Possible Manifestation of a Non-Pointness of the Electron in ee Annihilation Reaction at Centre of Mass Energies 55 - 207 GeV Yutao Chene Chih-Hsun Lina Minghui Liue Alexander S. Sakharovbc

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Possible Manifestation of a Non-Pointness of the Electron in e+e−

Annihilation Reaction at Centre of Mass Energies 55 - 207 GeV

Yutao Chen e, Chih-Hsun Lin a, Minghui Liu e, Alexander S. Sakharov b,c,

J¨urgen Ulbricht dand Jiawei Zhao e

aInstitute of Physics, Academia Sinica, Taipei, Taiwan 11529

bPhysics Department, Manhattan College

4513 Manhattan College Parkway, Riverdale, NY 10471, United States of America

cExperimental Physics Department, CERN, CH-1211 Gen`eve 23, Switzerland

dSwiss Institute of Technology ETH Zurich, CH-8093 Zurich, Switzerland

eChinese University of Science and Technology, USTC, Hefei, Anhui 230 029, China

Abstract

The experimental data from VENUS, TOPAS, OPAL, DELPHI, ALEPH and L3 collabora-

tions, collected from 1989 to 2003, are applied to study the QED framework through direct

contact interaction terms approach, using the annihilation reaction e+e−→γγ(γ). The analy-

sis involves performing of a χ2test to detect the presence of an excited electron e∗and evidence

of non-point like behavior in the e+e−annihilation zone. The results of the analysis indicate

a strong signal, with a conﬁdence level of approximately 5σ, for the presence of an excited

electron with a mass of 308 ±14 GeV, and a deviation from a point-like behavior of the charge

distribution of the electron. The radius of this deviation is 1.57 ±0.07 ×10−17 cm, which can

be interpreted as the size of the electron.

Keywords: QED; Contact interaction; Beyond standard model

March 2023

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 un-

derstanding 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.

arXiv:2210.11149v2 [hep-ex] 30 Mar 2023

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].

Three discoveries made in 1820 laid the groundwork for magnetism. Firstly, Hans Christian

Orsted demonstrated that a current-carrying wire produces a circular magnetic ﬁeld around

it [2]. Secondly, Andr’e-Marie Amp‘ere 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 [3].

Thirdly, Jean-Baptiste Biot and F’elix 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 [4].

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 Pl”ucker and Johann Wilhelm Hittorf [5]

and later named ”cathode rays” (Kathodenstrahlen) [6] by Eugen Goldstein in 1876. In 1897,

J. J. 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 [7].

Sir Joseph John Thomson credited in 1897 with the discovery of the electron, the ﬁrst

subatomic particle being 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 [8]. Finally, Millikan and Fletcher measured in 1909 the mass

and charge separately in the oil drop experiment [9].

After the discovery of the electron, Abraham [10] and Lorentz [11, 12] proposed the ﬁrst

models of the electron as an extended spherical electrical charged object with its total energy

concentrated in the electric ﬁeld, in 1903. However, these models were based on the assump-

tion 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 (like,

for example, the Poincare 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, Dirac [13] proposed a point-like model of the electron and rec-

ognized the appealing aspect of the Lorentz model [11] 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 in his paper [13] that although the electron can be

treated as a point charge to avoid diﬃculties with the 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 inves-

tigations of ferromagnetic substances with X-rays [14], he wrote: “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 magnetic moment µsis related to its spin S

through µs=−gsµBS/~, where gs≈2. The Stern-Gerlach experiment, ﬁrst proposed by Otto

Stern in 1921 and conducted by Walther Gerlach in 1922 [15], inferred the existence of quan-

tized 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 [15, 16].

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 beams [17] sources by electrostatic accelerators

such as Tandem accelerators, which can achieve a range of center-of-mass energies from 1.2

MeV [18] to 20 MeV [19]. There are three types of polarized beams that have been developed:

the atomic beam source, which uses the technique of the Stern-Gerlach experiment [20], the

Lamb-shift source [22] developed after the discovery of the Lamb shift in 1947 [21], and the

crossed-beam source [23].

Since 1926, various classical models of spinning point particles have been developed. How-

ever, 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 particle related to electron spin is the Schr¨odinger suggestion [24]

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 [25].

Other types of classical models of point-like spinning particles has been developed. The

Yang-Mills model, which 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 1+1 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 2+1

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. There are many reviews and textbooks that cover the diﬀerent classical models of

point-like spinning particles. Some examples include [26–30]. In general, such kind 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 [31] and then restricting undesirable eﬀects by applying geometrical [32] or

symmetry [33] constraints.

The discovery of the Kerr-Newman solution to the Einstein-Maxwell equations in 1965 led to

new possibilities for investigating the electron’s structure. Recently, this solution has been used

in [35] 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 [46], 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 [47].

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 [36–39] 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 [40] or a fermion and a boson [41]. In many models along this

line, quarks and leptons are composed of a scalar and a spin-1/2 preon. Composite models [36–

39] predict a rich spectrum of excited states [36–39, 42] 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 [43]. 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, we must rely on phenomenological studies of

substructure eﬀects, which can manifest in various reactions (see [44] 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. QED provides an ideal framework for studying potential substructure of

leptons. Any deviation from QED’s predictions in diﬀerential or total cross-section of e+e−

scattering can be interpreted as non-pointlike behavior of the electron or the presence of new

physics. Note that the case of excited quarks [45] is a direct generalization of the lepton case.

Theoretical predictions suggest that the transition mechanism of excited quarks is through

gluon emission, q∗→qg. 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 e+e−scattering experiment, the process of photon pair production e+e−→γγ

stands out due to its negligible contribution from weak interactions. Additionally, another

process of fermion pair production with e+e−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 the left panel of Fig. 1.

The two real photons in the ﬁnal state of the e+e−→γγ 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 reaction is highly sensitive to long range QED inter-

actions, 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 γand Z0. As

a pure annihilation reaction, the e+and e−in the initial state completely annihilate to two

photons in the ﬁnal state, making it easy to subtract the background signal.

The Bhabha reaction e+e−→e+e−is a mixed reaction that occurs via scattering in both

the s-channel and t-channel. At energies around the Z0pole, the Z0contribution dominates.

The elastic scattering and annihilation channel are superimposed since the e+and e−in the

initial and ﬁnal states are identical. Therefore, the e+e−→e+e−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 e+e−→γγ reaction measured by various e+e−storage ring experiments. Specif-

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PossibleManifestationofaNon-PointnessoftheElectronine+eAnnihilationReactionatCentreofMassEnergies55-207GeVYutaoChene,Chih-HsunLina,MinghuiLiue,AlexanderS.Sakharovb,c,JurgenUlbrichtdandJiaweiZhaoeaInstituteofPhysics,AcademiaSinica,Taipei,Taiwan11529bPhysicsDepartment,ManhattanCollege4513ManhattanCol...

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Possible Manifestation of a Non-Pointness of the Electron in ee Annihilation Reaction at Centre of Mass Energies 55 - 207 GeV Yutao Chene Chih-Hsun Lina Minghui Liue Alexander S. Sakharovbc

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