1. Introduction
The electron is one of the most fundamental building blocks of matter, and its discovery over a century ago revolutionized our understanding of the physical world. Since then, it has been the subject of countless investigations, revealing properties that continue to challenge our understanding of nature at the deepest level. A historical account of the electron’s discovery and subsequent study can provide valuable insights into the foundations of modern particle physics.
In 1785, the discovery of Coulomb’s law and in 1820, the discovery of magnetism paved the way for investigating charged particle beams with cathode-ray tubes. By 1869, this possibility was realized.
Charles-Augustin de Coulomb’s law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them [
1,
2].
Three discoveries made in 1820 laid the groundwork for magnetism. Firstly, Hans Christian
rsted demonstrated that a current-carrying wire produces a circular magnetic field around it [
3,
4]. Secondly, André-Marie Ampère showed that parallel wires with currents attract each other if the currents flow in the same direction and repel if they flow in opposite directions [
5,
6,
7]. Thirdly, Jean-Baptiste Biot and Félix Savart determined experimentally the forces that a current-carrying, long, straight wire exerts on a small magnet. They found that the forces were inversely proportional to the perpendicular distance from the wire to the magnet [
8].
Cathode rays, also known as electron beams or
e-beams, are streams of electrons observed in discharge tubes. They are produced by applying a voltage across two electrodes in an evacuated glass tube, causing electrons to be emitted from the cathode (the negative electrode). The phenomenon was first observed in 1869 by Julius Plücker and Johann Wilhelm Hittorf [
9,
10] and later named “cathode rays” (Kathodenstrahlen) [
10,
11] by Eugen Goldstein in 1876. In 1897, Joseph John Thomson discovered that cathode rays were composed of negatively charged particles, later named electrons. Cathode-ray tubes (CRTs) use a focused beam of electrons deflected by electric or magnetic fields to create images on a screen [
12].
Sir Joseph John Thomson is credited with the discovery of the electron in 1897, the first subatomic particle discovered. He showed that cathode rays were composed of previously unknown negatively charged particles, which must have bodies much smaller than atoms and a very large charge-to-mass ratio [
13,
14]. Finally, Robert A. Millikan and Harvey Fletcher measured in 1909 the mass and charge separately in the oil drop experiment [
15,
16,
17].
After the discovery of the electron, Max Abraham [
18] and Hendric Lorentz [
19,
20] in 1903 proposed the first models of the electron as an extended spherical electrical charged object with its total energy concentrated in the electric field. However, these models were based on the assumption of a homogeneous distribution of charge density. Although the model provided a means of explaining the electromagnetic origin of the electron’s mass, it also raised the problem of preventing the electron from flying apart under the influence of Coulomb repulsion. Abraham proposed a solution to this inconsistency by suggesting that non-electromagnetic forces (such as, for example, Henri Poincaré stress) were necessary to prevent the electron from exploding. One may say that at that time, modeling the electron within the framework of electromagnetism was deemed impossible. Later, Paul Dirac [
21] proposed a point-like model of the electron and recognized the appealing aspect of the Lorentz model [
19] regarding the electromagnetic origin of the electron’s mass. Nonetheless, at that time, this idea was found to be inconsistent with the existence of the neutron. Dirac highlighted [
21] that although the electron can be treated as a point charge to avoid difficulties with infinite Coulomb energy in equations, its finite size reappears in a new sense in the physical interpretation. Specifically, the interior of the electron can be viewed as a region of space through which signals can be transmitted faster than light.
Arthur Compton firstly introduced the idea of electron spin in 1921. In a paper on investigations into ferromagnetic substances with X-rays [
22,
23], he writes: “Perhaps the most natural, and certainly the most generally accepted view of the nature of the elementary magnet, is that the revolution of electrons in orbits within the atom give to the atom as a whole the properties of a tiny permanent magnet”. The electron’s spin magnetic moment,
, is related to electron’s spin amgular momentum,
S, through
, where the spin
g-factor,
,
is the Bohr magneton, and
ℏ is the reduced Planck constant. The Stern–Gerlach experiment, first proposed by Otto Stern in 1921 and conducted by Walther Gerlach in 1922 [
24], inferred the existence of quantized electron spin angular momentum. In the experiment, spatially varying magnetic fields deflected silver atoms with non-zero magnetic moments on their way to a glass slide detector screen, providing evidence for the existence of electron spin. The existence of electron spin can also be inferred theoretically from the spin-statistics theorem and the Pauli exclusion principle. Conversely, given the electron’s spin, one can derive the Pauli exclusion principle [
24,
25].
The existence of quantized particle spin allows for the possibility of investigating spin-dependent interactions by scattering polarized particle beams on different targets. In nuclear physics, scattering experiments use polarized beam [
26,
27] sources via electrostatic accelerators such as Tandem accelerators, which can achieve a range of center-of-mass energies from 1.2 MeV [
28,
29] to 20 MeV [
30,
31]. There are three types of polarized beams that have been developed: the atomic beam source, which uses the technique of the Stern–Gerlach experiment [
32,
33], the Lamb-shift source [
34,
35,
36] developed after the discovery of the Lamb shift in 1947 [
37], and the crossed-beam source [
27,
38].
Since 1926, various classical models of spinning point particles have been developed. However, these models face the challenge of constructing a stable point-like particle that includes a single repulsive Coulomb force over a range from zero to infinity.
One model of point-like particles related to electron spin is the Schr
dinger suggestion [
39] that connects electron spin with its Zitterbewegung motion—a trembling motion due to the rapid oscillation of a spinning particle about its classical worldline. The Zitterbewegung concept was motivated by attempts to understand the intrinsic nature of electron spin and involved fundamental studies in quantum mechanics [
40,
41,
42,
43,
44].
Other types of classical models of point-like spinning particles have been developed. The Yang–Mills model is a class of gauge theories that describe the strong and electroweak interactions in the Standard Model of particle physics. The Weyl model is a spinor field theory that describes massless spin-1/2 particles that do not follow the Dirac equation. The Thirring model is a
dimensional field theory that describes a system of Dirac fermions coupled to a massless bosonic field. It is exactly solvable and has been used as a toy model for studying many-body problems in condensed matter physics. The Gross–Neveu model is a
-dimensional field theory that describes a system of fermions with an interaction term that is quadratic in the fermion field. It is also exactly solvable and has been used to study critical phenomena in condensed matter physics. The Proca model is a relativistic quantum mechanical model that describes a massive vector boson. It is used to describe the massive vector bosons in the electroweak interaction and has been used extensively in the development of the Standard Model. Many reviews and textbooks consider various classical models of point-like spinning particles; some examples include Refs. [
45,
46,
47,
48,
49]. In general, these kinds of models encounter the problem of divergent self-energy for a point charge and approach this problem in the frame of various generalizations of the classical Lagrangian terms with higher derivatives or extra variables [
50,
51,
52,
53,
54,
55,
56,
57,
58,
59] and then restrict undesirable effects by applying geometrical [
60,
61] or symmetry [
62,
63] constraints.
The discovery of the Kerr–Newman solution [
64] to the Einstein–Maxwell equations in 1965 led to new possibilities for investigating the electron’s structure. Recently, this solution has been used in Ref. [
65] to propose a model that considers the interplay between electrodynamics and gravity in the electron’s structure. Coupling electrodynamics with gravity introduces the geometry of De Sitter spaces [
66,
67], which provides attractive/repulsive forces dependent on distance from the origin and can distinguish between Schwarzschild and De Sitter black holes, ensuring the electron’s stability from the Coulomb repulsion. The theoretical aspects of whether the electron is point-like or not are discussed in Ref. [
68].
The modeling of electron structure is driven by the desire to understand fundamental issues, such as the number of fermion families, fermion mass hierarchy and mixing properties, that the Standard Model cannot explain. For example, a natural consequence of the so-called composite models approach [
69,
70,
71,
72] to addressing the aforementioned questions is the assumption that quarks and leptons possess substructure. According to this approach, a quark or lepton might be a bound state of three fermions [
73] or a fermion and a boson [
74]. In many models along this line, quarks and leptons are composed of a scalar and a spin-1/2 preon.Composite models [
69,
70,
71,
72] predict a rich spectrum of excited states [
69,
70,
71,
72,
75] of known particles. Discovering the excited states of quarks and leptons would be the most convincing proof of their substructure. Assuming that ordinary quarks and leptons represent the ground states, it is natural to assign the excited fermions with the same electroweak, color and spin quantum numbers as their low-lying partners. Excited states are transferred to ground states through generalized magnetic-type transitions, where photons (for leptons) or gluons (for quarks) are emitted, as described in Ref. [
76]. When excited states have small masses, radiative transition is the main decay mode, but when their masses approach that of the
W boson, a large fraction of three-particle final states appear in the decay of excited states.
As discussed above, there is currently no fully predictive model capable of describing the substructure of quarks and leptons. Therefore, one must rely on phenomenological studies of substructure effects, which can manifest in various reactions (see [
77] for a review). The search for excited charged and neutral fermions has been ongoing for over 30 years, but to date there has been no success. Quantum electrodynamics (QED) provides an ideal framework for studying potential substructure of leptons. Any deviation from QED’s predictions in differential or total cross-section of
scattering can be interpreted as non-point-like behavior of the electron or the presence of new physics. Note that the case of excited quarks [
78] is a direct generalization of the lepton case. Theoretical predictions suggest that the transition mechanism of excited quarks,
, is through gluon,
g, emission,
. However, distinguishing this effect from the standard background of three-jet events poses a challenge. As a result, the lepton sector remains the most favorable field for searching for substructure effects from an experimental standpoint. Among the various channels in the
scattering experiment, the process of photon pair production
stands out due to its negligible contribution from weak interactions. Additionally, another process of fermion pair production with
final state, which is used as the luminosity meter at low angle, is also highly suitable for QED testing in the search for the electron’s substructure. Both processes are presented in
Figure 1, left.
The two real photons in the final state of the reaction are indistinguishable, so the reaction proceeds through the t- and u-channels, while the s-channel is forbidden due to angular momentum conservation. The s-, t-, and u-channels (Mandelstam variables) represent different ways particles interact and exchange energy and momentum. The reaction is highly sensitive to long-range QED interactions, and the two photons in the final state have left-handed and right-handed polarizations which results in total spin of zero, forbidding the s-channel with spin one for and . As a pure annihilation reaction, the and in the initial state completely annihilate into two photons in the final state, making it easy to subtract the background signal.
The Bhabha reaction is a mixed reaction that occurs via scattering in both the s-channel and t-channel. At energies around the pole, the contribution dominates. The elastic scattering and annihilation channel are superimposed since the and in the initial and final states are identical. Therefore, the reaction serves as a test for the superimposition of short-range weak interaction and long-range QED interaction.
In this paper, we analyze deviations from QED by combining data on the differential cross-section of the
reaction measured via various
storage ring experiments. Specifically, the VENUS Collaboration investigated this reaction in 1989 [
79] at the center-of-mass energies,
= 55 GeV–57 GeV, while the OPAL Collaboration [
80] studied it in 1991 at the
pole at
= 91 GeV. The TOPAS Collaboration also investigated this reaction in 1992 at
= 57.6 GeV [
81], while the ALEPH Collaboration studied it in 1992 at the
pole at
= 91.0 GeV [
82]. Moreover, the DELPHI Collaboration investigated the reaction from 1994 to 2000 at energies ranging from
= 91.0 GeV to 202 GeV [
83,
84,
85], while the L3 Collaboration studied it from 1991 to 1993 at the
pole, with
= 88.5 GeV to 93.7 GeV [
86]. The L3 Collaboration also studied the reaction in 2002 at
ranging from 183 GeV to 207 GeV [
87], and the OPAL Collaboration investigated it in 2003 at
ranging from 183 GeV to 207 GeV [
88]. Deviations from QED were investigated through the study of contact interactions and excited electron exchange displaced as in
Figure 1. Colleagues of some of the authors of this paper have reviewed experimental studies and models of deviations from QED in their Theses [
89,
90,
91]. For earlier review, see [
92].
The effective Lagrangian governing a contact interaction exhibits a dependence on the inverse cutoff scale,
, where the power of
is determined by the dimensionality of the fields involved. Additionally, this Lagrangian conserves the helicity of fermion currents. This ensures known particle masses are much less than
. Different helicity choices for the fields used in the Lagrangian result in different predictions for angular distributions and polarization observables in reactions where the contact interaction is present.
Figure 2b depicts the QED direct contact term, characterized by scale parameters
and
, which stand for the cutoff parameter in case of positive and negative interferences, as described in
Section 2.5.1. These parameters are subsequently interpreted as being indicative of an extended annihilation radius in the
reaction.
Figure 2c depicts Feynman diagrams that are sensitive to the mass of the excited electron,
, with the cutoff parameter,
, being a function of
.
In 1989, the VENUS Collaboration [
79] established initial limits of
81 GeV and
82 GeV. Table 11 in Ref. [
79] provides an overview of other Collaborations that have been studying the same subject. The significance of all analyses was below
(one standard deviation) and the fitted values of the parameters
and
were negative. In 2002, the L3 Collaboration [
87] established limits on
400 GeV,
300 GeV and
310 GeV, including negative fit parameters with a significance below
. In their latest publication on the subject, in 2013 the LEP (Large Electron-Positron collider) Electroweak Working Group [
93] conducted an analysis of data from the differential cross-section of all LEP detectors in the energy range of
= 133 GeV to 207 GeV. The group established limits of
431 GeV,
339 GeV and
366 GeV, which included negative fit parameters with a significance of nearly 2
.
Thus, comparing the results obtained by combining the data from all LEP2 Collaborations with those from L3 alone, it can be observed that the increased statistics have led to more confident results. Based on this observation, we conduct a global fit using data from all six research projects mentioned above to investigate
,
, and
for
ranging from 55 GeV to 207 GeV, including the corresponding luminosities. An initial attempt to perform the global fit, which involved some of the authors of this paper, has been previously described in detail in Refs. [
94,
95,
96]. It is noteworthy that the global fit revealed a significant deviation of the differential cross-section of the annihilation reaction
from QED predictions, with a statistical significance of 5
.
In the current paper, we scrutinize the global fitting procedure by examining all technical details used in the
-analysis.
Section 2 provides a detailed description of the theoretical framework, used for calculating the differential and total cross-sections of the
reaction in QED, including radiative corrections and modifications due to contact interactions and models with excited electrons.
Section 3 presents all the data used in the global fitting procedure, along with a description of the cross-section measurement procedure. The
-analysis, applied for the global fit, is described in
Section 4, and in
Section 5, we validate the
-procedure by inferring the total cross-section, which exhibits a similar significance of around 5
. The systematic uncertainties of the analysis are discussed in
Section 6. In
Section 7, we interpret the results of the global fit in the context of the non-pointness of the electron and present conclusions.
7. Concluding Remarks
The VENUS, TOPAS, OPAL, DELPHI, ALEPH and L3 Collaborations measured the differential cross-section of the
reaction to test QED. Except for ALEPH, all Collaborations observed a negative deviation from the QED, although with low significance. The total cross-section test of LEP2 and the comparison with measurements in
Figure 15 support these negative trends. We performed a thorough
analysis using all available data to search for evidence of an excited electron,
, and a finite annihilation length, using a direct contact term approach. By conducting a global analysis of the combined datasets, it was possible for the first time to establish a significance of approximately 5
on the mass of an excited electron, which is
GeV. A similar 5
significance effect was detected for a charge distribution radius of the electron,
cm. Therefore, combining the full statistical power of all available LEP and non-LEP high-efficiency experiments on measurements of the cross-section of the reaction of annihilation in
collisions allowed us to identify the signal of existence of excited electron and contact interaction at a high level of significance. Earlier analyses have restricted themselves to the data collected only with the LEP detectors and at not all and limited range LEP energies as well as with non-LEP detectors. Therefore, combining the full statistical power of all available LEP and non-LEP high-efficiency experiments on measurements of the cross-section of the reaction of annihilation in
collisions, we were able to recognize the signal of the existence of an excited electron and contact interaction at a high level of significance.
Extensive measurements and analyses were conducted to search for quark and lepton compositeness in contact interaction [
126], specifically in the Bhabha channel as shown in
Figure 1. A hint of axial-vector contact interaction was observed in the data on
scattering from ALEPH, DELPHI, L3 and OPAL at center-of-mass energies ranging from 192 to 208 GeV. The detection occurred at
TeV [
127].
At the Z
pole, the
reaction exhibits a suppression of the
s-channel, resulting in
, demonstrating very good agreement. Alternatively, in a Bhabha-like reaction (
), a different QED test is used to search for
, utilizing pair production in the s-channel through
and Z boson exchange, similar to Bhabha scattering as in
Figure 1. The mass values or limits for an excited electron depend on the test reaction used for its study and the theoretical interpretation of
values. For example, in the case of the
reaction, these values can be obtained from the Lagrangian (
28) (or Equation (
4) in Ref. [
87]). The L3 Collaboration [
128] set the lower limits on 95% CL (confidence level) for pair production of neutral heavy leptons, in the mass range
GeV to
GeV, depending on the model (Dirac or Majorana). L3 Collaboration also set the lower limits at 95% CL for pair-produced charged heavy leptons from
GeV to
GeV. Similarly, the OPAL Collaboration set the lower limits on 95% CL for long-lived charged heavy leptons and charginos by
GeV, as well as lower limits on the neutral (
) and charged (
heavy leptons [
129,
130]. The HERA H1 Collaboration searched for heavy leptons and obtained best-fit limits for
production in the HERA mass range in the
final state, with a composite scale parameter
excluding values below approximately 300 GeV [
131].
The CMS Collaboration is searching for long-lived charged particles in pp collisions [
132] using a modified Drell–Yan production process. This involves the annihilation of a quark and an antiquark from two different hadrons, producing a pair of leptons through the exchange of a virtual photon or
in the s-channel. The study excluded Drell–Yan signals with the charge,
, below masses of 574 GeV/
.
The latest experimental data from hadronic machines [
133,
134,
135,
136] do not provide evidence for excited leptons, setting an exclusion limit on the excited electron mass is
TeV for the reaction of single production like
(with
X denoting all particles), which is different from the double
production investigated in the
scattering reaction. The LHC experiments rigorously investigate lepton–quark contact interactions by analyzing high-mass oppositely charged lepton pairs produced through the
Drell–Yan process [
137,
138]. However, it is important to note that the result obtained from the Drell–Yan process analysis is not directly comparable to the result obtained from the
annihilation analysis. These two methods involve different experimental techniques and hence are sensitive to different types of cut-off parameters, leading to different interpretations in terms of an electron’s size. Future colliders with higher center-of-mass energy and luminosity are considered to continue the search for the excited leptons and contact interactions. The production of two photons at large angles in
annihilation has been suggested as a way to measure the luminosity of future circular and linear colliders. These colliders, including the Future Circular Collider, FCC-ee [
139], the Circular Electron Positron Collider, CEPC [
140], the International Linear Collider, ILC [
141], and the Compact Linear Collider, CLIC [
142], are designed to have polarized beams and can be used to test the accuracy of the Standard Model and search for signals of new physics.
The high-precision measurements of the electron’s magnetic dipole moment,
, provide a powerful tool for constraining the electron’s radius [
143,
144,
145]. If nonstandard contributions to
scale linearly with the electron mass, the estimated bound on the electron radius is on the order of
cm. However, if these contributions scale quadratically with the electron mass, as predicted by chiral symmetry [
144], the bound becomes weaker and is at the level of
cm. Importantly, this does not contradict the result obtained from the measurements of the direct contact term in the annihilation reaction, as obtained in our analysis.
The exchange of the excited electron does not produce non-zero polarization effects in the case of only one polarized beam [
146], at least in the lowest order of perturbation theory. This is because the reaction that produces the excited electron conserves space parity, which can be inferred from the expression for the Lagrangian (
28). From the other side, contact interaction affects the polarization observables in reaction
when the initial particles are polarized. In the general case, the contact interaction violates space parity [
146]. Therefore, non-zero observables arise only when one of the beams is polarized. The pure QED mechanism of this reaction, without taking radiative corrections into account, does not produce such polarization effects. However, electroweak corrections (at the one-loop level, as shown in Ref. [
147]) can introduce an additional term to the amplitude of this process that violates parity and, therefore, can lead to non-zero polarization observables. The future colliders, mentioned above, which employ polarized beams, offer a promising opportunity for experimental investigation into the polarization effects of contact interactions.
If one considers the question of whether the non-pointness of the electron is observed in the annihilation reaction, a speculative approach can be taken using a model suggested in Refs. [
65,
148] based on the superconducting cores [
149,
150,
151,
152]. The model proposes an electromagnetic spinning soliton for the electron, accompanied by a de Sitter vacuum disk that generates electric and magnetic fields. Within this framework, it is possible to construct a wave function for the electric field. When connected with the model of Ref. [
148], this wave function yields a Lorentz-contracted radius that agrees with experimental findings, approximately
cm [
148,
153,
154]. The numerical coincidence observed between the calculations in Refs. [
65,
148,
153,
154] and the experimental results suggests a potential manifestation of the non-point nature of the electron within the frameworks of studies in Refs. [
65,
148,
153,
154].
One can speculate that depending on the experimental tests, the electron may exhibit two types of extended interiors. Indeed, in the
reaction, only the QED long-range interaction is tested, while the weak interaction via
is suppressed by angular momentum conservation. As a result, the
of
GeV, obtained in our analysis is interpreted in terms of size of electron, which amounts to
cm. On the other hand, in the Bhabha reaction,
, the short-range weak and QED interactions are involved. Due to the much larger differential cross-section in the Bhabha channel compared to that in the pure QED channel, the Bhabha channel dominates. Moreover, the inclusion of the
contribution in the reaction results in a significantly higher
of
TeV compared to the
reaction, which in turn leads to an eight-fold reduction in extension if interpreted in terms of a radius. Based on the observed data and analysis, it is tantalizing to suggest that the electron may possess not just one, but two distinct interiors, an outer shell and an inner core. With these intriguing findings, we can tantalizingly speculate that the humble electron is not just a trivial point particle, but rather a complex entity with both an outer shell and an inner core. Could it be that two attributes are combined in this particle, as some theories have suggested [
65]? The possibilities are truly fascinating and open up new avenues for further exploration and discoveries in the field of particle physics.