Time-Like Proton Form Factors in Initial State Radiation Process

: Electromagnetic form factors are fundamental quantities describing the internal struc- 1 ture of hadrons. They can be measured with scattering processes in the space-like region and 2 annihilation processes in the time-like region. The two regions are connected by crossing symmetry. 3 The measurements of the proton electromagnetic form factors in the time-like region using the 4 initial state radiation technique are reviewed. Recent experimental studies have shown that initial 5 state radiation processes at high luminosity electron-positron colliders can be effectively used to 6 probe the electromagnetic structure of hadrons. The BABAR experiment at the B -factory PEP-II 7 in Stanford and the BESIII experiment at the τ -charm factory BEPCII in Beijing have measured 8 the time-like form factors of the proton using the initial state radiation process e + e − → p ¯ p γ . The 9 two kinematical regions where the photon is emitted from the initial state at small and large polar 10 angles have been investigated. In the ﬁrst case the photon is in the region not covered by the 11 detector acceptance and is not detected. The Born cross section and the proton effective form factor 12 have been measured over a wide and continuous range of the the momentum transfer squared q 2 13 from threshold up to 42 (GeV/c) 2 . The ratio of electric and magnetic form factors of the proton 14 has been also determined. In this report, the theoretical aspect and the experimental studies of 15 the initial state radiation process e + e − → p ¯ p γ are described. The measurements of the Born cross 16 section and the proton form factors obtained in these analyses near the threshold region and in the 17 relatively large q 2 region are examined. The experimental results are compared to the predictions 18 from theory and models. Their impact on our understanding of the nucleon structure is discussed. 19


Introduction
The concept of symmetry plays a key role for understanding the fundamental 22 building blocks of the universe and the forces binding them. Experiments employing 23 antimatter, a mirror or a counterpart of a subatomic particle with opposite charge 24 and right-or left-handed spin, provide important information about the elementary 25 units of which the universe is composed. Measurements of the exclusive production 26 of hadron pair in electron positron annihilation for example are of great interest for 27 understanding the internal structure of hadrons. Assuming the exchange of a virtual 28 photon of positive momentum transfer squared, q 2 , between the initial and the final 29 states, these processes contain direct information on the electromagnetic form factors 30 (EMFFs) in the time-like region. The number of independent EMFFs is determined by 31 of the hadronic final states with invariant masses below the c.m. energy of the collider 48 is provided by the initial state radiation (ISR) method. The ISR method, also called 49 radiative return technique, employs measurements the process e + e − → hadrons + nγ 50 where the emission of one or more high energy photons reduces the mass squared of the 51 hadronic system to below √ s. 52 The theoretical studies of the ISR processes started long time ago in the sixties of the 53 20 th century. The importance of the ISR approach has been outlined in the calculation 54 of the photon emission for muon pair production in Ref. [1] and for pions in Ref. [2]. 55 The possibility of using this approach at the high luminosity e + e − colliders has been 56 discussed later [3][4][5][6], in particular when the first generation of Φ and B factories came 57 in operation in the late nineties. It has been realized that the suppression factor α/π, 58 where α is the fine electromagnetic constant, due to the photon emission can be well is involved in this wide range coverage. 63 The ISR approach became a powerful tool for the analysis of experiments after the 64 development of dedicated Monte Carlo event generators for the ISR processes, including 65 higher order QED corrections, such as the PHOKHARA event generator [7,8] and the 66 AFKQED package (based on an early version of PHOKHARA called EVA [5,9] from threshold up to 42 (GeV/c) 2 considering both analyses. The ratio of electric and 82 magnetic form factors of the proton has been also determined in the LA-ISR analysis. 83 The hyperon final states ΛΛγ, ΛΣ 0 γ, ΣΣ 0 γ have been also investigated by BABAR [14]. 84 The |G E |/|G m | ratio for the Λ hyperon has been measured in two mass intervals between 85 2. 23  e + e − → π + π − and e + e − → K + K − have been provided by KLEO, BABAR, and BESIII 96 experiments [18][19][20][21][22][23][24][25]. These measurements cover a wide mass range and have a large 97 impact on the calculation of the hadronic contribution to the anomalous magnetic 98 moment of the muon.

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In this report, we describe the ISR analyses for the proton EMFFs measurements 100 at BABAR and BESIII experiments. In section 2, the theoretical calculations of the 101 differential and integrated cross section for the process e + e − → ppγ are described. The   In the e + e − c.m. frame, the differential cross section for the process of e + e − → pp 108 under the assumption of one virtual photon exchange ( Fig. 1) is [26], with y = πα β accounts for the electromagnetic interaction between the outgoing proton 115 and antiproton [12,27]. The cross section depends on the moduli of the magnetic and 116 electric FFs, which can be determined from the analysis of the proton angular distribution.

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By integrating the differential cross section (Eq. (1)), the total cross section for the process 118 e + e − → pp is obtained, An effective form factor can be introduced as a combination of |G M | 2 and |G E | 2 , 122 which is equivalent to |G M | determined from Eq. 2 under the assumption of |G M | = |G E |. 123 2.2. Cross section for the e + e − → ppγ process 124 The cross section for the lowest order ISR process shown in Fig. 2, i.e. including 125 only one photon emission from the initial state and integrated over the proton momenta, 126 can be written as [5,11,28], 128 where E * γ and θ * γ are the energy and the polar angle of the ISR photon, respectively, in 129 the e + e − c.m. frame. The radiator function W(s, x, θ * γ ) which describes the probability of 130 ISR photon emission, is written as follows [2,28,29],

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where m e is the electron mass. The integration of W(s, x, θ * γ ) over the ISR photon angle 133 is given for two practically cases [28,29]. The first one is given for a range of integration

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For the full range of the ISR polar angle (θ 0  for example the detection efficiency and the acceptance.

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In the simulations of the e + e − → ppγ process at large photon emission angle, the 153 BABAR collaboration has considered the additional photon radiation from the initial 154 state using the structure function method [31,32]. This correction takes into account Ref. [36].
where the superscript i represents the i th pp mass interval, N i pp is the number of pp 208 events, i and (1 + δ i ) are the detection efficiency and the radiative correction factor 209 determined by the MC simulation, and L i is the ISR differential luminosity calculated as, where q 2 is the four momentum transfer squared for the i th pp mass interval. In BABAR

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LA-ISR analysis, the process of e + e − → ppγ is simulated to extract the selection effi-212 ciency and radiative correction factor at the range: 20 frame, therefore a reduced phase-space factor cos θ 0 γ with θ 0 γ = 20 • was included to 214 calculate the differential luminosity by using the equation 6 [11,12]. In both analyses 215 from the BESIII experiment, the full radiative function W(s, x) (Eq. 7) was used for the 216 differential luminosity calculation, since the process was simulated in a full phase space   [17]. The experimental data presented also include measurements from direct annihilation process.
The effective form factor of the proton is extracted from the e + e − → pp cross section 223 measurements for both LA-and SA-ISR analyses at BABAR and BESIII experiments 224 according to Eq. 2 and 3. Figure 5 shows the results for the e + e − → pp cross section ( Fig.  5 (a)) and the proton effective FF (Fig. 5 (b)) from all the analyses with the ISR process at 226 BABAR and BESIII experiments, together with other available experimental data from 227 direct annihilation process. 228

The ratio of the proton FFs from the ISR process 229
Taking advantage of the strong boost of the ISR photon, a full angular distribution 230 of the proton can be obtained from the e + e − → ppγ process. This is in contrast to most 231 analyses using the direct annihilation process, which have to limit the angular range due 232 to detector acceptance. Therefore, the ISR method allows to extract the ratio of the proton    can be extended to the time-like region (for review see Refs. [37,38]), such as the models 255 based on dispersion relations [39] and vector meson dominance [40,41]. This region has isovector, or their orthogonal interference [43].  Figure 8. Periodic oscillation structure of the proton effective FF [17]. The dotted line represents the oscillation contribution, which is described by Eq. 12 and the value of fit parameters are taken from Ref. [45].

Discussions and conclusions
In 2015, Bianconi and Tomasi-Gustafsson studied the proton effective FF from 273 BABAR LA-ISR measurement, and found the data can be best reproduced by a three-274 pole function (F 3p (s)) with an additional periodic oscillation part (F osc [p(s)]) [44,45]. By 275 including more experimental data from BESIII (direct annihilation and SA-ISR analyses) 276 and CMD-3 experiments, this research was updated in 2021 [46].
where p(q 2 ) = q 2 (τ − 1) is the relative momentum of the proton in the frame where 278 the antiproton is at rest, the standard dipole parameter m 2 0 = 0.71 (GeV/c) 2 , and other 279 parameters are extracted by fitting to the experimental data, the values can be found in 280 the reference [46]. Figure 8 shows the periodic oscillation structure of the proton effective 281 FF described by Eq. 12, in which the values of parameters are taken from Ref. [45].

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Similar to the effective FF of the proton, a fit was performed in Ref. [46] on the ratio 283 of the proton EMFFs with the function including a monopole decrease and a damped 284 oscillation, As shown in Fig. 7, the fit (dotted line), performed in Ref. [46], was only applied   A full angular distribution of the proton improves the precision of the ratio of the  The disadvantage of the ISR process for the proton EMFFs measurement is obviously 326 that the statistics is much lower compared to the direct annihilation process, therefore 327 very high luminosity is necessary to compensate. The Belle-II experiment on the B-328 factory SuperKEKB is starting data collection and will be able to contribute for the 329 proton EMFFs study in the future. The Belle-II experiment has the goal to obtain a 330 total integrated luminosity of 50 ab −1 in the c.m. energy range from 9.46 to 11.24 GeV, 331 and it has planned to collect ∼5 ab −1 integrated luminosity in 2020 [50]. According 332 to a recent report, the plan is delayed and an integrated luminosity of 231 fb −1 was 333 accumulated until 2021 [51]. The BESIII experiment is continuing to collect data between 334 2.00 -4.95 GeV, where many large data sets might be helpful to improve measurements 335 of the proton EMFFs in a large q 2 range, and it is even possible to access other baryon-336 antibaryon channels. The BESIII experiment has accumulated large luminosity data sets 337 above 3.770 GeV. The data sets, which can be used for the baryon EMFFs measurements 338 with ISR technique, are estimated to be ∼22 fb −1 until to 2021 [52]. In next two years, 339 BESIII will collect data at √ s = 3.773 GeV to obtain a total luminosity up to 20 fb −1 340 (including the 2.9 fb −1 from previous data) [53,54]. Considering that one disadvantage 341 of the ISR technique is worse resolution of the pp invariant mass, BESIII experiment also 342 plans to explore the nucleon production threshold region through direct annihilation 343 process by taking data below 2.0 GeV [55].

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Regarding the expected luminosities from both experiments in the next few years, 345 researches for the EMFFs of the proton may achieve significant progress with the ISR 346 technique, especially for the EMFFs ratio of the proton close to the pp threshold region, 347 where the detection efficiency is almost zero for the low momentum proton/antiproton 348 tracks produced from the direct annihilation process. The ISR technique to analyze proton EMFFs is not only a complementary method, 351 but also has the advantages to access the proton-antiproton threshold and to improve 352 the precision of the ratio measurement with full proton angular distribution analysis. 353 The BABAR experiment started the proton EMFFs research with the ISR technique, and