# New fit of time-like proton electromagnetic form factors from $e^{+}e^{-}$
colliders

Egle Tomasi-Gustafsson<EMAIL_ADDRESS>CEA, IRFU, DPhN, Université Paris-
Saclay, 91191 Gif-sur-Yvette Cedex, France Andrea Bianconi
<EMAIL_ADDRESS>Dipartimento di Ingegneria dell ′Informazione,
Università degli Studi di Brescia and Istituto Nazionale di Fisica Nucleare,
Gruppo Collegato di Brescia, I-25133, Brescia, Italy Simone Pacetti
<EMAIL_ADDRESS>Dipartimento di Fisica e Geologia, and INFN Sezione di
Perugia, 06123 Perugia, Italy

###### Abstract

The data on the proton form factors in the time-like region from the BaBar,
BESIII and CMD-3 Collaborations are examined to have coherent pieces of
information on the proton structure. Oscillations in the annihilation cross
section, previously observed, are determined with better precision. The moduli
of the individual form factors, determined for the first time, their ratio and
the angular asymmetry of the annihilation reaction $e^{+}e^{-}\to\bar{p}p$ are
discussed. Fiits of the available data on the cross section, the effective
form factor, and the form factor ratio, allow to propose a description of the
electric and magnetic time-like form factors from the threshold up to the
highest momenta.

##  I Introduction

The understanding of the proton electromagnetic form factors (FFs), called
electric $G_{E}(q^{2})$ and magnetic $G_{M}(q^{2})$ Sachs FFs is the aim of
theoretical and experimental studies since decades, in the frame of a unified
view of the scattering and annihilation regions. Much progress has been done
recently, due, on one side, to new experiments that collected information with
better precision and/or in a wider kinematical range and, on the other side,
to theoretical efforts that extend models and parametrizations built in the
space-like (SL) region to the time-like (TL) region (for a review, see Ref.
Pacetti et al. (2015)). We discuss here the data on the
$e^{+}e^{-}\to\bar{p}p$ cross section, $\sigma_{e^{+}e^{-}\to\bar{p}p}$, from
the BaBar, BESIII and CMD-3 Collaborations, obtained either by direct
measurements of the annihilation process, or by means of the so-called initial
state radiation (ISR) technique, $i.e.$, by exploiting the three-body process
$e^{+}e^{-}\to\bar{p}p\gamma$, where the photon is radiated by one of the
initial leptons.

The emission of a real hard photon, leaving the radiating lepton in a “quasi-
real” state, allows extracting the cross section for the process
$e^{+}e^{-}\to\bar{p}p$ from the differential cross section of the three-body
process $e^{+}e^{-}\to\bar{p}p\gamma$. In such a kinematic domain,
$\sigma_{e^{+}e^{-}\to\bar{p}p}$ factorizes out in the expression of the ISR
differential cross section. In collinear kinematics, the ISR cross section
manifests a logarithmic enhancement as a consequence of the small mass of the
virtual electron that is almost on mass shell Baier et al. (1973). At fixed
energy colliders the ISR technique allows to extract values of the
$\sigma_{e^{+}e^{-}\to\bar{p}p}$ cross section at different transferred
momenta, $i.e.,$ different values of $q^{2}$ (being $q$ the four-momentum of
the virtual photon in the annihilation reaction $e^{+}e^{-}\to\bar{p}p$) by
tuning the kinematics of the real photon. The cost is a reduction of a factor
of $\alpha=e^{2}/(4\pi)\simeq 1/137$ (the electromagnetic fine constant) of
the number of events, that, however, can be compensated by the high luminosity
recently achieved at the experimental facilities.

By means of the ISR technique and detecting the radiated hard photon, the
BaBar Collaboration obtained data on the $e^{+}e^{-}\to\bar{p}p$ cross section
with an error lower than 10% in a wide energy region, from the production
threshold $\sqrt{s}=2m_{p}$ up to $\sqrt{s}\leavevmode\nobreak\
\simeq\leavevmode\nobreak\ 6$ GeV Lees et al. (2013a), where $s=q^{2}$ is the
total energy squared in the center of mass (CM) frame of the $\bar{p}p$-system
and $m_{p}$ is the proton mass. Recently, using the same technique but with
undetected initial photon, the BESIII Collaboration extracted 30 values of
$\sigma_{e^{+}e^{-}\to\bar{p}p}$ in the range $(2\leq\sqrt{s}\leq 3.8)$ GeV
Ablikim et al. (2019). The ISR photon is undetected, $i.e.$, it is mostly
emitted at small polar angles in a kinematical region uncovered by BESIII
acceptance. This method was also used by the BaBar Collaboration, where the
hard condition of the photon was insured by high energy of the colliding beams
Lees et al. (2013b).

The individual determination of the moduli of the FFs in the TL region was
done by the BESIII Collaboration, using the energy scan method Ablikim et al.
(2020), with a precision comparable to that of the data obtained in the SL
scattering region. The data in the SL region were mostly collected by the JLab
GEp Collaboration and published in a series of papers, summarized in Refs.
Puckett et al. (2017, 2012). The BESIII Collaboration has made the individual
measurement of $|G_{E}|$ and $|G_{M}|$, separately, in the TL region for the
first time ever.

In fact, before such a pioneering measurement, the few information on the FF
moduli in the TL region concerned a composed observable, namely their ratio
$R=|G_{E}|/|G_{M}|$, extracted from angular distribution measurements. Due to
luminosity limitations, only an ’effective form factor’ could be extracted
from the total cross section.

Let us stress that only the moduli of the FFs, which, in principle, have a
non-vanishing imaginary part in the TL region, can be extracted from a precise
large-statistics measurement of the angular distribution of the final-state
nucleons in the $e^{+}e^{-}$-CM frame. The underlying assumption is that the
reaction occurs through the one-photon exchange mechanism Zichichi et al.
(1962). No measurement of the relative phase between $G_{E}$ and $G_{M}$,
accessible through polarization observables Dubnickova et al. (1996), is
available yet for protons and neutrons .

Focussed on the threshold region, the CMD-3 Collaboration Akhmetshin et al.
(2019) measured the cross section for the reactions
$e^{+}e^{-}\to\bar{p}p\gamma$ and $e^{+}e^{-}\to\bar{n}n\gamma$. The scan of
the nucleon-antinucleon threshold energy region is done by measuring the beam
energy at 0.1 MeV precision by back-scattering laser light system. The energy
spread due to radiation and energy resolution is small enough to differentiate
the proton and neutron thresholds.

The aim of the present work is to scrutinize the recent data on proton FFs in
TL region, through the reaction $e^{+}e^{-}\to\bar{p}p(\gamma)$. Two
characteristics, earlier predicted or highlighted, can be confirmed or
infirmed by the new data: the finding of regular oscillations of the cross
section Bianconi and Tomasi-Gustafsson (2015) and the steeper
$q^{2}$-dependence of the electric FF ($G_{E}$) compared to the magnetic FF
($G_{M}$), as found in the SL region Jones et al. (2000); Puckett et al.
(2017). The suggestion of a similar $q^{2}$-dependence in space and TL regions
is based on analytical properties of the amplitudes Tomasi-Gustafsson and
Rekalo (2001) and illustrated in frame of a generalized definition of FFs
Kuraev et al. (2012).

Not all models developed in the SL region have the correct analytical
properties to be extended in the TL region, where FFs are of complex nature
Tomasi-Gustafsson et al. (2005). Models based on dispersion relations
Belushkin et al. (2007) or vector dominance Bijker and Iachello (2004); Lomon
and Pacetti (2012) have attempted a global description in SL and TL regions,
for a review, see Refs. Pacetti et al. (2015); Denig and Salme (2013). In this
paper we consider the new data and we propose a global fit from threshold up
to the maximum available transferred momentum. The individual TL FFs are
reproduced from a fit on the ratio $R$ and of the effective FF, allowing to
extrapolate their behavior at threshold, where $R$ is constrained to unity.

##  II  The $e^{+}e^{-}\to\bar{p}p(\gamma)$ cross section

As already pointed out, at fixed-energy $e^{+}e^{-}$-colliders, the
$e^{+}e^{-}\to\bar{p}p$ cross section, can be extracted from the data on the
differential cross section of the ISR process $e^{+}e^{-}\to\bar{p}p\gamma$,
where the photon is radiated by one of the initial electrons, over a range of
$\bar{p}p$-energies going from the threshold,
$\sqrt{s_{\mathrm{thr}}}=2m_{p}$, up to the full $e^{+}e^{-}$ CM energy,
$\sqrt{s_{e^{+}e^{-}}}$. Similar formalism can be applied for the annihilation
$e^{+}e^{-}\to\bar{n}n$.

In Ref. Lees et al. (2013a), based on the work of Ref. Bonneau and Martin
(1971), the differential cross section for the radiative process, integrated
over the nucleon momenta, was factorized into a function which depends on the
photon kinematical variables multiplied by the annihilation cross section of
interest, for the process $e^{+}e^{-}\rightarrow\bar{p}p$:

$\displaystyle\frac{d^{2}\sigma_{e^{+}e^{-}\to\bar{p}p\gamma}}{d\sqrt{s_{e^{+}e^{-}}}\,d\cos(\theta_{\gamma})}=\displaystyle\frac{2\sqrt{s}}{s_{e^{+}e^{-}}}W(s_{e^{+}e^{-}},E_{\gamma},\theta_{\gamma})\sigma_{e^{+}e^{-}\to\bar{p}p}(s)\,,\hskip
28.45274ptE_{\gamma}=\frac{s-s_{e^{+}e^{-}}}{2\sqrt{s_{e^{+}e^{-}}}}\,,$ (1)

where $\sqrt{s}$ and $\sqrt{s_{e^{+}e^{-}}}$ are the invariant masses of the
$\bar{p}p$ and $e^{+}e^{-}$ systems, $E_{\gamma}$ and $\theta_{\gamma}$ are
the energy and the scattering angle of the photon in the $e^{+}e^{-}$ CM
frame, while $W(s_{e^{+}e^{-}},E_{\gamma},\theta_{\gamma})$ represents the so-
called radiator function, it gives the probability that an initial photon with
energy $E_{\gamma}$ is emitted at the angle $\theta_{\gamma}$. In Eq. (1), the
factorization of the photon variables allows to single out the elementary
cross section $\sigma_{e^{+}e^{-}\to\bar{p}p}$ and extract the moduli of the
TL proton FFs. However, such a factorization does fail in describing the
scattering process when $\sin(\theta_{\gamma})\to 0$, $i.e.,$ when the photon
is radiated along the beam direction, because it neglects terms depending on
$(m_{e}^{2}/s_{e^{+}e^{-}})$, where $m_{e}$ is the electron mass, which become
important at small angles Benayoun et al. (1999); Baier et al. (1973). The
case of final state radiation (FSR), when the radiative emission is from the
final proton or anti-proton, was discussed in Ref. Bytev et al. (2011). It has
been found that also the ISR-FSR interference may spoil the factorization
hypothesis, if the detection is not symmetric around the colliding beams axis.

The differential cross section for the annihilation process
$e^{+}+e^{-}\to\bar{p}+p$ in Born approximation and in the CM frame is
Zichichi et al. (1962)

$\frac{d\sigma_{e^{+}e^{-}\to\bar{p}p}}{d\Omega}(s,\theta)=\frac{\alpha^{2}\beta\,{\cal
C}(\beta)}{4s}\left[\left(1+\cos^{2}(\theta)\right)|G_{M}(s)|^{2}+\displaystyle\frac{1}{\tau}\sin^{2}(\theta)|G_{E}(s)|^{2}\right]\,,$
(2)

where $s$ is the total energy squared of the $\bar{p}p$ system,
$\tau=s/(4m_{p}^{2})$ and $\beta=\sqrt{1-1/\tau}$ is the final particle
velocity. The function

${\cal C}(\beta)=\frac{y(\beta)}{1-e^{-y(\beta)}}\,,\hskip
28.45274pty(\beta)=\frac{\pi\alpha}{\beta}\sqrt{1-\beta^{2}}\,,$

represents the Coulomb correction that accounts for the $\bar{p}p$ final state
interaction Hoang (1996). It becomes effective ($\gg 1$) and divergent as
$\beta\to 0$. Such a divergency, that happens exactly at the production
threshold, $i.e.$, at $\beta=0$ or equivalently at $s=4m_{p}^{2}$, does cancel
out the phase-space factor $\beta$ by making finite and different from zero
the cross section at the threshold.

The even $\cos\theta$-angular dependence of the cross section of Eq. (2), in
particular the presence of the powers zero and two only, results directly from
the Born approximation, $i.e.,$ from the assumption of one-photon exchange and
the invariance of the electromagnetic interaction with respect to the parity
transformation.

Following Ref. Tomasi-Gustafsson and Rekalo (2001), in order to highlight the
angular dependence, the Born differential cross section given in Eq. (2) can
be written as

$\frac{d\sigma_{e^{+}e^{-}\to\bar{p}p}}{d\Omega}(s,\theta)=\sigma_{0}(s)\left[1+{\cal
A}(s)\cos^{2}(\theta)\right],\ \sigma_{0}(s)=\frac{\alpha^{2}\beta\,{\cal
C}(\beta)}{4s}\left(|G_{M}(s)|^{2}+\frac{1}{\tau}|G_{E}(s)|^{2}\right)\,,$

where $\sigma_{0}(s)$ is the differential cross section at $\theta=\pi/2$, and
the function ${\cal A}(s)$, assuming the one-photon exchange mechanism,
depends on the ratio of the FFs moduli $R(s)=|G_{E}(s)|/|G_{M}(s)|$, as

${\cal
A}(s)=\displaystyle\frac{\tau|G_{M}(s)|^{2}-|G_{E}(s)|^{2}}{\tau|G_{M}(s)|^{2}+|G_{E}(s)|^{2}}=\displaystyle\frac{\tau-R(s)^{2}}{\tau+R(s)^{2}}\,.$
(3)

It follows that ${\cal A}(s)$ represents an observable which is sensitive to
deviations of the differential cross section from linearity in
$\cos^{2}(\theta)$, in particular, a residual dependence on the scattering
angle $\theta$, $i.e.,$ a non null derivative $d\mathcal{A}/d\theta$, would
mean that, besides the one-photon exchange, other intermediate states do
contribute to the annihilation process $e^{+}e^{-}\leftrightarrow\bar{p}p$.
Similar studies can be made for the scattering processes $e^{-}p\to e^{-}p$,
in the SL region, by considering the deviation from linearity of the so-called
Rosenbluth plots, see Ref. Gakh and Tomasi-Gustafsson (2005) and references
therein.

The total cross section $\sigma_{e^{+}e^{-}\to\bar{p}p}(s)$, obtained by
integrating the differential cross section given in Eq. (2) over the solid
angle $d\Omega$, namely

$\sigma_{e^{+}e^{-}\to\bar{p}p}(s)=\frac{4\pi\alpha^{2}\beta\,{\cal
C}(\beta)}{3s}\left(|G_{M}(s)|^{2}+\frac{1}{2\tau}|G_{E}(s)|^{2}\right)\,,$
(4)

is proportional to an $s$-dependent combination of the moduli squared of the
FFs, which is commonly defined in terms of the effective FF, $F_{p}(s)$, whose
modulus squared is given by the normalized combination :

$|F_{p}(s)|^{2}=\displaystyle\frac{2\tau|G_{M}(s)|^{2}+|G_{E}(s)|^{2}}{2\tau+1}\,.$
(5)

Using such a unique effective FF is equivalent to consider the protons as a
spin-zero particle and hence, to assume
$|G_{E}(s)|=|G_{M}(s)|\equiv|F_{p}(s)|$ in Eq. (2). As a consequence of their
definitions in terms of the Dirac and Pauli FFs, $F_{1}(s)$ and $F_{2}(s)$ :

$G_{E}(s)=F_{1}(s)+\tau\,F_{2}(s)\,,\hskip
28.45274ptG_{M}(s)=F_{1}(s)+F_{2}(s)\,,\hskip 28.45274pt$

and the assumption of analyticity, the identity $G_{E}(s)=G_{M}(s)$ is
strictly valid only at the production threshold $s=4m_{p}^{2}$, $i.e.$,
$\tau=1$. This phenomenon can be also interpreted as a consequence of the
isotropy of the annihilation process $e^{+}e^{-}\to\bar{p}p$ just at the
production threshold, in the $\bar{p}p$ or $e^{+}e^{-}$ CM frame. In fact,
having no preferred direction, the amplitude must be independent on the
scattering angle, that implies $\mathcal{A}(4m_{p}^{2})=0$, see Eq. (3),
$i.e.,$ $G_{E}(4m_{p}^{2})=G_{M}(4m_{p}^{2})$.

Therefore, a measurement of the total cross section gives access to the
effective FF. The extraction of $R$ and/or $\mathcal{A}$ requires in addition
a precise measurement of the differential cross section. Even further
precision is required for a meaningful extraction of the individual FFs. It is
for this reason that it could be achieved only in the most recent experiments.

##  III Analysis of the results

###  III.1 Selected data sets

We consider four sets of data on the $\sigma_{e^{+}e^{-}\to\bar{p}p}$ cross
section.

  1. 1.

The set from the BaBar Collaboration, labeled as “BaBar”, has three sub-sets:

     * •

38 points, obtained with the ISR technique and detecting the initial photon,
in the range $(1.877\leq\sqrt{s}\leq 4.50)$ GeV, together with 6 points for
the ratio $R=|G_{E}|/|G_{M}|$ in the range $(1.877\leq\sqrt{s}\leq 3)$ GeV
Lees et al. (2013a).

     * •

13 points, obtained with the ISR technique and detecting the initial photon,
in the range $(1.8765\leq\sqrt{s}\leq 1.9625)$ GeV Lees et al. (2013a). These
data with a larger granularity overlap with the first four points of the above
series, that, therefore, are omitted in the analysis.

     * •

8 points, obtained with the ISR technique and not detecting the initial
photon, in the range $(3\leq\sqrt{s}\leq 5.50)$ GeV Lees et al. (2013b).

  2. 2.

Two sets from the BESIII Collaboration:

     * •

30 points, obtained with the ISR technique, in the range $(2.0\leq\sqrt{s}\leq
3.60)$ GeV Ablikim et al. (2019), labeled as “BESIII-ISR”.

     * •

22 points, obtained with energy scan, together with 16 points for the ratio
$R$, and for the disentangled moduli $|G_{E}|$ and $|G_{M}|$, in the range
$(2.0\leq\sqrt{s}\leq 3.08)$ GeV Ablikim et al. (2020), labeled as “BESIII-
BS”.

  3. 3.

A set from CMD-3 of 11 points, obtained with energy scan, in the range
$2m_{p}<\sqrt{s}\leq 2.006)$ GeV Akhmetshin et al. (2019). They belong to a
sub-set of the published data, that includes only those points lying above the
production threshold $\sqrt{s}=2m_{p}$. Indeed, the complete set covers an
energy interval that, as a consequence of experimental limits of the energy
resolution, extends also below the physical threshold. This is the second
measurement of the $\sigma_{e^{+}e^{-}\to\bar{p}p}$ cross section performed by
the CMD-3 Collaboration and it improves the first one Akhmetshin et al. (2016)
by enhancing the precision and extending the energy range. As numbers are not
given in the original paper, the points (red squares in Fig. 4 of Ref.
Akhmetshin et al. (2019)) have been read from the figure. This set of data is
labeled as CMD-3.

###  III.2 Confirmation of the oscillations

In Ref. Bianconi and Tomasi-Gustafsson (2015) it was pointed out that the
cross section of $e^{+}e^{-}\to\bar{p}p$ measured by the BaBar Collaboration
Lees et al. (2013a) shows evidence of structures. These structures become
regular when plotted as a function the 3-momentum $p$ of one of the two
hadrons in the frame where the other one is at rest and it is proportional to
the relative velocity $\beta=\sqrt{1-1/\tau}$.

The BaBar data on the modulus of the proton effective FF Lees et al. (2013a),
extracted from the $e^{+}e^{-}\to\bar{p}p$ total cross section by means of the
formulae given in Eq. (4) and (5), in the range $(2m_{p}<\sqrt{s}<6)$ GeV, are
well reproduced by the function Bianconi and Tomasi-Gustafsson (2015)

$F_{p}^{\rm fit}(s)=F_{\rm 3p}(s)\ +\ F_{\rm osc}(p(s))\,.$ (6)

It is the sum of two contributions: a dominant three-pole (3p) $F_{\rm
3p}(s)$, and a damped oscillatory component $F_{\rm osc}(p(s))$, whose
expressions are

$\displaystyle F_{\rm 3p}(s)$ $\displaystyle=$
$\displaystyle\frac{F_{0}}{\left(1+\frac{s}{m_{a}^{2}}\right)\left(1-\frac{s}{m_{0}^{2}}\right)^{2}}\,,$
(7) $\displaystyle F_{\rm osc}(p(s))$ $\displaystyle=$ $\displaystyle
Ae^{-Bp}\cos(Cp+D)\,.$ (8)

The explicit expressions of the variables $p=s(p)$ and $p=p(s)$, as well as of
the functions in terms of $s$ and $p$ are explicited in the Appendix.

The 3p function $F_{\rm 3p}(s)$, that describes the smooth behavior (ignoring
small-scale oscillations) of the effective FF, is the product of a free
monopole, depending on two free parameters: the adimensional $F_{0}$ and the
mass $m_{a}$, and the standard dipole with $m_{0}^{2}=0.71$ GeV2.

The oscillatory contribution $F_{\rm osc}(p(s))$, reproduces the GeV-scale
oscillations in the $p$ variable. These irregularities are treated as small
perturbations of the dominant smooth behavior, $i.e.,$ $|F_{\rm osc}(p(s))|\
\ll\ |F_{\rm 3p}(s)|$. Moreover, due to their regular periodic nature, they
have a vanishing mean effect

$\langle F_{\rm osc}(p(s))\rangle_{\Delta p}\mathop{\longrightarrow}_{\Delta
p\geq 1\,{\rm GeV}}0\,.$

Here we show that the recent data on $F_{P}$ from the BESIII Collaboration
Ablikim et al. (2019, 2020) are compatible with those from the BaBar
Collaboration Lees et al. (2013a, b) and confirm the previous findings of Ref.
Bianconi and Tomasi-Gustafsson (2015). This is proved by the consistency of
the fit parameters, obtained by including the data sets BESIII-ISR and BESIII-
SC, besides the BaBar one, compared to the parameters obtained by fitting the
BaBar data only (Table 1).

Table 1:  Fit parameters from Eq. (7)  Ref. | Exp. | N | $F_{0}$ |  $m_{a}^{2}$ (GeV2)
---|---|---|---|---
Lees et al. (2013a, b); Bianconi and Tomasi-Gustafsson (2015) | BaBar | 85 | 7.7 $\pm$ 0.3 | 15 $\pm$ 1
Ablikim et al. (2019, 2020); Lees et al. (2013a, b) | BaBar,BESIII-ISR,BESIII-SC | 107 | $8.9\pm 0.2$ | $8.8\pm 0.6$

In Fig. 1a the cross section data are plotted as a function of $p$. The result
of the fit using Eq. (7) is then subtracted from the data. The obtained
residue ${\cal D}$ (data minus $F_{\rm 3p}(s)$) displayed in Fig. 1 shows a
damped and periodic oscillatory behavior, that has been fitted with the four-
parameter function of Eq. (8). The values of the parameters are reported in
Tables 1, 2, together with those obtained by fitting the BaBar data only.

Table 2:  Fit parameters from Eq. (8) and corresponding values of the normalized $\chi^{2}$.  Ref. | Data set | $A\pm\Delta A$ | $B\pm\Delta B$ | $C\pm\Delta C$ | $D\pm\Delta D$ | $\chi^{2}/n.d.f$
---|---|---|---|---|---|---
|  |  | (GeV-1) | (GeV${}^{-1})$ |  |
Lees et al. (2013a, b); Bianconi and Tomasi-Gustafsson (2015) | BaBar | $0.05\pm 0.01$ | $0.59\pm 0.2$ | $5.6\pm 0.1$ | $0.2\pm 0.2$ | 57/(55-4)= 1.1
Ablikim et al. (2019, 2020); Lees et al. (2013a, b) | BESIII-ISR,SC,BaBar | $0.07\pm 0.01$ | $0.93\pm 0.09$ | $5.9\pm 0.1$ | $0.1\pm 0.2$ | 227/(107-4)=2.2
Figure 1: (a): TL proton generalized FF as a function of $p$ from the data of
BaBar, Ref. Lees et al. (2013a) (black circles), BESIII-ISR Ablikim et al.
(2019) (blue squares) and BESIII-SC Ablikim et al. (2020) (green triangles),
with the regular background fit with Eq. (7) (black solid line ); (b): data
after subtraction, fitted with Eq. (8) (black solid line). For comparison the
fit from Ref. Bianconi and Tomasi-Gustafsson (2015) (red dashed lines) is also
shown.

As shown in Fig. 1, even with a slightly worse normalized $\chi^{2}$, the new
fit (black solid line) follows closely the one on the only BaBar data Bianconi
and Tomasi-Gustafsson (2015) (red dashed line). Let us note that the
consistency of the data obtained with different methods, beam scan and ISR,
rules out the possibility that the oscillations could be an artefact of the
ISR technique or of the photon detection.

##  IV Global Fit of the data

The cross section or the effective FF data can also be directly fitted with
the six-parameter function $F_{p}^{\rm fit}(p)$ of Eq. (6). The parameters are
reported in Table 3 and the fit is illustrated in Fig. 2 as a function of the
relative momentum $p$ (black solid line), together with the result from Ref.
Bianconi and Tomasi-Gustafsson (2015).

Table 3:  Six-Parameters fit, Eq. (6), of the annihilation cross section $\sigma_{e^{+}e^{-}\to\bar{p}p}$ as a function of relative momentum $p$ for the BaBar, BESIII and CMD-3 data. Ref. | $F_{0}$ | $m_{a}^{2}$ | $A$ | $B$ | $C$ | $D$ | $\displaystyle\frac{\chi^{2}}{\rm n.d.f.}$
---|---|---|---|---|---|---|---
(GeV2) |  (GeV-1) |  (GeV-1)
Ablikim et al. (2019, 2020); Lees et al. (2013a, b); Akhmetshin et al. (2019) |  9.7 $\pm$ 0.3 |  7.1 $\pm$ 0.5 |  0.073 $\pm$ 0.007 |  1.05 $\pm$ 0.07 |  5.51$\pm$ 0.09 |  0.04 $\pm$ 0.1 | $\displaystyle\frac{278}{118-6}=2.5$

Extending the data sets does not change essentially the fit, worsening the
$\chi^{2}$. The inclusion of the CMD-3 data heightens the curve in the near
threshold region. The blue dash-dotted line corresponds to a constant fitted
in the range $0.1<\sqrt{s}<0.9$ GeV, that gives the average value of the cross
section $\bar{\sigma}=0.87\pm 0.02$ nb. Such a value is close to the cross
section at the production threshold for a structureless fermions Baldini et
al. (2009), as for instance that of the reaction
$e^{+}e^{-}\to\mu^{+}\mu^{-}$.

Fig. 3 shows the data on the effective FF together with the curves that
represent the corresponding fit functions.

Figure 2: Born cross section for $e^{+}+e^{-}\to p+\bar{p}$ as a function of
the momentum $p$. The data are from CMD-3 Akhmetshin et al. (2019) (red
stars), BaBar Lees et al. (2013a, b) (black circles) BESIII-ISR Ablikim et al.
(2019) (blue squares) and BESIII-SC Ablikim et al. (2020) (green triangles)
are shown together with the six-parameter fit from Eqs. (6,7,8) and Table
3(black solid line), compared to the fit from Ref. Bianconi and Tomasi-
Gustafsson (2015) (red dashed line). The blue dash-dotted line corresponds to
a constant, fitted in the range $0.1<\sqrt{s}<0.9$ GeV.  Figure 3: Same as
Fig. 2 but for the TL proton generalized FF. The blue dash-dotted line is the
expectation for a constant cross section $\sigma=0.87$ nb.

###  IV.1 Analysis of the form factor ratio $R$

The comparison between the absolute values of the electric and magnetic FFs in
the TL and SL regions can be more easily done by considering their ratio.
Exploiting the Akhiezer-Rekalo recoil proton polarization method Akhiezer and
Rekalo (1968, 1974), that represents a unique and very powerful technique to
extract directly the FF ratio $G_{E}/G_{M}$ from the longitudinal to
transverse recoil proton polarization in the elastic scattering process
${\vec{e}}^{\,-}p\to e^{-}\vec{p}$, the JLab-GEP Collaboration obtained very
precise values of $R$ in a wide region of transferred momenta Puckett et al.
(2017, 2012). Note that the individual FFs can not be determined by this
method. Therefore it is assumed that the magnetic FFs is well known from the
unpolarized cross section measurements.

In the TL region, the present data from the BESIII Collaboration bring new
information on the ratio of the FFs moduli with comparable precision as in the
scattering region. The data from BaBar and BESIII are plotted in Fig. 4 as a
function of $|q^{2}|$. The choice of this variable does allow to show on the
same graph SL and TL values of the FF ratio and of their moduli respectively
Tomasi-Gustafsson and Rekalo (2001).

While the SL data (red squares in Fig. 4) show a monotone decrease, the TL
ones (green triangles in Fig. 4) Ablikim et al. (2020), decrease too, but show
the presence of oscillations, not contradicting the results from BaBar Lees et
al. (2013a) (black circles in Fig. 4). One can see a minimum in the TL range
$(5-6)$ GeV2, in correspondance to a little dip in the SL region, that should
be confirmed, because it lies just at the square momentum transfer
corresponding to the kinematical limits of two experiments of the JLab-GEP
Collaboration.

The SL and TL values of the FF ratio, move away with a smooth decrease from
$1/\mu_{p}$ ($\mu_{p}$ is the proton magnetic moment in units of the Bohr
magneton) at $q^{2}=0$, and from unity at the production threshold
$q^{2}=4m_{p}^{2}$, respectively. These are the values expected from the
definitions given above, as well as, at large transferred momenta, from the
QCD quark counting rules Matveev et al. (1973); Brodsky and Farrar (1973).
This is an indication that the perturbative domain has not been reached and
corroborates the predictions from Ref. Kuraev et al. (2012). Following a
similar approach as for the effective FF, we fit the ratio in the TL region
with a function $F_{R}$ reproducing a monopole decrease and a damped
oscillation:

$F_{R}(\omega(s))=\displaystyle\frac{1}{1+\omega^{2}/r_{0}}\left[1+r_{1}e^{-r_{2}\omega}\sin\left(r_{3}\omega\right)\right],\
\omega=\sqrt{s}-2m_{p}\,,$ (9)

where the unitary normalization at the production threshold,
$F_{R}(4m_{p}^{2})=1$, is imposed. The curve representing the fit function,
Eq. (9), obtained with the parameters reported in Table 4, is shown as a black
line in Fig. 4, together with the corresponding data on the TL ratio $R$
(black circles and green triangles). The monopole and the oscillatory
components are also shown.

Table 4:  4-parameters fit for $R$ as a function of $q^{2}$.  $r_{0}$ | $r_{1}$ | $r_{2}$ | $r_{3}$ | $\displaystyle\frac{\chi^{2}}{\rm n.d.f}$
---|---|---|---|---
(GeV2) |  (GeV-1) |  (GeV-1)
3 $\pm$ 2 |  0.5 $\pm$ 0.1 |  1.5 $\pm$ 1.2 |  9.3 $\pm$ 0.5 | $\displaystyle\frac{14}{22-4}=0.8$

Figure 4: Ratio $R=|G_{E}|/|G_{M}|$ as a function of $|q^{2}|$ (left) and $p$
(right) from BaBar (black circles) Lees et al. (2013a, b) and BES-SC (green
triangles) Ablikim et al. (2020). The solid black line is the fit from Eq.
(9), decomposed in the monopole component (green dashed line) and the
oscillatory component (black dotted line - shifted up by 0.5). The SL ratio
from the JLab-GEp Collaboration Puckett et al. (2017) is also shown (red
squares), together with its constrained monopole fit (red long-dashed line).

The red long-dashed line in Fig. 4 visualizes a one-parameter monopole
function, constrained to $1/\mu_{p}$ at $q^{2}=0$. Let us remind that in the
space-like region the electric FF is normalized to 1 (in unit of electric
charge) and the magnetic FF is normalized to $\mu_{p}$ at $q^{2}=0$ .

In Ref. Kuraev et al. (2012) it was suggested that a faster decreasing
behavior of the electric FF compared to the magnetic FF in the SL, as well as
in the TL region, is expected as a consequence of the presence of an inner
volume inside the nucleon that is electrically neutral (short distances
corresponding to large transferred momenta). The consequence is a dipole
behavior for the magnetic FF and an additional monopole decrease for the
electric FF, so that the ratio decreases like a monopole.

###  IV.2 Zero crossing of the angular asymmetry ${\cal A}$

A further possibility to illustrate these results, knowing the ratio $R$ and
the fit function, is to calculate the angular asymmetry, ${\cal A}(s)$, from
Eq. (3). By definition, it assumes values in the range $[-1,1]$, being null at
the production threshold, $i.e.,$ ${\cal A}(4m_{p}^{2})=0$. The data and the
fit on ${\cal A}$ are shown in Fig. 5.

It has been previously pointed out that, when extracted directly from the
cross section, the relative error on this variable is equivalent to an error
on $R^{2}$, being therefore preferable for the extraction of the individual
FFs Singh et al. (2016); Tomasi-Gustafsson and Dbeyssi (2014).

One can see that ${\cal A}(s)$ crosses zero at $s=(4.62\pm 0.07)$ GeV2,
meaning that, also at this squared momentum transferred, the modulus of the
ratio is equal to one, and hence, $|G_{E}|=|G_{M}|$. The uncertainty is
obtained by varying the function in a $\pm 5\%$ range (dashed black lines).
The determination of the zero crossing gives a precise experimental constrain
on FF models.

Figure 5: Angular asymmetry as a function of $|q^{2}|$ (left) and of $p$
(right) from BaBar (black circles), BESIII-SC (green triangles). The dash-
dotted blue curve corresponds to a constant and unitary ratio, $i.e.,$
$|R|=1$, while the solid black curve is related to the fit function
$F_{R}(s)$, with the dashed black curves representing a $\pm 5\%$ variation of
the function.

###  IV.3 Individual form factors $|G_{E}|$ and $|G_{M}|$

The monopole background, used to fit the FF ratio is consistent with Ref.
Kuraev et al. (2012) where it was suggested that the magnetic FF would follow
a dipole dependence, whereas an additional monopole factor would induce a
faster decrease of the electric FF in both SL and TL regions.

Overlapping the data for $|G_{E}|$ and $|G_{M}|$, extracted separately for the
first time by the BESIII Collaboration Ablikim et al. (2020), from
$e^{+}e^{-}\to\bar{p}p$ differential cross section data, the different
behavior of the two FFs becomes visible and sizeable, as shown in Fig. 6.
Surprisingly, $|G_{E}|$ and $|G_{M}|$ are also different at smaller $q^{2}$,
(even though they should coincide at the production threshold) and seem to
converge towards small values or to zero at large $q^{2}$.

One may inquire if the oscillations that are present in the cross section and
in the effective FF are also visible in the individual FFs, and, in this case,
if they have to be attributed to the electric or the magnetic FF, or to both
of them. The modulus of the electric FF, $|G_{E}|$, shows larger deviations
from a smooth behaviour, in particular it has a dip around 5-6 GeV2, whereas
$|G_{M}|$ follows closely a $(q^{2})^{-2}$ decrease. The relations between the
pairs of functions ($|G_{E}|$, $|G_{M}|$) and ($R$, $F_{p}$) are:

$|G_{E}(s)|=F_{p}(s)\sqrt{\frac{1+2\tau}{R^{2}(s)+2\tau/R^{2}(s)}}\hskip
28.45274pt|G_{M}(s)|=F_{p}(s)\sqrt{\displaystyle\frac{1+2\tau}{R^{2}(s)+2\tau}}.$
(10)

By means of these expressions, the moduli of the electric and magnetic FFs can
be calculated using for the ratio $R$ and the effective FF $F_{p}$ their fit
function $F_{R}$, Eq. (9), and $F_{p}^{\rm fit}$, Eq. (6), respectively. The
resulting curves are shown in Fig. 6. This procedure gives by construction a
smooth description of the individual moduli of the two FFs, from threshold up
to the highest experimentally accessible values of $s$ and represents a
particular interest to illustrate the near threshold behavior, as the
extrapolation of the FF data is constrained by the condition
$R(s=4m_{p}^{2})=1$.

The result is shown in Fig. 6. Oscillations characterize both FFs, although
they are more smooth on $|G_{M}|$. By the definition of the $F_{R}(s)$ fit
function, the convergence of the two electric and magnetic FFs, and hence,
also of the effective one, to a common value at the production threshold is
implied and we find:
$|G_{E}(4m_{p}^{2})|=|G_{M}(4m_{p}^{2})|=|F_{p}(4m_{p}^{2})|\equiv F_{\rm
th}\simeq 0.48$.

Note that the QCD model fitted to the cross section data Tomasi-Gustafsson et
al. (2005), when extrapolated back to the threshold, gives a common value for
the FFs equal to $F_{\rm th}\simeq 0.34$. On the other hand, the vector meson
dominance (VMD) model of Ref. Bijker and Iachello (2004) gives $F_{\rm
th}\simeq 0.29$. The comparison between the data and these models is shown in
Fig. 7. The QCD extrapolation provides, by definition, the same prediction for
the two FFs, as it depends on the number of the quarks involved in the
process. The VDM model of Ref. Bijker and Iachello (2004) predicts a steeper
behavior for $|G_{M}|$ and a number of resonances occurring in the unphysical
region, i.e., the portion of the TL region lying below the production
threshold. Such a region is accessible through the reaction $\bar{p}p\to
e^{+}e^{-}\pi^{0}$ Adamuscin et al. (2007) and can be investigated in next
future at the PANDA@FAIR facility Ritman (2005).

Figure 6: $|G_{E}|$ (red circles) and $|G_{M}|$ (blue squares) from BESIII.
The dashed red (dash-dotted blue) line is the calculation of $|G_{E}|$ and
$|G_{M}|$ from the fits of the effective FF $F_{p}$ and the ratio $R$. Figure
7: (a) $|G_{E}|$ (red circles) and (b) $|G_{M}|$ (blue squares) from the
BESIII Collaboration. The dashed blue (solid green) curve is the calculation
of $|G_{E}|$ and $|G_{M}|$ from the QCD extrapolation Tomasi-Gustafsson et al.
(2005) and from the VMD model of Ref. Bijker and Iachello (2004).

##  V  Discussion and Conclusions

We have considered the recent data on TL proton FFs from Ref. Ablikim et al.
(2019, 2020); Akhmetshin et al. (2019). These data confirm the regular
oscillations found in Ref. Bianconi and Tomasi-Gustafsson (2015, 2017). We
present a general fit of these data, that includes and updates the previous
analysis. A more precise determination of the oscillation parameters has been
done. In particular the oscillation period is a relevant parameter since it
has been related to sub-hadron scale processes Bianconi and Tomasi-Gustafsson
(2015, 2017). A similar behavior would be shown by the future $e^{+}e^{-}\to
n\bar{n}$ data S.Ahmed (2020), in this case the oscillation parameters should
bring information on the dynamics underlying the formation from the vacuum of
quark-diquark states, the quark having different flavor.

Our analysis does confirm a faster average decrease of the electric FF
compared to the magnetic one, following a similar behavior as in the SL
region. It is in agreement with the predictions of Ref. Kuraev et al. (2012),
where such a decreasing behavior was attributed to the existence of an
electrically neutral inner region in the proton. It is also compatible with
the VMD model of Ref. Bijker and Iachello (2004), that slightly overestimates
the magnetic FF. This appears also from the fact that the QCD behavior, that
does not differentiate the two FFs, overestimates $G_{E}$, reproducing better
$G_{M}$.

The new proton data, together with the future neutron data, will require a
revision of the phenomenological models based on fitting procedures, as the
parameters were determined in TL region, from the effective FF only. This will
be the object of a future work.

##  VI Appendix: expressions of the fit functions

The change of variables $s=s(p)$, as well as $p=p(s)$, follows from the
relations

$s=2m_{p}\left(m_{p}+\sqrt{p^{2}+m_{p}^{2}}\right)\,,\hskip
28.45274ptp=\sqrt{s\left(\frac{s}{4m_{p}^{2}}-1\right)}\,.$ (11)

Therefore Eqs. (7,8) can be rewritten as:

$\displaystyle F_{\rm 3p}(s)$ $\displaystyle=$
$\displaystyle\frac{F_{0}}{\left(1+\frac{s}{m_{a}^{2}}\right)\left(1-\frac{s}{m_{0}^{2}}\right)^{2}}=$
(12) $\displaystyle=$
$\displaystyle\frac{F_{0}}{\left(1+\frac{2m_{p}\left(m_{p}+\sqrt{p^{2}+m_{p}^{2}}\right)}{m_{a}^{2}}\right)\left(1-\frac{2m_{p}\left(m_{p}+\sqrt{p^{2}+m_{p}^{2}}\right)}{m_{0}^{2}}\right)^{2}}\,,$
$\displaystyle F_{\rm osc}(p(s))$ $\displaystyle=$ $\displaystyle
Ae^{-B\sqrt{s\left(\frac{s}{4m_{p}^{2}}-1\right)}}\cos\left[C\sqrt{s\left(\frac{s}{4m_{p}^{2}}-1\right)}+D\right]=Ae^{-Bp}\cos(Cp+D)\,.$
(13)

Equation (9) can be expressed as a function of $s$ as

$F_{R}(\omega(s))=\displaystyle\frac{1}{1+(\sqrt{s}-2m_{p})^{2}/r_{0}}\left[1+r_{1}e^{-r_{2}\left(\sqrt{s}-2m_{p}\right)}\sin\left(r_{3}\left(\sqrt{s}-2m_{p}\right)\right)\right]\,,$
(14)

taking into account that $\omega=\sqrt{s}-2m_{p}$.

## References

  * Pacetti et al. (2015) S. Pacetti, R. Baldini Ferroli, and E. Tomasi-Gustafsson, Phys. Rept. 550-551, 1 (2015).
  * Baier et al. (1973) V. N. Baier, V. S. Fadin, and V. A. Khoze, Nucl. Phys. B65, 381 (1973).
  * Lees et al. (2013a) J. Lees et al. (BaBar Collaboration), Phys. Rev. D87, 092005 (2013a).
  * Ablikim et al. (2019) M. Ablikim et al. (BESIII), Phys. Rev. D99, 092002 (2019), eprint 1902.00665.
  * Lees et al. (2013b) J. Lees et al. (BaBar Collaboration), Phys. Rev. D88, 072009 (2013b).
  * Ablikim et al. (2020) M. Ablikim et al. (BESIII), Phys. Rev. Lett. 124, 042001 (2020), eprint 1905.09001.
  * Puckett et al. (2017) A. J. R. Puckett et al., Phys. Rev. C96, 055203 (2017), [erratum: Phys. Rev.C98,no.1,019907(2018)], eprint 1707.08587.
  * Puckett et al. (2012) A. Puckett et al., Phys. Rev. C 85, 045203 (2012), eprint 1102.5737.
  * Zichichi et al. (1962) A. Zichichi, S. Berman, N. Cabibbo, and R. Gatto, Nuovo Cim. 24, 170 (1962).
  * Dubnickova et al. (1996) A. Dubnickova, S. Dubnicka, and M. Rekalo, Nuovo Cim. A 109, 241 (1996).
  * Akhmetshin et al. (2019) R. R. Akhmetshin et al. (CMD-3), Phys. Lett. B794, 64 (2019), eprint 1808.00145.
  * Bianconi and Tomasi-Gustafsson (2015) A. Bianconi and E. Tomasi-Gustafsson, Phys. Rev. Lett. 114, 232301 (2015).
  * Jones et al. (2000) M. Jones et al. (Jefferson Lab Hall A), Phys. Rev. Lett. 84, 1398 (2000), eprint nucl-ex/9910005.
  * Tomasi-Gustafsson and Rekalo (2001) E. Tomasi-Gustafsson and M. Rekalo, Phys. Lett. B 504, 291 (2001).
  * Kuraev et al. (2012) E. Kuraev, E. Tomasi-Gustafsson, and A. Dbeyssi, Phys. Lett. B712, 240 (2012).
  * Tomasi-Gustafsson et al. (2005) E. Tomasi-Gustafsson, F. Lacroix, C. Duterte, and G. Gakh, Eur. Phys. J. A 24, 419 (2005), eprint nucl-th/0503001.
  * Belushkin et al. (2007) M. Belushkin, H.-W. Hammer, and U.-G. Meissner, Phys. Rev. C 75, 035202 (2007), eprint hep-ph/0608337.
  * Bijker and Iachello (2004) R. Bijker and F. Iachello, Phys. Rev. C69, 068201 (2004).
  * Lomon and Pacetti (2012) E. L. Lomon and S. Pacetti, Phys. Rev. D 85, 113004 (2012), [Erratum: Phys.Rev.D 86, 039901 (2012)], eprint 1201.6126.
  * Denig and Salme (2013) A. Denig and G. Salme, Prog. Part. Nucl. Phys. 68, 113 (2013), eprint 1210.4689.
  * Bonneau and Martin (1971) G. Bonneau and F. Martin, Nucl. Phys. B27, 381 (1971).
  * Benayoun et al. (1999) M. Benayoun, S. I. Eidelman, V. N. Ivanchenko, and Z. K. Silagadze, Mod. Phys. Lett. A14, 2605 (1999), [Frascati Phys. Ser.15(1999)], eprint hep-ph/9910523.
  * Bytev et al. (2011) V. V. Bytev, E. A. Kuraev, E. Tomasi-Gustafsson, and S. Pacetti, Phys. Rev. D84, 017301 (2011), eprint 1103.4470.
  * Hoang (1996) A. H. Hoang, in _Electroweak interactions and unified theories. Proceedings, 31st Rencontres de Moriond, Leptonic Session, Les Arcs, France, March 16-23, 1996_ (1996), pp. 129–134, eprint hep-ph/9606288.
  * Gakh and Tomasi-Gustafsson (2005) G. Gakh and E. Tomasi-Gustafsson, Nucl. Phys. A 761, 120 (2005), eprint nucl-th/0504021.
  * Akhmetshin et al. (2016) R. Akhmetshin et al. (CMD-3), Phys. Lett. B 759, 634 (2016), eprint 1507.08013.
  * Baldini et al. (2009) R. Baldini, S. Pacetti, A. Zallo, and A. Zichichi, Eur. Phys. J. A 39, 315 (2009), eprint 0711.1725.
  * Akhiezer and Rekalo (1968) A. Akhiezer and M. Rekalo, Sov. Phys. Dokl. 13, 572 (1968).
  * Akhiezer and Rekalo (1974) A. Akhiezer and M. Rekalo, Sov. J. Part. Nucl. 4, 277 (1974).
  * Matveev et al. (1973) V. Matveev, R. Muradyan, and A. Tavkhelidze, Teor. Mat. Fiz. 15, 332 (1973).
  * Brodsky and Farrar (1973) S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153 (1973).
  * Singh et al. (2016) B. Singh et al. (PANDA), Eur. Phys. J. A 52, 325 (2016), eprint 1606.01118.
  * Tomasi-Gustafsson and Dbeyssi (2014) E. Tomasi-Gustafsson and A. Dbeyssi (PANDA), EPJ Web Conf. 66, 06024 (2014).
  * Adamuscin et al. (2007) C. Adamuscin, E. Kuraev, E. Tomasi-Gustafsson, and F. Maas, Phys. Rev. C 75, 045205 (2007), eprint hep-ph/0610429.
  * Ritman (2005) J. Ritman (PANDA), Int. J. Mod. Phys. A 20, 567 (2005).
  * Bianconi and Tomasi-Gustafsson (2017) A. Bianconi and E. Tomasi-Gustafsson, Phys. Rev. C 95, 015204 (2017), eprint 1611.02149.
  * S.Ahmed (2020) S.Ahmed (BESIII), in _3th European Research Conference on Electromagnetic Interactions with Nucleons and Nuclei, 27/10 – 02/11 2019,Paphos, Cyprus_ (2020).