Experimentally, the Born cross section of e+ e− → p p¯ is measured by following expression
Jμ=
q2>0
ekμ=(Ee, k)
63
(7)
In principle, one may obtain both |G E | and |G M | respectively by fit the measured distribution of the proton
Nobs − Nbkg , L · ε · (1 + δ)
(9)
where Nobs is the observed number of candidate events, Nbkg the number of background events estimated by MC simulations, L the integrated luminosity of the data sample, ε the detection efficiency determined with MC generator, and (1 + δ) the radiative correction factor. Comparisons of σBorn and |G| measured with BESIII data to the previous experimental measurements are shown in Fig. 2. Compared to the BaBar results [19], the precision of the Born cross section measured √ with BESIII is improved by 30% for data sets with s ≤ 3080.0 MeV, and the precision of the FFs is improved. Both R and Nnorm (G M (s)) can be extracted by fitting the efficiency corrected cos θ p distributions. The polar angular distributions in cos θ p are shown in Fig. 3 for √ s = 2232.4 MeV and 2400.0 MeV, as well as √ for a combined data sample with sub-data samples at s = 3050.0 MeV, 3060.0 MeV and 3080.0 MeV. The distributions are fitted with the Eq. (8), and the fit results are shown in Fig. 3. The ratios R = |G E /G M | are shown in Fig. 4, and the results from the previous experiments are also presented on the same plot for comparison. 4. Measurement about e+ e− → π+ π− The Born (bare) cross section√ of e+ e− → π+ π− at effective center-of-mass energy s can be expressed in term of the form factor Fπ (s ) [8, 9], πα2 β3π |Fπ (s )|2 . (10) 3s In experiment, the measured cross section contains the vacuum polarization effect, but the final state radiation (FSR) correction considered, which is called as the . Bare and dressed cross sections dressed one, σdressed ππ are related by following relation [10]: 2 σdressed α(0) bare dressed σ = = σππ · , (11) α(s) |1 − Π(s)|2 σ0ππ (s ) =
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Figure 2: Comparison of (a) the Born cross section and (b) the effective FF |G| between this measurement and previous experiments, shown on a logarithmic scale for invariant p p¯ masses from 2.20 to 3.70 GeV/c2 .
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Figure 3: Efficiency corrected distributions of cos θ p and fit results for data at c.m. energies (a) 2232.4, (b) 2400.0 MeV and (c) a combined sample with c.m. energy at 3050.0, 3060.0 and 3080.0 MeV. The dots with error bars represent data. The solid line (black) represents the overall fit result. The dot-dashed line (in red) shows the contribution of the magnetic FF and the dashed line (in blue) of the electric FF.
where Π(s) is the vacuum polarization factor [11], the explanations can be found in Ref.[12]. |Fπ |2 can be ex= σ(e+ e− → π+ π− ), tracted using the measured σdressed ππ which contains the vacuum polarization, but corrected for FSR effects. 4.1. The methods The Born cross section of e+ e− → π+ π− is measured by two independent normalization schemes with BESIII data [2]. 4.1.1. The first method The following formula is used to measure the cross section and extract the FFs: σbare ππ(γFSR ) =
Nππγ · (1 + δππ FS R )
ππγ L · global · H(s) · δvac
,
(12)
where Nππγ is the number of signal events in data, L the luminosity of the data set, and H the radiator function, ππγ the global efficiency of the signal determined by global MC, δvac the vacuum polarization correction, δππ FSR the
final state radiation correction factor. The tree-level diagrams for ISR and FSR contribution to e+ e− → π+ π− γ can be found in Ref. [13]. 4.1.2. The second method As a cross check, the following formula is used to measure the cross section and FFs: μμγ μμ Nππγ global 1 + δFSR bare bare σμμ . (13) σππ(γFSR ) = ππγ Nμμγ global 1 + δππ FSR In this way, the common factors L, H and δvac for e+ e− → π+ π− γIS R and e+ e− → μ+ μ− γIS R can be canμμγ is the global efficiency of dimuon celed. Where, global μμ selection, δFSR the FSR correction factor to μ+ μ− final state, σbare μμ the bare cross section of dimuon given by 4πα2 βμ (3 − βμ ) σbare , (14) = μμ 3s 2 √ where s is the effective center-of-mass energy of 2
μ+ μ− system in ISR return, βμ = 1 − 4m2μ /s the muon speed, and mμ the mass of muon.
final state correction is considered. The result for |Fπ |2 is shown in Fig. 7, which includes vacuum polarizations, but final state radiation effects are excluded, and the exactly the same fit formula and fit procedure are applied as described in Ref. [14]. Fig. 8 shows the difference between fit and data. The comparisons between the BESIII fit and the previous measurements, BaBar [14], KLOE [15, 16, 17], are illustrated in Fig. 9.
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Figure 6: The measured bare e+ e− → π+ π− (γFSR ) cross section. Only the statistical error are shown.
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Figure 5: Comparison between the results using luminosity (black) and normalization by σμμ (blue). The lower panel shows the ratio of these results together with a liner fit (blue line) to quantify their difference.
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)) [nb]
Figure 4: The measured ratio of electric to magnetic FFs |G E /G M | at different c.m. energy from BESIII (filled circles), BaBar at SLAC (open crosses) and PS170 at LEAR/CERN (open circles).
)) [nb]
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4.2. Data analysis Events of the type e+ e− → π+ π− γISR in the invariant mass range 600 < mππ < 900 MeV are selected from √ the data sample taken at center-of-mass energy s = 3773 MeV and with integrated luminosity of 2.93 fb−1 [2]. The integrated luminosity of the data set is measured with the larger angle Bhabha scattering events, L = (2931.8 ± 0.2stat ± 13.8sys )pb−1 with a relative uncertainty of 0.5%. The track-based muonpion separation based on the Artificial Neural Network (ANN), which is trained using π+ π− γ and μ+ μ− γ MC samples, is used to suppress dominant background e+ e− → μ+ μ− γISR . The possible remaining backgrounds are subtracted using MC method. 4.3. Results The results for σbare (e+ e− → π+ π− (γFSR )) as a func√ tion of s = mππ is illustrated in Fig. 6. The cross section is corrected for vacuum polarization effect, and the
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Figure 7: The measured squared pion form factor |Fπ |2 , only statistical errors are shown, the red line represents a fit using the GounarisSakurai parametrization.
4.4. Contribution to (g − 2)μ The contribution of BESIII cross section measurement to the hadronic contribution of (g−2)μ is computed according to the following integral: 1 (0.6−0.9GeV) = 3 aππ,LO μ 4π
(0.9GeV) 2
ds K(s )σbare ππ(γ) (s ), (15)
(0.6GeV)2
where K(s ) is the kernel function [18]. As summarized in Fig. 10, the results from KLOE 08 [15], 10 [16], 12 [17], BaBar [14] also showed for a comparison.
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Figure 8: Relative difference of the form factor squared from BaBar and the BESIII fit. Statistic and systematic uncertainties are included in the data points. The with of the BESIII band shows the systematic uncertainty only.
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Figure 10: The contribution of BESIII cross section measurement to the hadronic contribution of (g − 2)μ , the band shows the 1σ range of the BESIII result.
t ρ(770) mass region between 600 MeV and 900 MeV, and the form factor |Fπ | is fitted. The two-pion contri(600 − 900 bution to (g − 2)μ is determined to be aππ,LO μ MeV) = (370.0 ± 2.5stat ± 3.3sys ) · 10−10 . It is found that a deviation of more than 3σ between the SM prediction of (g − 2)μ and its direct measurement is confirmed.
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References
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Figure 9: Relative difference of the form factor squared from KLOE and the BESIII fit. Statistic and systematic uncertainties are included in the data points. The with of the BESIII band shows the systematic uncertainty only.
5. Summary The Born cross section of e+ e− → p p¯ is measured, and the effective form factor |G| is extracted under the assumption |G E | = |G M | using data between 2232.4 MeV and 3671.0 √ MeV. The precision of the Born cross section with s ≤ 3.08 GeV is between 6.0% and 18.9% which is much improved comparing with the previous results from BaBar experiment [19], and the precision is comparable with those of previous results at √ s > 3.08 GeV. The |G√E /G M | ratios and |G M | are extracted at the energies s = 2232.4 MeV and 2400.0 MeV, and a combined data sample with energies of 3050.0 MeV, 3060.0 MeV and 3080.0 MeV. The measured |G E /G M | ratios are close to unity which are consistent with the BaBar experiment in the same q2 region. Using the method of ISR return, a new measurement of the cross section σbare (e+ e− → π+ π− (γFSR )) is performed with an accuracy of 0.9% in the dominan-
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