Measurement of proton electromagnetic form factors in the time-like region using initial state radiation at BESIII

The electromagnetic process ee ! p p is studied with the initial-state-radiation technique using 7.5 fb 1 of data collected by the BESIII experiment at seven energy points from 3.773 to 4.600 GeV. The Born cross section and the effective form factor of the proton are measured from the production threshold to 3.0 GeV/c using the p p invariantmass spectrum. The ratio of electric and magnetic form factors of the proton is determined from the analysis of the proton-helicity angular distribution.


Introduction
The investigation of nucleon structure through electromagnetic probes plays a central role in the understanding of strong interactions. Space-like (SL) photons (momentum transfer squared q 2 < 0) in elastic electron-nucleon scattering experiments allow an accurate description of the three-dimensional structure of the nucleon through the study of the electromagnetic form factors (FFs). The electric FF G E and the magnetic FF G M are assumed to be analytic functions of q 2 [1], and thus are also defined for the time-like (TL) kinematic domain q 2 > H. Furthermore, it is possible to relate SL and TL FFs through dispersion relations [2]. In the TL region, nucleon FFs can be associated with the time evolution of the charge and magnetic distributions inside the nucleon [3].
Compared to the SL sector, where a percent level precision has been achieved [4], the knowledge of the TL proton FF is rather limited. Proton FFs in the TL region have been studied by various experiments in the direct annihilation processes e + e 3 p" p [5][6][7][8][9][10][11][12][13][14][15][16] and p" p 3 e + e [17][18][19][20], and in the initial-state-radiation (ISR) process e + e 3 p" p [21][22][23][24]. Due to low statistics, many previous experiments have only determined the absolute value of the effective FF of the proton from the cross-section measurement. More recent measurements [14,15,18,21,22] have been able to determine the ratio of the proton FF absolute values (R em a jG E j=jG M j) in the p" p invariant mass (M p p ) region below 3.08 GeV/c 2 . The best determination of R em , with a precision of around IH7, has been achieved by BESIII [16].
The ISR technique with an undetected photon has been used in our previous study [24] of the process e + e 3 p" p to measure the TL proton FFs. In that analysis, events were selected where the ISR photon was emitted at small polar angles (SA-ISR), and hence the threshold region below 2 GeV/c was not accessible due to the limited angular acceptance of the BESIII tracking system. In this Letter we extend our previous study to the case where the ISR photon is emitted at large polar angles (LA-ISR) and is detected. This al-lows access to the threshold region and provides measurements of the proton helicity angle p in the full M p p range, in contrast to the analysis of the SA-ISR events.
By analyzing the distribution of p , defined in the process e + e 3 p" p as the angle between the proton momentum in the p" p rest frame and the momentum of the p" p system in the e + e c.m. frame, it is possible to determine the ratio of the proton FFs. In the e + e center-of-mass (c.m.) frame, the differential cross section for the process e + e 3 p" p under the assumption of one virtual photon exchange ( where q 2 is equal to the square of the p" p invariant mass M p p , % 1 137 is the fine structure constant, a p I I= is the velocity of the proton with a q 2 =Rm 2 p and m p the proton mass, and # is the polar angle of the proton in the e + e c.m. frame where the z-axis points along the direction of the positron momentum. The Coulomb factor g a y 1 e y with y a accounts for the electromagnetic interaction between the outgoing proton and antiproton [22,26]. The cross section depends on the moduli of the magnetic and electric FFs, which can be determined from the analysis of the proton angular distribution. The precise knowledge of the FFs in a wide kinematic region probes the transition region, from non-perturbative to perturbative QCD (pQCD).
By integrating the differential cross section (Eq. (1)), the total cross section for the process e + e 3 p" p is obtained, p p @q 2 A a R 2 g Qq 2 jG M j 2 C jG E j 2 P : An effective FF is introduced as a linear combination of jG M j 2 and jG E j 2 , jG e j a r PjG M j 2 C jG E j 2 P C I ; (3) which is equivalent to jG M j determined under the assumption of jG M j a jG E j. A complementary approach to study the e + e 3 p" p process is provided by the ISR technique. This technique makes use of the emission of at least one high energy photon off the beam particles ( Figure 2) which reduces the invariant mass of the p" p system in the final state.
where E £ is the energy of the ISR photon in the e + e c.m. frame, W @s; xA [27] is the radiator function which gives the probability of ISR photon emission, and m e and p s are the electron mass and the c.m. energy of the beams, respectively. In the study of the e + e 3 p" p process, the cross section for the process e + e 3 p" p and the ratio of the proton FFs can be measured over the full M p p range from the p" p threshold to p s.

BESIII experiment and data sets
The BESIII experiment collects data at the BEPCII electron-positron collider, which operates at c.m. energies between p s = 2.0 and 4.7 GeV. The baryon TL FFs can be measured at BESIII both in ISR and in direct annihilation processes [28]. In this letter the investigation of the ISR process e + e 3 p" p is reported. The data sets used in this analysis [29,30] have been collected by BESIII at seven c.m. energies between 3.773 and 4.600 GeV with a total integrated luminosity of 7.5 fb 1 (see Table 1).
The cylindrical BESIII detector [31]  software [34], are used to optimize the event selection criteria, estimate the background contamination and determine the selection efficiency. The signal process e + e 3 p" p is generated with the event generator PHOKHARA 9.1 [35], which includes radiative corrections of ISR up to next-to-leading order, final-stateradiation and vacuum polarization. Inclusive MC samples generated with BesEvtGen event generator [36] are used to simulate all the hadronic final states containing u, d and s quarks. The dominant background channel, e + e 3 p" p 0 , is generated exclusively using the phase space CONEXC [36] generator. 3. Selection of e + e 3 p" p events is then assumed to be the ISR photon candidate.
After the event reconstruction, a four-constraint (4C) kinematic fit is performed requiring the fourmomentum conservation between the initial e + e system and the final p" p system. Events are selected as e + e 3 p" p candidates if they fulfill the requirement 2 4C < SH. The background from the process e + e 3 p" p 0 can not be completely removed by means of the kinematic fitting, and a dedicated background evaluation is performed, as described in the next section. Figure 3 shows the combined M p p spectrum for p" p candidates selected at the seven energy points. The residual e + e 3 p" p 0 background discussed in Section 4 is also shown (blue histogram in Figure 3). A clear peak from the resonance decay of J= 3 p" p is seen in the spectrum. By fitting the J= peak using a Breit-Wigner function convolved with a Gaussian, the number of resonance decays J= 3 p" p @N J= A for each data sample is determined. The branching fraction J= 3 p" p can be calculated for each data sample individually as follows [37]: e + e ¢ f@J= 3 p" pA a sm J= IP 2 N J= J= W @s; x J= Av ; (5) where m J= is the mass of the resonance, W @s; x J= A is the radiator function (x J= a I M 2 J= =s), e + e is the electronic width of the J= [38] and v is the in-

Background estimation
For the processes e + e 3 + , K + K , e + e and + no MC events survive the selection cuts described in Section 3. The residual background from these sources can be neglected, given the number of generated MC event exceeds the number of events expected in data.
The main source of background for the process under study is e + e 3 p" p 0 . To ensure a good description of this background in the simulation, the MC distri-

Ratio of proton form factors
The ratio of the electric and magnetic FFs R em is determined by analyzing the distribution of os p dN d os p a ep M @os p ; M p p AC R 2 em P p E @os p ; M p p A: (7) where N is the number of selected e + e 3 p" p candidates after e + e 3 p" p 0 background subtraction. The shapes of the magnetic contribution p M @os p ; M p p A and the electric contribution p E @os p ; M p p A are determined from the MC simulation, which includes the radiative corrections. The distributions obtained for p M and p E in a given M p p interval are approximately proportional to I C os 2 p and sin 2 p , respectively, as follows from Eq. (1). The factor 1 2 arises from the normalization of p M and p E to the same integral, and the parameter e is an overall normalization factor. In the fit function, is calculated as the mean value over the p" p mass interval.
The ratio of the proton FFs is determined by fitting the os p distribution in six M p p intervals from threshold to 3.0 GeV/c 2 , with M p p reconstructed from the measured tracks of the proton and antiproton candidates. For each data set and M p p interval, the estimated background from the process e + e 3 p" p 0 is subtracted from the number of signal candidates.
The remaining signal is corrected for the selection efficiency calculated with corresponding MC samples and the efficiency-corrected distributions from all data sets are combined. Figure 4 shows the os p distribution for six M p p intervals, and the results of the fits using Eq. (7). Table 2 summarizes the R em ratios obtained from the fits.  The total systematic uncertainties for the measurement of the ratio of the proton FFs are listed in Table 2.   with the results from previous experiments: BABAR [22], PS170 [18], CMD-3 [15], and BESIII [14,16,24]. Both statistical and systematic uncertainties are included in all the results.
6. Cross section for the process e + e 3 p" p and proton effective form factor The Born cross section for the process e + e 3 p" p is calculated in each M p p interval i and for each data sample j (j a I; P; :::; U) as follows: ij a x ij x bkg ij ij @I C ij Av ij ; (8) where x ij is the number of selected e + e 3 p" p candidates, x bkg ij is the number of e + e 3 p" p 0 background events, ij is the detection efficiency, @I C ij A is the radiative correction factor calculated from the MC simulations and v ij is the ISR binned integrated luminosity. The index j runs over the seven c.m. energies.
The binned integrated luminosity v ij is calculated as: v ij a Z W @s j ; x ij Av j dx ij ; x ij a I q 2 ij s j ; (9) where W @s j ; x ij A [Eq. (4)] is a function of the c.m. energy squared s j (j a I; P; :::; U) and the energy fraction x ij , and v j is the integrated luminosity collected at the c.m. energy p s j ( Table 1). The integration in Eq. (9) is performed over the width of the selected M p p interval.
The sum over the seven energy points for the binned integrated luminosity (v i ), the selected e + e 3 p" p candidates (x i ) and the background events (x bkg i ) in each M p p interval is given in Table 4. The averages over the seven c.m. energy points of the selection efficiencies (" i ) and the radiative correction factors @I C i A calculated as: " i a ¦ j @ ij v ij A=v i ; @I C i A a ¦ j @@I C ij Av ij A=v i ; (10) are also listed in  The experimental resolution on M p p is typically eight times smaller than the width of M p p intervals and the event migration across the intervals is relatively small, well below the total uncertainty of the measured cross sections. The migration effect is taken into account using the MC simulation.
Several sources are considered as contributing to the systematic uncertainties. The uncertainties from tracking and PID efficiencies and the E=p requirement are each 1.0% per track for all the p" p mass intervals [14].
The systematic uncertainty of the luminosity measurement is 1.0% for all the data sets [29,30]. The systematic uncertainties due to the 4C-kinematic fit and the background estimation are determined using the same methods as described in Section 5. The radiative correction factor @ICA is calculated in PHOKHARA generator with a theoretical uncertainty of 1%. The uncertainty from the energy dependence of the Born cross section used for the @ICA calculation is determined by varying the line shape of the cross section from PHOKHARA event generator within the errors of the measured cross section. The systematic uncertainties listed above are added in quadrature and are summarized in Table 3.
In Table 4 [14,16,24], BABAR [22,23], E835 [19,20], Fenice [9][10][11], PS170 [18], E760 [17], DM1 [6], DM2 [7,8], BES [13], CLEO [12], and ADONE73 [5]. The blue dashed curve shows the parameterization from Ref. [41] based on Eq. (12). results from this analysis for the e + e 3 p" p cross section and the proton effective FF, respectively, together with results from previous experiments.  Figure 7: The effective FF of the proton, after subtraction of the smooth function described by Eq. (12), as a function of the relative momentum p. The data are from the present analysis (red points) and previous measurements of BESIII [16,24] and BABAR [22,23]. The data on the TL effective FF are best reproduced by the function proposed in Ref. [41], jG e j a e @I C q 2 =m 2 a AI q 2 =q 2 0 2 ; q 2 0 a H:UI @GeV/cA 2 ; where e a U:U and m 2 a a IR:V @GeV/cA 2 are the fit parameters obtained previously in Ref. [43]. This function is illustrated in Fig. 6b by the blue dashed curve and reproduces the behavior of the effective FF over the full q 2 range. However, the measurements indicate oscillating structures which are clearly seen when the residuals are plotted as a function of the relative momentum p of the final proton and antiproton [42]. Figure 7 shows the values of the proton effective FF as a function of p after subtraction of the smooth function described by Eq. (12). The black curve in Fig. 7 describes the peri-odic oscillations and has the form [42] F p a A osc exp@ B osc pA os@C osc p C D osc A; (13) where A osc a H:HS, B osc a H:U @GeV/cA 1 , C osc a S:S @GeV/cA 1 and D osc a H:H have been obtained from a fit to the BABAR data [43].

Summary
Using seven data sets with a total integrated luminosity of 7.5 fb 1 collected by the BESIII experiment at p s between 3.773 and 4.600 GeV, the ratio of the proton electromagnetic FF absolute values, the Born cross section for the process e + e 3 p" p and the effective FF of the proton are measured from the p" p threshold to 3.0 GeV/c 2 through the ISR process e + e 3 p" p.
This measurement confirms an enhancement of the ratio of FFs in the M p p region below 2.2 GeV/c 2 previously observed by BABAR and BESIII and differs from the behavior reported by PS170 [18]. Close to the threshold, the observed ratio is compatible with unity within the uncertainties. The results on the Born cross section for the process e + e 3 p" p and the proton effective FF presented in this work are in a good agreement with the measurements from the previous experiments [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]24]. In particular, we reproduce the structures seen in the BABAR and previous BESIII measurements of the proton effective FF. The origin of these oscillating structures can be attributed to an interference effect involving rescattering processes in the final state [43] or to independent resonant structures, as in Ref. [44]. The precision of the measurements obtained in this work are comparable to or lower than that achieved in previous BESIII studies [14,16,24] using the direct annihilation and SA-ISR processes which benefit from higher statistics. The analysis described here shows the possibility to use the LA-ISR technique at BESIII to perform independent and complementary measurements of the proton FFs down to the production threshold. Larger sampes that are currently being collected by BESIII [28] will enhance the precision of these measurements.