Measurements of the electric and magnetic form factors of the neutron for time-like momentum transfer

We present the first measurements of the electric and magnetic form factors of the neutron in the time-like (positive $q^2$) region as function of four-momentum transfer. We explored the differential cross sections of the reaction $e^+e^- \rightarrow \bar{n}n$ with data collected with the BESIII detector at the BEPCII accelerator, corresponding to an integrated luminosity of 354.6 pb$^{-1}$ in total at twelve center-of-mass energies between $\sqrt{s} = 2.0 - 2.95$ GeV. A relative uncertainty of 18% and 12% for the electric and magnetic form factors, respectively, is achieved at $\sqrt{s} = 2.3935$ GeV. Our results are comparable in accuracy to those from electron scattering in the comparable space-like (negative $q^2$) region of four-momentum transfer. The electromagnetic form factor ratio $R_{\rm em}\equiv |G_E|/|G_M|$ is within the uncertainties close to unity. We compare our result on $|G_E|$ and $|G_M|$ to recent model predictions, and the measurements in the space-like region to test the analyticity of electromagnetic form factors.

We present the first measurements of the electric and magnetic form factors of the neutron in the time-like (positive q 2 ) region as function of four-momentum transfer. We explored the differential cross sections of the reaction e + e − →nn with data collected with the BESIII detector at the BEPCII accelerator, corresponding to an integrated luminosity of 354.6 pb −1 in total at twelve center-of-mass energies between √ s = 2.0 − 2.95 GeV. A relative uncertainty of 18% and 12% for the electric and magnetic form factors, respectively, is achieved at √ s = 2.3935 GeV. Our results are comparable in accuracy to those from electron scattering in the comparable space-like (negative q 2 ) region of four-momentum transfer. The electromagnetic form factor ratio Rem ≡ |GE|/|GM | is within the uncertainties close to unity. We compare our result on |GE| and |GM | to recent model predictions, and the measurements in the space-like region to test the analyticity of electromagnetic form factors.
Proton and neutron are the fundamental building blocks of atomic nuclei. Their complex internal struc-ture emerges from Quantum Chromodynamics (QCD) but is not accessible from ab initio calculations in the non-perturbative regime of QCD governed by quark confinement [1] and gluon self-coupling [2]. On the other hand, measurements of the electromagnetic form factors (EMFFs) of the nucleon are straightforward [3]. EMFFs have long since served as a testing ground for the understanding of QCD at low momentum transfer q 2 . Since Hofstadter's groundbreaking measurements [4], various experiments at different facilities successfully measured nucleon EMFFs [5] in electron scattering with increasing precision, providing important input for theoretical calculations of nucleon properties [6,7], in particular the size of the neutron charge radius [8]. A spin-1 2 particle, such as the nucleon, is described by two EMFFs, G E (q 2 ) and G M (q 2 ), which are Fourier-transforms of the intrinsic electric and magnetic distributions of the nucleon in the Breit frame [9]. Depending on the sign of q 2 of the virtual exchange photon, we can distinguish two types of reactions. The space-like (SL) region of negative q 2 can be accessed in lepton scattering, the time-like (TL) region of positive q 2 in annihilation reactions (see Fig. 1). Here, γ * represents the virtual photon transferring the fourmomentum squared q 2 of the reaction. The gray circle represents the internal nucleon structure parameterized by the EMFFs.
Precise measurements of scattering of leptons with neutrons [10][11][12][13] are far more difficult than with protons due to the absence of free neutron targets. Possible alternatives like deuterons inevitably introduce uncertainties from nuclear binding corrections. On the other hand, free protons and neutrons are directly accessible in the TL region. Measurements of the TL EMFFs of the proton have significantly gained precision over recent years [14,[16][17][18][19][20]. However, data with large statistics from e + e − annihilation reactions has been rare. The first measurements of the neutron TL EMFFs were reported in the 1990s by the FENICE [21] and the DM2 [22] experiments and two other measurements were reported by the SND experiment [23,24]. A precise measurement of the neutron effective form factors |G| was recently published by the BESIII collaboration [1]. However, so far no separate result for G E and G M is available due to the difficulties in (anti-)neutron detection and efficiency calibration. These difficulties have prevented a detailed analysis of angulardependent differential cross sections.
In this letter we report the first measurement of separate neutron EMFF moduli |G E | and |G M | in the TL regime by exploring the differential cross sections of the reaction e + e − →nn with the data collected by the BESIII [26] experiment at the BEPCII [27] collider. Compared with the total cross sections, the differential cross sections provide more information to determine scattering amplitudes of distinguished partial waves [28] in order to verify the analyticity of EMFFs and to test various nucleon models. The data analyzed in this work corresponds to a total integrated luminosity L int = 354.6 pb −1 [29,30] at twelve center-of-mass energies (c.m.) between √ s = 2.0 and 2.95 GeV, and is grouped into five energy intervals to extract the EMFFs. An additional dataset with 10087 ± 44 million J/ψ events [31] has been used for a precise data-driven calibration of the n(n) detection efficiency with the processes J/ψ →pπn(pπn), and an investigation of identification and reconstruction of neutral particles using the processes J/ψ → nn, J/ψ → π + π − π 0 (→ γγ) and e + e − → γγ.
The moduli of the electric and magnetic form factors of the neutron can be obtained by comparing the theoretical prediction to the experimentally accessible differential Born cross section as discussed in [32]: Here, s ≡ q 2 represents the c.m. energy squared of the electron-positron system, M n the neutron mass, β = √ 1 − τ its velocity, and C the Coulomb enhancement factor accounting for the electromagnetic interaction between the outgoing baryons, which is equal to 1 for neutral baryons. Furthermore, cos θn is the cosine of thē n polar angle along the positron beam direction in the electron-positron c.m. frame, |G M | the modulus of the magnetic form factor, and R em = |G E |/|G M | the ratio of the moduli of the electric and magnetic form factors. The differential Born cross section Eq. (1), is calculated using the luminosity L, the signal reconstruction efficiency E M C and its correction E cor , the next-to-leading-order radiation and vacuum-polarization correction (1+δ), and the signal yield N bin per cos θn bin. We integrate over the bin width in accordance to histograms of angular distribution as will be discussed below.
The final state of the signal process contains one antineutron and one neutron. Hence our data analysis strategy is based on the rejection of events with charged tracks in the multilayer drift chamber (MDC). For each event, the most energetic shower in the electromagnetic calorimeter (EMC) within the polar angle range of | cos θ| < 0.7 and a deposited energy within (0.5, 2.0) GeV is considered as an candidate. This shower is further associated with a response in the time-offlight (TOF) system aggregating all hits within an azimuthal angle span of 6 TOF plastic scintillators along then momentum. To avoid a potential bias and to provide a cross-check, events are classified into three categories (i = A,B,C) depending on the detector responses to nn particles. Events with responses from knockoff protons interaction in the TOF plastic scintillators from both particles and one associated hadronic shower registered in the EMC from the antineutron are classified as category A. Events with showers in the EMC from both particles, but only one measured knockoff proton interaction in the TOF from the anti-neutron are assigned to category B. Events lacking any TOF responses but with reconstructed hadronic showers in the EMC from both particles are classified as category C. Each event belongs to not more than one category. These categories not only provide a reliable cross-check but also guarantee a better precision using an inverse-variance weighting technique. The selection method for the three categories is described in details in Ref. [1].
With surviving events passing aforementioned selections per category, cos θn of the antineutron is filled in histograms of 7 equidistant bins within −0.7 < cos θn < 0.7. For category A, signal events are characterized by the time difference between the time measured with the TOF and the expected flight time calculated from the neutron's momentum and flight path, respectively. For category B and C, signal events are characterized by the opening angle n n between neutron and antineutron as measured with the EMC. Since the surviving events still contain contributions coming mainly from the beamrelated background and the e + e − → γγ background, we use a composite model taking into account the background (M b ) and signal (M s ) distributions to fit data and determine the number of reconstructed signal events N bin i (cos θn) (i = A,B,C) at a given cos θn bin: The signal (background) distribution are modeled with the MC simulation samples of the signal (background) process generated with CONEXC event generator or control samples. The fit optimization is performed for each category by minimizing a global negative loglikelihood (NLL) with the MIGRAD [33] routine, taking into account local NLLs from each data set (i=A,B,C) and using a modified version of the David-Fletcher-Powell method [34]. Following, a HESSE and a MINOS algorithm [33] improve the uncertainty determination of the model parameters, which describe the yields of both signal and background. Note that, background distributions are shared among all energies for the same category. The asymmetrical uncertainties associated to N bin i are from the Poisson distribution to take into account the low statistics at most bins. As an example, Fig. 2 shows the extraction of signal yield N bin i=A at √ s = 2.1250 GeV for category A.
The signal reconstruction efficiency E i (cos θ) for each category is determined by a dedicated Monte Carlo (MC) simulation for the process e + e − → nn at each √ s using the MC generator Conexc [35] up to the next-to-leading order, followed by a Geant4 [36]-based simulation procedure, which mimics the response of various particles in the BESIII detector. A data-driven efficiency calibration C dm i (cos θ) is achieved for both n andn by using the process J/ψ →pπn(pπn). A trigger correction C trg i (cos θ) describing the probability of EMC-based online trigger capturing neutral processes is also applied. In addition, (1 + δ) i (cos θ) is the initial state radiation and vacuum polarization correction. All corrections are multiplied to be E cor i (cos θ) = C dm i · C trg i . The details about these corrections are given in Ref. [1].
With the above numbers for the three categories, |G M | and R em are determined by minimizing the negative logarithm of the likelihood function (NLL) based on the Poisson probability density function. For example, Fig. 3 illustrates a simultaneous fitting to data of cos θndependent cross sections at √ s = 2.3864 and 2.396 GeV. The fitting results and associated parameters are summarized in Table I. Note that data collected at 12 c.m. energies is grouped into five c.m. energy intervals to maximize the statistical precision of the results. The details on the fitting procedure and values of differential cross sections for the other energy points are listed in Ref. [37]. Various sources of systematic uncertainties are considered for the determination of |G E |, |G M |, and R em , including the uncertainties from the luminosity, the category-specific signal event selections, the MC model, the Born cross section input for the signal efficiency determination, the trigger efficiency, and the fit procedure. The uncertainty from the integrated luminosity is determined with large-angle Bhabha scattering to be 1% [29,30]. The uncertainty from the signal event selection is taken into account by using the efficiency corrections for the differences between data and signal MC. By varying C dm within 1σ, the difference on the differential cross section is taken as the systematic uncertainty. The uncertainty in the efficiency determination stemming from the form factor input model to the MC Born cross section is reduced to 1% by iterative efficiency determination and fitting. The uncertainty from the signal event extraction arises from three sources: the signal shape, the background shape, and the fitting range. By changing the signal and background shapes and varying the fitting range, the systematic uncertainty from the signal yield extraction is determined as the largest deviation from the nominal results. The uncertainty from the trigger efficiency is studied with a different parametrization of the detector response, as discussed in Ref. [1]. The TABLE I. The integrated luminosity L, form factor ratios Rem = |GE|/|GM |, electric |GE| and magnetic |GM | form factors. The first uncertainties are statistical and the second systematic. The nominal energy for each energy interval is weighted by luminosity and c.m. energy of the corresponding data sample. The lower (upper) energy uncertainty is taken as the difference between the nominal energy and the lowest (highest) energy among the group. overall systematic uncertainties at each bin are summarized in Ref. [37]. The variation of mean values of R em and |G M | with/without including systematic uncertainties during fitting angular distributions is taken as a systematic uncertainty. In total, three categories are used to determine the final results of |R em | and |G M | by taking into account correlations of systematic uncertainties at different bins. The uncertainties of |G E | are propagated from uncertainties of R em , |G M | and their correlations. Table I lists values of the total systematic uncertainties.
In conclusion, values of |G E |, |G M |, and R em have been extracted at five c.m. energy intervals in the TL region. The results for R em are consistent to unity considering systematic uncertainties in a wide range of q 2 .
Compared with the FENICE results, the values for |G M | from this work are smaller by a factor of ∼ 2-3 in the range of √ s = 2.0 ∼ 2.5 GeV, as shown in Fig. 4. The measured |G E | and |G M | can be used to test various nucleon models to provide a more comprehensive picture of the nucleon structure. Among models such as a parametrization obtained from the pQCD [38], a modified dipole model based on the quark counting rule and analytical extension (MD) [39], a Vector Meson Dominance model (VMD) [40], and a model based on Dispersion Relations (DR) [41][42][43], our results show the best agreement with the DR-based model (long-dashed line). Note that the modified dipole parametrization (dot-dashed line) is re-analyzed with the experimental results from this work. The free parameters of the DRbased model are optimized by a fit to the TL |G M | data, which are extracted for the neutron under the hypothesis that |G E | = |G M |. The free parameters of the MD model are optimized with a fit to the TL effective FFs. The pQCD based parametrization was initially developed for |G E | and |G M |, legitimizing the use of these models also for a comparison with |G E |. In contrast, the VMD model predicts different values for |G E | and |G M |.  The EMFFs derived from data of unpolarized experiments empirically scale like G E,M ∼ (−q 2 ) −2 [44] in case of q 2 → −∞ in the SL region. It is interesting to check whether the TL form factors show any asymptotic behavior. Our results show |G n E,M | ∼ (q 2 ) −2 in the TL region. It is important to test the analyticity of EMFFs as a direct consequence of micro causality and unitarity. As stated in the Phragmèn-Lindelöf (P-L) theorem [45], EMFFs in the TL region can be extended to any direction of the q 2 complex plane. As a result, the numerical values of EMFFs should approach each other Fig. 5 shows that the TL |G E | (|G M |) has no intersections with the SL G E (G M ), using an extrapolation with current fitting parameters for the neutron. The measured ratios are R E = 5.18 ± 1.18 > 1 for the electric form factors and R M = 1.72 ± 0.14 > 1 for the magnetic form factors. We further check the analyticity of effective form factors |G| as shown in Fig. 6. The neutron TL |G n | have no intersection with its SL G n M multiplied by its magnetic moment µ n when following the parameterized formula F 1 in Ref. [37]. Note that the illustrated fitting (black dot-dashed line) makes use of data of effective FFs from both BESIII (black dots) and SND (green circles). A refit with the parameterized formula F 3 shows an in- tersection around 47±41 GeV 2 . The proton TL |G p |, which have been normalized to the reciprocal of a dipole FF, has an intersection around 18±6 GeV 2 with its SL G p M multiplied by its magnetic moment µ p when using the parameterized formula F 3 . Note that the illustrated fitting (black dot-dashed line) used only BABAR data. The related fitting results are listed in table II and III.  In summary, we have separated |G E | from |G M | for the neutron within a wide range of q 2 from 4 to 9 GeV 2 with relative uncertainty around 12% for the modulus of the magnetic form factor. This is comparable in accuracy to results from electron scattering in a similar SL region of four-momentum transfer. In the future, further efforts will be made not only at electron accelerators [47,48] but also at electron-positron [49] and proton-antiproton colliders [50] to obtain a global picture of all data in the TL and SL regions which will further deepen our understanding of the nucleon structure. Supplemental Material for "Measurements of the electric and magnetic form factors of the neutron for time-like momentum transfer The predicted cross-section σ pred i at i-th bin for each category X (=A,B,C) can be obtained as below: with the theoretical formula, where τ = 4m 2 n s , |G M | and R em ≡ |G E |/|G M | are two free parameters shared by 3 categories to be determined with the following minimization.
We minimize a negative logarithm likelihood as below: where N obs X,i and N pred X,i represent number of observed events and number of predicted events, L is the luminosity, E M C is the signal efficiency, (C dm , C trg , C ee , C muc ) are four types of efficiency corrections, (1 + δ) is the initialstate-radiation and vacuum polarization correction determined by the signal MC generator, at i-th bin for category X.
The systematic uncertainty at each bin is calculated to be ∆ T,i = ∆ 2 uc,i + ∆ 2 c,i , where ∆ uc = N obs * R 2 dm + R 2 isr represents the uncorrelated uncertainty, and ∆ c = N obs * R c the correlated one. Here, R c is the relative systematic uncertainties which are quoted from Ref. [1]. R dm is directly calculated with C dm values, for example C dm = 1.23±0.10 yields a R dm = 0.10 1.23 , so are the others. Thus the negative logarithm likelihood becomes: using a convolution between a Poisson probability (P a ) and a Gaussian probability (P b ) which describes the systematic uncertainties per bin per category.
(B1)      V. The form factor ratios Rem = |GE|/|GM |, electric |GE| and magnetic |GM | form factors. The results are obtained by fitting angular distributions of the antineutron w/o systematic uncertainties at each bin. The center shifts of results w/o considering systematic uncertainties per cos θn during the fitting is also taken as a systematic uncertainty added to final results. The label "stat. only" represents results obtained from the fitting to data of antineutron angular distributions with statistical errors. The label "stat. + sys." represents results obtained from the fitting to data of antineutron angular distributions with both statistical errors and systematic errors. Appendix D: DATA