###### Contents

  1. Preface
  2. 1\. Historical introduction: I
  3. 2\. Some early works on cosmic rays
  4. 3\. Historical introduction: II
    1. 3.1 On Landé separation factors
    2. 3.2 On Field Theory aspects of AMM
    3. 3.3 Experimental determinations of the lepton AMM: a brief historical sketch
      1. 3.3.1 On the early 1940s experiences
      2. 3.3.2 Some previous theoretical issues
      3. 3.3.3 Further experimental determinations of the lepton AMM
  5. 4\. Towards the first exact measurements of the anomalous magnetic moment of the muon
  6. List of some publications of A. Zichichi
  7. List of some publications of R.L. Garwin
  8. Bibliography

## Preface

Most of the work of leading scientists has always been characterized both by
an initial theoretical setting and analysis of the given problem under
examination and by the related experimental arrangement, and vice versa,
taking into account the main Galileian paradigm of scientific knowledge,
essentially given by the dialectic and inseparable relationships between
experimental bases and theoretical-formal structures from which arise the
rational thought. These scientists have always been interested both to
theoretical aspects and experimental data, like Jun John Sakuari (1933-1982)
as remembered by John S. Bell in (Sakurai 1985, Foreword). On the other hand,
just due to its Galileian nature, no history of theoretical physics can be
disjoined from experimental context, and vice versa. We have herein tried to
adopt a new way of doing history of science: namely, trying to delineate a
technical (or internal) history of a certain field of knowledge through the
life and the work of those people who have, at international level,
significantly and permanently contributed to it, along their life. Amongst
them, we shall consider, for example, some of the main works of A. Zichichi
and R. Garwin, namely those which have led to the first exact measurements of
the anomalous magnetic moment of the muon, one of the first precise test of
QED.

To be precise, in drawing up this work, we adopt that unique possible
historiographical methodology which has to be followed to pursue a correct and
objective historic-biographical report of the work of a given author under
examination, that is to say, the one consisting in giving primary and absolute
priority to the study and to the analysis of the original papers and works of
the author under examination (primary literature). Only subsequently it will
be then possible also to take into account the related already existent
secondary literature. This for trying to minimize, as much as possible, the
distortions and mystifications due to the unavoidable personal equation111This
concept has its own history which starts from Astronomy to Freudian
Psychoanalysis and Jungian Analytical Psychology. Here, we shall mean such a
term in the latter wider psychological meaning (see (Galimberti 2006) and
(Thomä and Kächele 1990, Vol. 1., Chap. 3, § 3.1)). Following (Carotenuto
1991, Chapter X), the personal equation is an unavoidable subjective factor
which influences on the evaluation of objective data, leading to different
visions of the phenomenological fields under examination. It is determined by
the individual history, by constitutional and typological elements and by
social-cultural factors. It acts as a perceptive filter, or rather as an
internal transducer, which redefines, according to personal parameters, the
reality, shaping the knowledge’s act. For instance, the various mythological
deformations and biases are mainly due to its action, so that, as regards
historical sciences, we agree with that historiographical method which gives
priority to the study of the primary literature of a given author, like her or
his works. (biases, complexes, mythicizations, etc) which is implicitly
presents in everyone of us. As picturesquely recalled by Vittorio de Alfaro in
(De Alfaro 1993, Introduction, p. 3), <<the historical reconstruction is
everything except a ’fractal’: indeed, the level of enlargement with which we
treat a historical process, greatly shall influence the conclusions that we
deduce>>. A similar case is besides also recalled by Bruno Rossi in (Rossi
1964, Preface), in which he warns on the impartiality with which himself has
written the history of cosmic rays, since he was directly involved, as a
leading actor, in the international research on this field, so that he does
not exclude to have given a major load to the work made by his research group.
On the other hand, this historiographical methodology is just that advocated
by Benedetto Croce himself (see (Croce 1938)) to obtain, through a rational
analysis of the sources, an impartial historical judgement devoid of any
biased or partial mystification.

There are non-negligible historiographical questions about the general history
of science, which we wish to outline too as an apology for the method used in
carrying on this work; in exposing such a historiographical problematic, we
mainly follow (Piattelli Palmarini 1992, Chap. I, §§ 2.I, 2.II and 2.III). Let
us say it immediately: such a problematic derives from the far from being
trivial questions existing between myth and science, which embed their roots
in the crucial historical passage from mythological to philosophical thought.
The relationships between myth and science are far from being ancient and
negligible: in this regards, it is enough to remember as Wolfgang Pauli
himself, after a long period of collaboration with Carl Gustav Jung, put much
attention to the possible links and intersections between epistemology and
analytical psychology, writing many interesting works on these arguments (see
(Jacob 2000), (Tagliagambe and Malinconico 2011) and references therein). The
French anthropologist Pierre Smith claims that the myth is always in nuce
(that is to say, implicitly) presents in the way in which each of us tells of
herself or himself, above all as regards her or him own past, this, in turn,
implying unavoidable distortions and mystifications which give rise to a
mythic production. In short, the myth is an efficient way to organize and to
coordinate the individual and collective memory. The Smith’s schoolmaster,
Claude Lévi-Strauss, said that the myth is a story continuously transformed by
who believe only of repeating it and to which, instead, he or she gives ”an
excess of meaning” whenever it is re-evoked.

Also in science there is an organization of collective and individual memory
where the mythical element may appear. For instance, Thomas S. Kuhn,
collecting a great number of interviews carried out with the founders of
modern physics, frequently noticed many inaccuracies and inconsistences as
concerns their biographies which resulted to be strangely logical, linear and
educationally edifying, but contrasting with the real facts and the original
sources; hence, Kuhn finished to conclude that the real history of science is
not so perfectly constructed and have not the direct pedagogical function of
those a little mythical stories told by handbooks and protagonists. Also
Gerald Holton has experienced an emblematic case of the same type: precisely,
interviewing in old age, Einstein convinced himself to have developed his
special theory of relativity on the basis of the results of the famous
Michelson-Morley experiment, building up, in such a manner, a logical
motivation to the birth of his theory222Also (Brown and Hoddeson 1983) confirm
that ¡¡people cannot be totally objective about the events in which they
participate; we tend unconsciously to reinterpret history in terms of present-
day values¿¿. But, in doing so, often the historical reality may go lost..
Indeed, such an experiment was yes carried out few years before the
Einsteinian publications, but Einstein did not know it when he formulated his
ideas. The Einstein reconstruction was logic and educationally efficacy even
if historically false; bona fide, he self-convinced himself that the things
were just gone so. In these cases, we should consider, according to Claude
Lévi-Strauss and François Jacob, the myth as an excess of meaning needed for
organizing the memory and for giving a logical and instructive meaning to its
contents, often to detriment of the real historical and chronological truth.
All this makes particularly difficult to do history of science; it may be
included in the wider unavoidable problematic concerning the already mentioned
personal equation, which would shed a certain shadow of discredit on the
history of science if it weren’t taken into the right account. For these
reasons, we think that the more correct historiographical method for carrying
out a scientific biography is that consisting at first in analyzing directly
the primary related literature, trying to prevent the non-objective
deformations given by the effects of this personal equation, almost desiring
to aspire to emulate the coldness or indifference of a psychoanalyst which
must simply reflect like a clean mirror (see (Thomä and Kächele 1990, Vol. 1.,
Chap. 3, § 3.1)) with the highest objectivity degree333The psychoanalysts try
to attain this by means of the so-called didactic analysis, which is strictly
correlated to the dualistic and dialectic interaction between transfer and
countertransfer phenomena (see (Thomä and Kächele 1990, Vol. 1., Chap. 3, §
3.1))..

This methodology, moreover, is the only way which permits us to may infer the
formation and evolution of the thought of a given author, undergoing his
creative process (as in this case), along her or his social-cultural and
scientific career. Such a historiographical method resembles, in a certain
sense, that already adopted by Francesco Giacomo Tricomi in (Tricomi 1967)
from the mathematics history side. On the other hand, all this had already
been highlighted by Sir Patrick M. S. Blackett (1897-1974) who remembered to
what distortions may lead the above recalled mythical production, by the so-
called scientific divulgation, if one does not take the right position respect
to the author under examination. The first and the most frequent
methodological error made by the historians of science concerns the location
of the own Ego, in the sense that often he puts herself or himself as main
subject rather than the author under examination. This decentralization of the
Ego is a primary epistemological process whose importance has been highlighted
by Sigmund Freud himself: indeed, following (Vegetti Finzi 1986, Chap. I, §
2), the scientific knowledge has reached its highest levels in concomitance to
real narcissistic wounds, as those occurred with the Copernican revolution,
the Darwinism and the Freudian psychoanalysis, each of these having just
reappraised the human Ego, self-limiting this. Such a self-limitation of the
Ego therefore corresponds to a general criterion of further improvement and
completion of knowledge, as highlighted too by Max Planck himself in (Planck
1964) (see also (Straneo 1947, Introduction)).

This Ego decentralization also plays a non-trivial role in historiography as
regards the position of the historian compared to the object under attention.
This fact, for instance, has been emblematically recalled by one of the most
important Italian mathematicians of the last century, Bruno Pini444For some
brief biobibliographical notes on the life and works of Bruno Pini
(1918-2007), see (Cavallucci and Lanconelli 2011) and (Lanconelli 2012)., who
had to say that <<sometimes, when one is called to commemorate someone, it
goes end up to overly speak of herself or himself>> (see (Lanconelli 2008));
this simple consideration may be extended to the general biographical
studies555The opposite case to this is that related to the deifications, like
those present in many hagiographies.. This is simply due to the human
weakness, scientist or not who he or she be, even turned toward the own
egocentric accomplishments (from which it follows, for some respects, the
well-known Latin maxim <<tot capita, tot sententiæ>>). Therefore, it turns out
clear what methodological importance has the examination of the original
scientific production of every author under examination, as we just hope to do
in any case herein analyzed, to avoid any possible mystification.

Finally, since, according to Chen Ning Yang (see (Yang 1961, Preface)), <<a
concept, especially a scientific one, have not full meaning if it is not
defined respect to that knowledge context from which it derived and has
developed>>, each examined original work or paper of the author under
consideration shall be even contextually laid out into the related theoretical
framework of the time, so that, where possible, a brief historical recognition
should be mentioned as a contextual story meant as follows. In a certain
sense, we might say to follow an epistemological path analogous to that
outlined by Stephan Hartmann in (Hartmann 1999) where he claims on the
importance, above all in hadron physics, to consider a theoretical model as
the result of an interpreted formalism plus a story, this last being meant
both as a narrative but rigorous told around the formalism of the given model
and as a complement of it, hence an integral part of the model; the
relationships between formalism and story are then placeable out into the
wider class of relationships subsisting between the syntactic and semantic
parts of a general physical model which are unavoidable just in Physics.
Therefore, our work might be considered as a sort of making a story to certain
groups of works of the author under examination in order to get an overall
historical view of the subject matter in which her or him worked. For these
reasons, it is also indispensable refers us to the general technical-
scientific literature to support what said. Only doing so, it will be possible
to pursue the highest objectivity degree and historical correctness in
descrying the scientific figure of an author, trying to avoid the above
mentioned irrationality elements. At the same time, in dependence on the
scientific level of the treated author, with this method it shall be possible
to outline a history of the related work area.

Another confirmation of the validity of the above mentioned work program
follows from some epistemological considerations about the foundations of
science, due to the modern French school which goes from G. Bachelard, A.
Koiré and G. Canguilhem to the structuralists J. Lacan, C. Lévi-Strauss, L.
Althusser, M. Foucault, F. Regnault, A. Badiou and F. Wahl. Indeed, following
(Cressant 1971, Introduction), the scientific activity should be looked at as
a constructive process which pulls out the truth or the essence of the real
objects that will constitute the central core upon which building up the
corresponding knowledge object, trying to separate ideological questions from
the mere scientific contents. Read an arbitrary work just means try to
separate the general ideological and philosophical context from the scientific
one; it means to analyze the problematic frame within which this work has been
conceived, rebuilding up the prime structural causes from which it shall
develop. In doing so, a passive and sterile lecture will be replaced by an
active and productive re-enact (see (Wahl 1971)), almost analogously to what
foreseen both by the Robin Collingwood historicism (see (Kragh 1990) and
(Iurato 2013)) and by Wilhelm Dilthey methodological hermeneutics, according
to which any written source should be laid out into the proper original
historical context, according to the Zeitgeist of the time. Following (Schultz
1969, Chapter I) and (Wertheimer 1979, Chapter 1), the ideology is always an
unavoidable judgment component of human being, hence also of every historian,
since it is a common perspective to conceive the history as chiefly due to the
subjective idiosyncrasies and to the preconceptions which will play the role
of selective mental grid of what to consider or not and of how to interpret
this. Contrarily to what one could thought, the ideology is also an
unavoidable component of the normal scientific context: in this regards, see
(Boudon 1991, Chapter VIII).

## 1\. Historical introduction: I

Mario Gliozzi, in666The Enciclopedia delle Matematiche Elementari e
Complementi has been the most important and notable Italian encyclopedic
handbook on mathematical sciences and their applications, reviewed abroad as
one of the main encyclopedic work made in this context, as valuably remarked
by (Archibald 1950) and (Miller 1932). This article of Mario Gliozzi was the
first systematic attempt to outline a brief history of physics. It was later
retaken as a first core for drawing up another more extended article published
in the 1962 Nicola Abbagnano treatise on the history of science, in turn
posthumously enlarged and revised by the sons of Mario Gliozzi, in the new and
definitive 2005 edition (Gliozzi 2005), which is one of the most complete
textbook on the history of physics. Herein, we have mainly followed (Gliozzi
1949) because of its conciseness which is functional to the aims of this
section, referring to (Gliozzi 2005) for a more complete and in-depth view.
(Gliozzi 1949, § 30), outlines the main features of the experimental physics
through the last 19th Century decades to the 1940s. This was an almost unique
period for the history of physics since, from the new results of atomic
physics of the 19th Century end, appears, in all its complexity, the new
submicroscopic Weltbild to whose knowledge inextricably taken part
philosophical, theoretical and experimental physical questions above all
characterized by the crucial passage from the classical determinism to the
modern probabilism as recalled by (Pignedoli 1968, Chap. I) which gives a
clear and synthetic historical summary of this critical epistemological step.
Above all, the experimental physics had needed for new methods, techniques and
tools to approach and to examine this unexpected world so closed to our direct
perceptions, this, in turn, implying the formulation of new theories to
explain it at the light of these experimental results which arose from the
discovery of cathodic and anodic emissions, channel and X rays, and
radioactivity777The spontaneous radioactivity was discovered by H. Becquerel
in 1896 under advice of H.J. Poincaré. Indeed, the latter suggested to the
former to investigate on the possible relationships between optical
fluorescence and X rays, which revealed to be fake, but that led, for
serendipity, to the discovery of radioactivity (see (Segrè 1999, Chap. 1)).
(see (Born 1976, Chap. 2)). In this regards, in 1897, Charles T.R. Wilson
discovered that the ions produced in air by ultraviolet and X rays as well as
by radioactive radiations, acted as condensation nuclei of water steam
suitably supersaturated by rapid adiabatic expansion. This notable discovery
was at the basis of the so-called cloud chamber, one of the first valuable
displaying particle detector, first set up at the Cambridge Cavendish
Laboratory in 1896 and subsequently improved by Wilson (see (Wilson and
Littauer 1965, Chap. 3) and (Yang 1969, Chap. 1)), so that it is often called
too Wilson chamber; it will play a fundamental role in experimental atomic
physics, even to be said ”an open window on the world” (E. Persico). The
particle detectors may be classified into two main categories, namely the
displaying detectors and the optical (or electronic) detectors; the first ones
comprise the Čerenkov and scintillation counters, the Wilson (or cloud),
bubble, spark and photographic emulsion chambers, whereas the second ones
include the ionization chamber, the Geiger-Müller, the proportional and solid-
state counters (see (Segrè 1999, Chap. 3), (Tolansky 1966, Chap. 17) and
(Chiavassa, Ramello and Vercellin 1991, Chap. 2)).

Following (Segrè 1999, Chap. 1), after the discovery of electron in 1897, the
first atomic models due to J.J. Thomson, E. Rutherford and N.H. Bohr at the
beginnings of 20th Century, together with the introduction of quanta by M.
Planck and A. Einstein as regards the electromagnetic radiation, led to the
formulation of quantum mechanics which succeeded to explain many atomic
phenomena. At the same time, after the discovery of atomic nucleus in 1911,
the new quantum theories gradually opened the way to nuclear physics with the
first $\alpha$ particle bombardment phenomena which led, after the 1919
pioneering Rutherford discovery of the proton, to the definitive 1924-25
experimental ascertainment of such a particle by P.M.S. Blackett, who was a
Rutherford’s pupil (see (Gliozzi 1949, § 30, footnote 239)) and (Gamow 1963,
Chap. VIII)). Thereafter, on the basis of the previous works made by R.J. Van
de Graaff888Nevertheless, the principle of the method upon which relies the
running of such machines is quite similar to one already studied by A. Righi
at the end of 19th Century (see (Gliozzi 1949, § 30, footnote 241)))., J.D.
Cockcroft, E.T.S. Walton, H. Greinacher and R. Wilderöe, in 1933 E.O. Lawrence
and S. Livingston built up the first particle accelerator, the so-called
cyclotron (see (Wilson and Littauer 1965) and (Segrè 1976, Chap. XI)), based
on the resonant acceleration method. Independently of each other, in 1944 the
Russian physicist V.I. Veksler proposed a new particle accelerator based on
the phase stability method, while in 1945 E.M. MacMillan proposed an analogous
particle accelerator which will be called synchrotron. See (Lee 2004) for a
complete and masterful updated knowledge on accelerator physics, in which
there are also interesting historical notes.

Retaking into account some previous experiences made by W. Bethe and H.
Becker, the spouses I. Curie and F. Joliot discovered a new particle, already
suggested by Rutherford in 1920 and whose exact nature was subsequently
experimentally ascertained by J. Chadwick who called it neutron (see (Hughes
1960)); the Curie’s experiences given rise to the first artificial
radioactivity phenomena. In the years 1932-1934, a new particle was observed,
almost at the same time, by many scientists: amongst them, by I. Curie and F.
Joliot in collision phenomena with $\alpha$ particles, by C.D. Anderson in the
United States and by P.M.S. Blackett with G. Occhialini in England, in
experiences concerning cosmic rays (see (Gliozzi 1949, § 30, footnote 243))),
which was called, by C.D. Anderson, positive electron, or positron. Such a
particle had already been theoretically provided by P.A.M. Dirac with his
elegant 1930s electron theory, which, inter alia, established too the so-
called charge conjugation invariance principle; this new particle was
experimentally determined having mass almost equal to the electron one but
with positive charge. The discovery of positron was a celebrated experimental
confirmation of Dirac’s electron theory, which was besides unknown to Anderson
but not to Blackett and Occhialini which made their above researches at the
Cavendish Laboratory of which Dirac was a member, at that time (see (Rossi
1964, Chap. VI)).

In the years 1933-1934, taking into account the previous works of the Curie-
Joliot spouses, E. Fermi was the first to use neutrons as collision particles,
in place of $\alpha$ particles: indeed, he rightly argued that neutrons were
more suitable to this, due to the lack of electrostatic repulsion respect to
an atomic nucleus; slow neutrons turned out to be very efficient in breaking
the atomic nucleus. Such ingenious intuitions were put in practice in Rome, by
E. Amaldi, O. D’Agostino, B. Pontecorvo, F. Rasetti and E. Segrè, where it was
carried out the celebrated experiences with slow neutrons (see (Gliozzi 1949,
§ 30, footnotes 245), 246)) which will lead to the discovery of nuclear
fission and to the subsequent chain reactions, all this at the World War II
eve (see (Gliozzi 1949, § 30, footnotes 247)-251))). It was the beginning of
the nuclear physics with the use and applications of the nuclear energy by E.
Fermi in 1942, in this, the Italian school having been leader in the
international research framework of the time. In this regards, from a
historical viewpoint, it is enough to give a glance to the fundamental works
(Wick 1945; 1946) to witness all this, which represented the first treatise on
the new neutron physics; this unique two-volume treatise is the most valuable
historical source which exposes the ”state of the art” of that time as regards
this new chapter of nuclear physics.

In the decade 1920s to 1930s, the building of quantum mechanics was achieved,
with the elegant and rigorous formulation given by P.A.M. Dirac in his
celebrated textbook (Dirac 1958), whose first edition date back to 1930 and
that is still the classical and definitive treatise on the subject with its
last 1958 fourth edition; in it, the chapters on the new quantum
electrodynamics were updated till the results of 1950s. Once discovered the
neutron, one of the main problem of the new nuclear physics was to determine
the interaction forces among the constituents of the atomic nucleus, that is
to say (d’après W. Heisenberg, D. Ivanenko and E. Majorana) protons and
neutrons, which have been interpreted as two states of the same particle,
called nucleons, having different values of a well-determined numerical
parameter called isospin. This last quantum number is related to the formal
description of the notion of isotopic (or isobaric) invariance, that was first
introduced by W. Heisenberg in 1932, then used by B. Cassen and E.U. Condon in
1936 and by E.P. Wigner in 1937 (see (Landau and Lifšits 1982, Chap. XVI, §
116)) and subsequently applied to the classification of other subnuclear
particles, as we will see later. The next twenty years will see the birth of
the so-called quantum field theory (QFT), before all with the new quantum
electrodynamics999For this fascinating story, see (Schweber 1983; 1994), (De
Alfaro 1993) and (Weinberg 1999, Vol. I, Chap. 1). In (Schweber 1983; 1994)
there is also an extensive history of quantum field theory of 20th Century,
both from an internal and external historical standpoint. (QED), which
develops, according to the Galileian scientific method, in close concomitance
with the related experimental physics contexts, above all those concerning the
radioactive emissions and the cosmic radiation, which will play a fundamental
role in developing the nuclear and subnuclear physics; within the theoretical
framework given by the incoming QFT, they will flow into the dawning of
particle physics. The quantum electrodynamics started with the works of W.
Heisenberg, W. Pauli and P.A.M. Dirac, culminating in the Dirac’s radiation
theory in which the photons (already experimentally determined by E. Mayer and
W. Gerlach in 1914 - see (Born 1976, Chap. 4, § 24)) are the quanta of the
electromagnetic field, this theory having been taken as main model for
building up any further quantum field theory, like the electronic-positronic
field, the nucleonic and the mesonic ones, and so on (see (Fermi 1963, Chap.
1, § 1)). In the decade from 1940s to 1950s, the electromagnetic field has
been successfully quantized starting from the Maxwell’s equations, while the
electronic-positronic field has been treated starting from the Dirac’s
electron theory with a new formal process introduced by E.P. Wigner and W.
Pauli in 1928, called second quantization, which is a modification of the
previous quantization procedures to account for supervened spin statistic
problems. The situation concerning the electronic-positronic and nucleonic
fields was instead much more complex (see also (Weinberg 1999, Chap. 1, §
1.2)).

For our historical ends, we are more interested towards those aspects of
particle physics history regarding both radioactive decays and cosmic rays,
which, as already said, have played a very fundamental role in the dawning of
particle physics and whose historical paths often have intertwined each other.
Indeed, following (Weinberg 1999, Chap. 1, § 1.2), despite significant
successes achieved by QFT (in primis, the Dirac’s ones), a certain
dissatisfaction held towards it for all the 1930s, above all due to its
apparent incapacity to explain many new phenomena coming from the cosmic
radiation as well as all the new type of particles contained in it. On the
other hand, as regards the various proposed theories explaining the
radioactive emissions, above all that regarding the $\beta$ decay to have
played a fundamental role both in understanding the nuclear structure and in
developing QFT. Following (Persico 1959, Chaps. XI, XII and XIII), (Segrè
1999, Chap. 8), (Castagnoli 1975, Chap. 3, § 3.7), (Tolansky 1966, Chap. XV),
(Born 1976, Chap. 7, § 53) and (Friedlander and Kennedy 1965, Chaps. 6 and 7),
the theory of $\alpha$ emission was successfully achieved by G. Gamow, R.
Gurney and E. Condon in 1928-29, as the first attempt to apply the new quantum
theories to nuclear structure, while the development of the theory of $\gamma$
emission was parallel to that of quantum theory of radiation which started
with an interpretative theory analogous to the first atomic radiative
emissions and continued, through 1920s to 1950s, with the works of E.
Rutherford, H. Robinsson, W.F. Rowlison, E.N. Da Costa Andrade, L. Meitner,
H.J. Von Baeyer, O. Hahn, C.F. Von Weizsäcker, H.A. Bethe, W. Heitler, O.
Klein, Y. Nishina, J.P. Thibaud, E. Feenberg, H. Primakoff, E.P. Wigner, M.
Goldhaber, J.M. Blatt, V.F. Weisskopf, E. Wilson, K.T. Bainbrige, A.W. Sunyar,
P.B. Moon, R.L. Mössbauer, and others (see (Heitler 1953)).

Instead, the $\beta$ emission, in both its $\beta^{-}$ and $\beta^{+}$
components, shown to have particular difficulties to be laid out into a
coherent theoretical description, above all in relation to the interpretation
of the related continuum electron velocity spectrum which was one of the most
serious theoretical nuclear physics problems of the time. The main theoretical
problem concerned an apparent non-validity of the energy and spin conservation
laws at every elementary emission act, that Bohr attempted to justify invoking
a sort of mean validity of it. Nevertheless, in analogy with the case of
$\gamma$ emission (which was experimentally excluded to be associated with a
$\beta$ emission), E. Fermi proposed an alternative and more valid
quantitative interpretation based on the possible contemporaneous emission of
a new particle, called neutrino, together the electron involved in each
elementary $\beta$ emission act. The neutrino was a particle first
theoretically predicted by H. Weyl in 1929 (see (Weyl 1931) and references
therein) on the basis of Dirac’s electron theory, but his hypothesis was at
once refused since it did not verify the parity symmetry. It however will be
reconsidered later in 1957 after the work of T.D. Lee and C.N. Yang on parity
violation. Thereafter, in 1930, W. Pauli proposed to consider such a new
particle to explain the lack of validity of the above mentioned conservation
laws as regards $\beta$ decay, which was later so named by E. Fermi in 1934.
It was supposed to have zero mass and spin one-half. The quantitative theory
of $\beta$ emission, first proposed by E. Fermi in 1934 and later improved by
E.J. Konopinski, H.J. Lipkin, G. Uhlenbeck and H. Yukawa, is a very general
one which may be also applied to other type of interactions. Taking into
account previous theoretical studies, as already said mainly due to W.
Heisenberg, E.U. Condon and E.P. Wigner, this theory assumes proton and
neutron as two distinct quantum states of a unique particle, the quantum of
the nucleonic field, called nucleon, which can go either into one, or into the
other, of these two states just through a $\beta$ decay, emitting one
positive/negative electron and one neutrino101010In this historical account,
it is no possible to consider the problem of helicity of the neutrino as well
as the related questions inherent the existence or not of the antineutrino,
the alternative theory of E. Majorana, and so on. See (Fermi 1963, Chap. 1)
and (Yang 1969, Chap. 1).. To be precise, we have a $\beta^{-}$ emission in
the transformation of a neutron ($n$) into a proton ($p$) according to a decay
process of the type $n\rightarrow p+e^{-}+\nu$, with the emission of one
(negative) electron ($e^{-}$) and one neutrino ($\nu$); we have a $\beta^{+}$
emission in the transformation of a proton into a neutron according to a decay
process of the type $p\rightarrow n+e^{+}+\nu$, with the emission of one
positive electron ($e^{+}$) and of one neutrino. Nevertheless, as it will be
proved later by L.M. Lederman and co-workers as well as by other workers,
there exists another type of neutrino different from the one produced by the
above proton and neutron decays (denoted by $\nu_{e}$), namely the neutrino
produced by $\mu$ and $\pi$ meson decays (denoted by $\nu_{\mu}$). On this
last point, we shall return later.

For a certain time, it was supposed that the nuclear forces could be explained
through a nucleonic field associated to the pair electron-neutrino (see
(Polara 1949, Chap. IV, § 7), (Segrè 1976, Chap. X) and (Weinberg 1999, Chap.
1, § 1.2)) and whose quantum was the neutrino. It was hypothesized that proton
and neutron were linked together by means of an exchange of one neutrino, like
in case of the ionized molecule $H_{2}^{+}$ where the force between the atom
$H$ and the ion $H^{+}$ became attractive at distances of the magnitude of
$10^{-15}$ just thanks to a periodic exchange of the unique available
electron111111Indeed, the notion of exchange force comes from the quantum
theory of chemical bond (see (Slater 1980)). It will be later extended first
to nuclear physics thanks to the work of E.P. Wigner (see (Eisenbud and Wigner
1960, Chaps. 5 and 11)), then to the particle physics context.. Nevertheless,
in this way it wasn’t possible to account for nucleus stability questions, so
that this hypothesis (which had also been considered by E. Fermi in his 1934
theory) had to be rejected. All that, however, will be one of the starting
points of the subsequent pioneering 1935 Yukawa’s work (see (Yukawa 1935)).
Following (Castelfranchi 1959, Chaps. XX and XXI, §§ 231, 262), the positive
electrons $e^{+}$ are emitted only in artificial radioactive decays and were,
almost at the same time, discovered in 1932-33, by many researches,
independently of each other, amongst whom C.D. Anderson, R.A. Millikan and
P.M.S. Blackett with G. Occhialini in experiences with cosmic rays, as well as
by I. Curie with F. Joliot, by L. Meitner, C.Y. Chao, H.H. Hupfeld, J.R.
Richardson and J. Chadwick in experiments on induced radioactivity. It was
later observed, above all by Blackett and Occhialini, that these positive
electrons just were the antielectrons expected by the Dirac’s electron theory.
At the same time in which it was carried out the above experiences on $\beta$
decay, a prominent role began to have the study of a new type of very high
energy radiation, most highly penetrating, that is to say, the cosmic
radiation. Between the 1940s and 1950s, the unique available high energy
sources were the cosmic rays, until the coming of particle accelerators in the
1950s which allowed more controllable energetic sources. In that period, there
was a research competition between experiences made on cosmic rays and those
through particle accelerators.

Following (Polara 1949, Chap. V), (Castelfranchi 1959, Chap. XXIII), (Rossi
1964), (Tolanski 1966, Chap. 18), (Born 1976, Chap. 2, § 15), (Brown and
Hoddeson 1983), (Schlaepfer 2003) and (Carlson and De Angelis 2010), the
cosmic radiation was discovered, in the early years of 20th Century, by J.
Elster with H. Geitel in German and by C.T.R. Wilson in England, from the
observation of a weak residual electrification in perfectly isolated
electroscopes. Some year later, E. Rutherford with H. Lester and H.L. Cooke,
together to J.C. McLennan and E.F. Burton, showed that 5 cm of lead reduced
this mysterious radiation by 30% while an additional 5 tonnes of unrefined
lead failed to reduce the radiation further. Such a phenomenon was immediately
attributed to a not well identified external strong penetrating radiation,
maybe coming from the Earth. Thanks to a new electroscope made by T. Wulf in
1907, it was possible to observe that this external radiation did not decrease
with the altitude but, in some cases, even increased, so that it could not
come from the Earth as it was later confirmed first by A. Gockel in 1909-10,
then both by V. Hess with W. Kolhörster in 1911-14 and by D. Pacini in 1911,
the former with a series of experiments made with flight balloons equipped
with electroscopes, and the latter121212Very few are the textbooks which quote
the Italian physicist Domenico Pacini (1878-1934) as one of the pioneers of
cosmic-ray research; amongst them (Castelfranchi 1959, Chap. XXIII) - where,
inter alia, it is also possible to find interesting historical notes
throughout the text - and (Gliozzi 2005, Chap. 16, Section 16.12). For this
historical case, and for a modern general historical revisitation of the
cosmic ray story, see (Carlson and De Angelis 2010). by means of deep sea
immersions of electroscopes. The World War I interrupted the researches on
this strange type of penetrating radiation, then retaken later in 1920s and
1930s with the experiences of A. Millikan, E. Regener, G. Pfotzer, I.S. Bowen,
H.V. Neher, H. Tizard, A. Piccard, M. Cosyns, W. Bothe, J. Clay, A. Corlin, D.
Hoffmann, D.V. Skobeltzyn, E. Steinke, G.H. Cameron, P.S. Gill, G.L. Locher,
E.J. Williams, C.F. Von Weizsäcker, L. Nordheim, J.B. Street, H. Kulenkampff,
E.C. Stevenson and others, continued until the 1940s and 1950s pioneering
experimental works of T.H. Johnson, L.W. Alvarez, A.H. Compton, C.D. Anderson,
D.A. Glaser, S. Neddermeyer, G.E. Roberts, R.E. Marshak, H.A. Bethe, B. Rossi,
F. Rasetti, G. Bernardini, S. De Benedetti, C. Størmer, G. Lemaître, M.S.
Vallarta, V. Bush, G. Clark, P. Bassi, M. Schein, H.L. Bradt, B. Peters,
P.M.S. Blackett, G. Occhialini, C.F. Powell, C.M.G. Lattes, M. Conversi, E.
Pancini, O. Piccioni, H. Muirhead, J.F. Carlson, J.R. Oppenheimer, P. Auger,
P. Ehrenfest, L. Leprince-Ringuet, S.I. Tomonaga, G. Araki, G.D. Rochester,
C.C. Butler and others (see (Brown and Hoddeson 1983, Part III) and (Rossi
1964)). As regards the related experimental techniques employed, the first
group of researches were conducted by means of ionization chambers and, above
all, Wilson chambers, these last first used by D. Skobeltzyn in 1929, in the
version improved by P.M.S. Blackett and under the action of strong magnetic
fields. In the second group of researches, instead, besides Wilson chambers,
sequential Geiger-Müller counters were also used as well as photographic
emulsion chambers in the version improved by C.F. Powell and G. Occhialini on
the basis of the previous 1937 works made by M. Blau and H. Wambacher on
nuclear emulsions.

From the experimental data provided by all these notable works, in particular
from the various absorption curves related to this cosmic radiation and
related geomagnetic effects, it was possible to identify two secondary
components, departing from a primary one, which have different nature
according to the results of the azimuthal and latitude effects, both then
characterized, but in a different manner, by the so-called east-west asymmetry
phenomenon which is closely connected with an asymmetry related to cosmic ray
intensity distributions, in turn related to the geometry of the so-called
Størmer cones which give the allowed trajectories of cosmic rays under the
action of the geomagnetic field. For this, it was identified both a hard
component, much penetrating, and a soft component, little penetrating. The
terrestrial magnetic field and the atmosphere, constitute two protective
layers against the reaching of cosmic rays on the Earth’s surface: once that
high energy primary cosmic rays hit terrestrial magnetosphere (involving too
the Van Allen belts - see (Rossi 1964, Chap. XIII)) and the upper atmosphere,
they interact with the encountered atoms (above all, the nitrogen and oxygen
ones), the resulting collisions producing fragment’s stars (that is to say,
multiple traces outgoing from a same collision point) and atmospheric showers
of many particles, at that time most unknown, according to multiple production
processes theorized by W. Heisenberg and G. Wataghin. Theoretical attempts to
explain the related phenomenology led to the so-called cascade theory of
cosmic showers which was worked out, in 1930s and 1940s, mainly by J.F.
Carlson with J.R. Oppenheimer in the United States and by M. Blau, H.
Wambacher, L. Jánossy, H.A. Bethe, W. Heitler, J. Hamilton, H. Peng and H.J.
Bhabha in Europe, but also with notable contributions by L.D. Landau, I.E.
Tamm, V.L. Ginzburg, S.Z. Belenky, H.S. Snyder, R. Serber, W.H. Furry and S.K.
Chakrabarty. Thanks to this theory, it was possible to ascertain that the soft
component of cosmic radiation was mainly made by high energy electrons and
photons, whereas the determination of the particle composition of the hard
component was more difficult to achieve; and, at this point, the cosmic
radiation and $\beta$ decay research pathways meet.

Following (Polara 1949, Chaps. V and VI), (Rossi 1964), (Muirhead 1965, Chap.
1), (Yang 1969, Chap. 2), (Segrè 1976, Chap. XII), (Born 1976, Chap. 2),
(Zichichi 1981), (Brown et al., 1989), (Segrè 1999, Part III), (Zichichi 2000,
II.1-3a-II.1-4b) and (Gliozzi 2005, Chap. 16, Section 16.12), whilst the
(local) cosmic radiation soft component was ascertained to be mainly made by
high energy electrons and photons, many perplexities yet held as regards the
hard component whose constituents seemed do not belong to the set of
elementary particles then known. Indeed, the latter appeared to possess either
positive or negative electric charge, while the analysis of experimental data
strongly indicated the existence of a particle having mass intermediate
between that of the proton and of the electron, and probably in the region of
100-200 electron masses ($m_{e}$). This suspicion was verified by S.H.
Neddermeyer with C.D. Anderson and by E.C. Stevenson with J.C. Street, in the
years between 1936 and 1938, who photographed quite instable particles with
masses estimated (above all by R.B. Brode and co-workers) to be about 200-240
$m_{e}$, stopping in a cloud chamber. These particles were generically called
mesotrons by C.D. Anderson or mesons by W. Heisenberg, because of their mass
value; it ended then to prevail the second name131313Following (Gamow 1966,
Chap. VIII), the name mesotron was discarded because, under advise of
Heisenberg father, a professor of classical language, the right etymology of
the term was inclined towards the term meson.. As said above, notwithstanding
many difficulties subsisted in the research field devoted to cosmic radiation,
the tenacity of researchers led to the conclusion that this type of radiation
(namely, the hard one) was formed by new particles both positively and
negatively charged with mass intermediate between the electron and proton
ones. Nevertheless, these authors did not know the 1935 Yukawa work and what
there was predicted141414The discovery of the meson, as well as that of the
positron, has been preceded by theoretical forecasts respectively due to H.
Yukawa in the first case, and to P.A.M. Dirac in the second one., mainly due
to the fact that it was published in a journal not widely known outside Japan
(see (Kragh 2002, Chap. 13)). In such a paper, starting from the Heisenberg
work on nuclear structure and from the Fermi theory on $\beta$ decay, Yukawa
supposed that proton and neutron could interact through a quantized field (the
mesonic one) of which, in analogy with the electromagnetic case, he computed
too the main physical properties of the related quantum.

Yukawa made a further suggestion about the properties of his hypothetical
particle: indeed, in order to simultaneously account for nuclear $\beta$ decay
and for the fact that the meson had not been observed (at that time), he
suggested that it decayed spontaneously into one electron and one neutrino in
a time which was estimated to be about $10^{-7}$ sec. In 1938, with some first
experiences made by H. Kuhlenkampff, the question related to meson decay was
one of the most debated of the period between the 1930s and the 1940s, which
had, as main protagonists, W. Heisenberg, P.M.S. Blackett, H. Euler, A.H.
Compton, B. Rossi, D.B. Hall, N. Hilberry, J.B. Hoag, W.M. Nielsen, H.V.
Neher, M.A. Pomerantz, G. Bernardini, G. Cocconi, O. Piccioni, M. Conversi and
others. An apparent verification of this property was obtained by E.J.
Williams and G.E. Roberts in 1940, observing a $\beta$ decay of a particle of
mass about 250 $m_{e}$ into a cloud chamber. In this period, attempts to
identify the generic meson observed in the cosmic radiation by C.D. Anderson
and co-workers, with the Yukawa’s particle were done, notwithstanding that
will reveal out to be false151515The occurred mistaken particle’s
identifications will be mainly due to the experimental difficulties to
identify the related spin values which are the only ones that allow to discern
between particles having equal mass and charge values (see (Villi et al. 1971,
Introduction)).. Furthermore, an apparent experimental evidence for such an
identification was provided by the measurements of the meson lifetimes by F.
Rasetti in 1941 and by many others, amongst whom B. Rossi, N.G. Nereson, K.I.
Greiser, R. Chaminade, A. Freon and R. Maze. At the same time, the comparison
of the Yukawa nucleonic theory with the cosmic radiation one (mainly due to
J.R. Oppenheimer and J.F. Carlson) in the light of the obtained experimental
data, above all those made by M. Conversi, E. Pancini and O. Piccioni in Italy
in the years 1943-1945 and by R. Chaminade, A. Freon and R. Maze in France in
1945, led R.E. Marshak and H.A. Bethe to suggest in 1947 the possible
existence of two different types of mesons, also on the wake of what
previously envisaged by E. Fermi, E. Teller and V.F. Weisskopf. Nevertheless,
due to the World War II circumstances, the Japanese physicists worked in an
almost full isolation and most of their researches of that time were
recognized only later. Indeed, S.I. Tomonaga, Y. Tanikawa, S. Sakata and T.
Inoue, already in 1943 had proposed the hypothesis of the possible existence
of two different types of meson. The main conclusion of the above mentioned
experiences was that the negative and positive mesons differently interacted
with matter: in fact, measuring the related capture rates $\lambda_{c}$, the
positive ones decayed as they were more or less free, whereas the negative
ones were attracted by the nuclei, reacting in a strong manner with heavy
nuclei and in a weak manner with the light ones, and this wasn’t what
predicted by S.I. Tomonaga and G. Araki in 1940 on the basis of Yukawa’s
theory.

The clarification of such a question came from the technological developments
which have even been historically connected with the scientific progress of
ideas. Indeed, starting from the previous experimental techniques due to S.
Kinoshita and C. Waller, it was set up new nuclear emulsion detectors in 1940s
by the Bristol group made by C.F. Powell, G. Occhialini, C.M.G. Lattes and H.
Muirhead, thanks to which it was possible to effectively identify two types of
mesons. This conclusion was further confirmed, in 1948, by other experiments
run both by the above Bristol group with also Y. Goldschmidt-Clermont, D.T.
King and D.M. Ritson, and, at Berkeley, by E. Gardner and C.M.G. Lattes with a
particle accelerator. These two types of mesons, detected by the above
fundamental experiences, led to identify two first classes of mesons: in one,
it was included those mesons at first called primary mesons, then meson $\mu$
or muon; in the other, it was included lighter mesons at first called
secondary mesons, then meson $\pi$, or pion. In the former falls the meson
foreseen by C.D. Anderson with S.D. Neddermeyer and detected by one of the
celebrated Conversi-Pancini-Piccioni experiences, while in the latter it
should fall the Yukawa’s one. Thus, the $\pi$ meson provided the glue for
nuclear forces and undergone to the following main decay-chain-reaction
$\pi\rightarrow\mu\rightarrow e$ where the first decay scheme
$\pi\rightarrow\mu+\nu_{\mu}$ was first studied, in 1948, by U. Camerini, H.
Muirhead, C.F. Powell and D.M. Ritson as well as by J.R. Richardson. Then, the
various experiences performed on negative and positive counterparts of cosmic
rays led to the conclusion according to which both these two types of meson
may be either positively or negatively charged, denoted by $\mu^{\pm}$ and
$\pi^{\pm}$, the Yukawa’s one being of the type $\pi^{-}$. The $\pi$ meson
significatively and strongly interacts with atomic nuclei, contrarily to the
$\mu$ meson which is mainly subjected to weak interactions; the former has
mass about 270-300 $m_{e}$ while the latter has mass about 200-210 $m_{e}$.
The $\pi$ mesons decay in $\mu$ mesons and these, in turn, decay in electrons,
by means of reactions of the type

$\pi^{+}\rightarrow\mu^{+}+\nu_{\mu},\ \pi^{-}\rightarrow
e^{-}+\bar{\nu}_{\mu},\ \mu^{+}\rightarrow e^{+}+\nu_{\mu}+\bar{\nu}_{\mu},\
\mu^{-}\rightarrow e^{-}+\nu_{\mu}+\bar{\nu}_{\mu}$

where $\nu_{\mu}$ denotes the neutrino and $\bar{\nu}_{\mu}$ the related
antineutrino. As said above, the neutrino $\nu_{e}$ was determined to be
different from the neutrino $\nu_{\mu}$ by experiences made by G. Danby, J.M.
Gaillard, K. Goulianos, L.M. Lederman, N. Minstry, M. Schwartz and J.
Steinberger in 1962 at Brookhaven. Nevertheless, the possible existence of two
different types of neutrino was first theoretically proposed by G. Puppi in
1948 in studying the universality of Fermi weak interactions (Puppi triangle)
on the basis of decay processes involving $\mu$ and $\pi$ mesons: indeed,
following (Hughes and Wu 1977, Vol. I, Chapter I), the most important
contribution resulting from the study of the muon so far, was probably the
revelation of the close relationships between muon decay, muon capture and
nuclear beta decay, just known as the three sides of the Puppi triangle.
Moreover, further experiences made by F. Reines and C.L. Cowan Jr. in 1959, by
M.G. Inghram and J.H. Reynolds in 1950, by C.S. Wu in 1960 and by G.
Bernardini161616See (Zichichi 2008) for brief recalls on the work of Gilberto
Bernardini. and co-workers in 1964, showed that
$\nu_{e}\neq\bar{\nu}_{e},\nu_{\mu}\neq\bar{\nu}_{\mu},\nu_{e}\neq\nu_{\mu}$.

Afterwards, to explain the experimental evidence for the charge independence
of nuclear forces as well as to account for the soft component in cosmic
radiation, independently of each other, N. Kemmer in 1938, H. Tamaki in 1942
and H.W. Lewis, J.R. Oppenheimer with S.A. Wouthuysen in 1947, pointed out
that in addition to Yukawa’s charged meson, a neutral meson had to exist,
whose experimental evidence was obtained in the years 1950-1951 by A.G.
Carlson, J.E. Hooper and D.T. King, by R. Bjorkland, W.E. Crandall, B.J. Moyer
and H.F. York and by W.K.H. Panofsky, R.L. Aamodt, H.F. York and J. Hadley.
Such a meson, denoted by $\pi^{0}$, decays according to a law of the type
$\pi^{0}\rightarrow\gamma+\gamma$, being $\gamma$ a photon. The $\mu$ and
$\pi$ mesons will be generically called too L mesons. At this point, it was
generally felt that the neutral pion discovery marked the end of particle
searches, whereas the decay $\pi^{0}\rightarrow\gamma+\gamma$ marked instead
the opening of new horizons in subnuclear and theoretical physics: for
instance, on the basis of this decay process, J. Schwinger formulated the so-
called partial conservation of the axial current (PCAC) hypothesis in quantum
electrodynamics and quantum chromodynamics, opening new fruitful chapters in
current algebra theory. Indeed, studying in-depth the nuclear interactions of
the particles of cosmic rays, it was possible to discover other elementary
particles. As said above, the $\mu$ mesons weakly interact with matter whereas
the $\pi$ mesons are nucleary active particles together other ones which were
discovered at high altitudes in many mountain laboratories located in
different world areas, amongst which those at Aiguille du Midi in the French
Pyrenees (Chamonix), at Testa Grigia and Plateau Rosa in the Italian Alps, at
Chacaltaya in the Bolivian Andes, at Mount Evans in Colorado, at Jungfraujoch
in the Bernese Alps, and so on. Thanks to nuclear emulsion techniques set up
by the Bristol group headed by C.F. Powell and by further experiences made by
R.H. Brown, U. Camerini, P.H. Fowler, H. Muirhead, C.F. Powell and D.M.
Ritson, in the years 1947-1950 new particles having mass intermediate between
the $\pi$ meson mass and the proton one, were detected. They observed the
decay of a charged particle into three charged mesons, one of these appearing
to be a $\pi$-particle. The parent particle was called a $\tau$ meson and its
mass was estimated to be about 1000 $m_{e}$, the first heavy meson. So, these
researchers identified two classes of new heavy unstable particles, that of
heavy mesons (lighter than the protons and heavier than the $\pi$ mesons) and
that of hyperons (heavier than the protons), which may be electrically charged
or neutral and never isolated. The heavy mesons were also generically called K
mesons or kaons, while the hyperons were also generically called Y mesons, so
that a new hierarchy of mesons had to present. It was also customary to
indicate the nature and number of the decay products by subscripts; thus, for
example, the $\tau$ meson was also called a $K_{\pi 3}$ due to one of its
decay schemes $\tau^{+}\rightarrow\pi^{+}+\pi^{+}+\pi^{-}$. In the 1950s,
besides the usual experimental research on cosmic radiation, with the advent
of particle accelerators new experiences begun too, in such a manner that new
particles were discovered and considerable further researches were
accomplished to classify these new particles according to masses, lifetimes
and decay schemes. All this represented one of the most important period of
the physics of 1950s.

The first heavy mesons and hyperons were observed by L. Jánossy in the years
1943-1946 at Dublin and by G.D. Rochester and C.C. Butler in 1947 at
Manchester. At first, such new particles were variously called k particles or
$V$ particles, due to the V-shaped tracks leaved by the non-neutral decay
particles observed into cloud chambers. Amongst these, there were those
particles which will be later called $K^{0},\Lambda^{0}$, $\tau$ and $\theta$
particles, these last two could be either neutral or electrically charged. As
recalled above, the Bristol group, headed by C.F. Powell, detected in 1949 the
first positively charged heavy meson, at first called $\tau^{+}$ meson, then
$K^{+}$ meson, that undergo different decay processes, amongst which
$K^{+}\rightarrow\pi^{+}+\pi^{0}$, whose experimental evidences were obtained,
in the years 1951-1954, by C. O’Ceallaigh, by the Paris group of B.P. Gregory,
A. Laggarigue, L. Leprince-Ringuet, F. Muller and Ch. Peyrou, by J. Crussard,
M.F. Kaplon, J. Klarmann and J.H. Noon and by A.L. Hodson, J. Ballam, W.H.
Arnold, D.R. Harris, R.R. Rau, G.T. Reynolds and S.B. Treiman. Further
researches made in cloud chambers with magnetic fields will show that both
$K^{+}$ and $K^{-}$ mesons exist. Then, the neutral heavy meson, at first
called $\theta^{0}$ particle, then $K^{0}$ particle, was first observed by C.
O’Ceallaigh in 1950. At the same time, R. Armenteros, K.H. Barker, C.C. Butler
and A. Cachon as well as L.M. Lederman K. Lande, E.T. Booth, J. Impeduglia and
W. Chinowsky in 1956, were able to show that at least two types of neutral
particles existed, one is the $\Lambda^{0}$ hyperon decaying according to the
scheme $\Lambda^{0}\rightarrow p+\pi^{-}$, and the other probably decayed as
follows $\theta^{0}\rightarrow\pi^{+}+\pi^{-}$. At that time, the Bristol
research group discovered two heavy mesons, called $\tau$ and $\theta$ mesons,
which initially seemed to be the same particle because they had same mass and
mean lifetime, but underwent distinct decay processes and had different
parity, so that, in the years 1953-1956, d’après R.H. Dalitz, it spoke of a
$\theta-\tau$ puzzle. The analysis of further experimental data led T.D. Lee
and C.N. Yang to assume in 1956 a parity violation of weak interactions,
thanks to which it was possible to establish that $\tau$ and $\theta$ mesons
are the same particle, thereafter called K particle. The discovery of the
breaking of the symmetry operators parity (P) and charge (C) received first
experimental evidence by C.S. Wu, E. Ambler, R.W. Hayward and D.D. Hoppes in
1957. Subsequent 1957 works made by R. Garwin, L. Lederman and M. Weinrich and
by J.J. Friedman and V.L. Telegdi, showed further evidence for a non-
conservation of parity and charge in the decay of kaons and hyperons,
attaining a deeper theoretical knowledge on the C, P and T invariance
properties. Subsequently, many decay modes for kaons were found even if, at
first, it was not realised that they represented alternative decay modes of
the same particle. Then, other types of hyperons were also found in cosmic
radiation: amongst these, C.M. York, R.B. Leighton and E.K. Bjornerund, in
1952, were able to experimentally ascertain a new type of hyperon, called
$\Sigma^{+}$ particle, which decays according to a reaction of the type
$\Sigma^{+}\rightarrow n+\pi^{0}$. A further confirmation of the existence of
$\Sigma$-hyperons was obtained by A. Bonetti, R. Levi-Setti, M. Panetti and G.
Tomasini in 1953, who also identified the alternative decay mode
$\Sigma^{+}\rightarrow n+\pi^{+}$, while the negative counterpart $\Sigma^{-}$
was observed by W.B. Fowler, R.P. Shutt, A.M. Thorndike and W.L. Whittemore at
Brookhaven in 1954. Another hyperon of mass $\sim 2600m_{e}$, called $\Xi^{-}$
meson, was detected by E.W. Cowan in 1954, which decays according to the
scheme $\Xi^{-}\rightarrow\Lambda^{0}+\pi^{-}$. The neutral $\Xi^{0}$ and
$\Sigma^{0}$ hyperons were then experimentally revealed by L.W. Alvarez, P.
Eberhard, M.L. Good, W. Graziano, H.K. Ticho and S.G. Wojcicki in 1959, only
after having been theoretically predicted as follows. This was the situation
around middle of 1950s, where the emphasis shifted from work using cosmic
radiations to work on large accelerators. The attention was also focused on
the classification of the various particles so far discovered according to
their masses, lifetimes and decay schemes.

It was observed that the various kaons and hyperons discovered in the 1950s
(above all the $\Lambda^{0}$), had a strange behavior respect to their decay
and production processes, so that they were given the collective appellation
of strange particles. Namely, following (Yang 1961, Chapter III) and (Muirhead
1965, Chapter 1, Section 1.4), it was experimentally found that these strange
particles have production times of about $10^{-23}$ sec and decay times of
about $10^{-10}$ sec, so that the forces involved in their production
processes were stronger than those present in the decay ones; furthermore,
this strange fact did not occur when such particles were isolated, in which
only weak interactions taken place. To explain this incompatibility between
experimental data and theoretical framework, A. Pais proposed in 1952 the
hypothesis of associated production according to which the decay and
production processes are not inverses of each other, but rather they differ
for the presence of another (associated) particle, or rather, at least two
strange particles should be involved in the production process in order that a
strong interaction could occur, whilst a weak interaction occurs if only one
strange particle is presents, as in the decay process. The Pais hypothesis
received experimental confirmation with the 1953-55 works of W.B. Fowler, R.P.
Shutt, A.M. Thorndike and W.L. Whittemore. At this point, it is necessary to
reconsider the above mentioned notion of isospin and related conservation law.
Following (Muihread 1965, Chapter 1, Sections 1.3 and 1.4), the concept of
conservation of isotopic spin (isospin) is associated with the experimental
evidence for the principle of charge independence of nuclear forces according
to which, at identical energies, the forces between any of the pairs of
nucleons n-n, n-p and p-p, depend only on the total angular momentum and
parity of the pair and not upon their charge state. So, B. Cassen and E.U.
Condon, in 1936, showed that the principle of charge independence could be
elegantly expressed by the isospin concept. The isospin of a system is
formally similar to angular momentum but is linked to the charge states of the
system. If a group of nuclei or particles exist in n charge multiplets, then
the isospin number T for this group is given by $2T+1=n$. The charge state of
a particle or nucleus in the multiplet is related to the third (along the z
axis) component of an isospin operator via the relations
$Z={Q}/{e}=(T_{3}+\frac{1}{2}A)$ for nucleons and nuclei, and $Q/e=T_{3}$ for
pions, where $Z$ is the atomic number, $Q$ the total charge (hypercharge) and
$A$ the atomic weight of the system. For instance, if $\chi_{p}$ and
$\chi_{n}$ denote the isospin functions for the proton and the neutron
respectively, then $T_{3}$ has eigenvalues $1/2$ and $-1/2$ respectively, that
is to say $T_{3}\chi_{p}=(1/2)\chi_{p}$ and $T_{3}\chi_{n}=-(1/2)\chi_{n}$.
The isospin quantum numbers were assigned to the strange particles produced,
independently by M. Gell-Mann and by T. Nakano and K. Nishijima, in the years
between 1952 and 1956. In regards to a cluster of elementary particles, they
observed invariance properties when the charge center of the multiplet of
strange particles was displaced respect to the center of the multiplet of non-
strange particles. Furthermore, they considered a scheme assuming that the
conservation laws for $T$ and $T_{3}$ were conserved or broken in dependence
on the given interaction: to be precise, $T$ and $T_{3}$ were conserved in
strong interactions, $T$ was broken whilst $T_{3}$ was conserved for
electromagnetic interactions and, finally, $T$ and $T_{3}$ were broken in weak
interactions. The satisfactory nature of Gell-Mann, Nakano and Nishijima
scheme lays in the fact that it predicted the existence of two new elementary
particles which were later experimentally found.

Later, again following (Muirhead 1965, Chapter 1, Section 1.4), it was pointed
out by Gell-Mann and Nishijima, independently of each other in the years
1955-56, that a more elegant classification of the strongly interacting
particles than that based on isospin alone, could be made if a new parameter
$S$, called the strangeness number, was introduced and defined by the relation
$Q/e=T_{3}+(B+S)/2$ where $B$ is the baryon number; baryon is a generic name
for nucleons and hyperons. Such a relation shows that the associated
production phenomena and the isospin symmetry are related together. As it has
been said above, the success of the Gell-Mann, Nakano and Nishijima scheme
lies in the fact that it predicted, on the basis of isospin and strangeness
conservation laws, various charge multiplets; in particular, new particles
were predicted like the $\Sigma^{0}$, $\Xi^{0}$, $K^{+}$, $K^{-}$ and related
antiparticles. All these theoretical assumptions received experimental
confirmation by the works of Y. Nambu, K. Nishilima, Y. Yamaguchi and W.B.
Fowler in the years 1953-1955, as well as many of the above predicted
particles (besides the $\Xi^{0}$ and $\Sigma^{0}$ above remembered) were
detected by W.B. Fowler, R.P. Shutt, A.M. Thorndike, W.L. Whittemore and W.D.
Walker, in the years 1955-1959. However, the elegant semi-empirical
classification scheme of Gell-Mann, Nakano and Nishijima, envisaged the
existence of many other new particles which will be later discovered, whose
static and dynamic properties will be of fundamental importance for the
subsequent 1960s theoretical work of M. Gell-Mann and Y. Ne’eman (eihgtfold
way). In conclusion, following171717Such a leading textbook has been one of
main references herein followed. (Muirhead 1965, Chapter 1, Section 1.5), the
discoveries of new particles have occurred sometimes as a result of
theoretical insights and sometimes by accident, the most strange particle
falling into the latter category: from what has been said above, at this
stage, we have that the weak interactions are associated with electrons, muons
and neutrinos $(e,\mu,\nu)$, collectively called leptons (which are not
subject to strong interactions) and with certain decay processes for mesons
and hyperons; the nucleons and hyperons $(n,p,\Lambda,\Sigma,\Xi)$ are then
collectively called baryons. Following (Roman 1960, Introduction), the
particles recalled so far may be also classified into two classes according to
their spin values, identifying a first class in which fall particles with
integer spin $(\gamma,\pi,K)$, said to be bosons, and a second one in which
fall particles with half-integer spin $(\nu,e,\mu,p,n,\Lambda,\Sigma,\Xi)$
said to be fermions. The fermions again fall into two rather distinct groups,
namely leptons and baryons. Apart from the photons, the bosons fall too into
two groups: the lighter $\pi$-mesons (or pions) and the heavier $K$-mesons (or
kaons). Thus, our classification scheme is tentatively Photon-Leptons-Pions-
Kaons-Baryons, and represents one of the main achievement of the physics in
the decade 1950-1960.

## 2\. Some early works on cosmic rays

In this section, according to what has been said in the Preface, in
delineating some early works on cosmic rays we consider the first research
papers of A. Zichichi and co-workers, dating back to 1950s, which concerned
physics of $K$ mesons, with particular attention to the experimental context.
To be precise, as a research fellow at CERN of Geneva, he joined the heavy
meson decay research group which, in turn, belonged to the wider investigation
activity on cosmic rays, according to the research program policy of that
period of this worldwide renowned research institution. The first papers of
Zichichi on cosmic rays mainly concern with the various properties of strange
particles.

As said in the previous section 1, in that period there were still many
unsolved questions concerning the so-called strange or $V$ particles. In 1.,
it is just discussed, together the related experimental arrangements, certain
peculiar features showed by only two out of twenty events observed into a
cloud chamber located at Testa Grigia laboratories (3,500 m.a.s.l.) in which
about 150 $V^{0}$-particles related to decay of a single charged unstable
fragment (like an unstable hydrogen isotope, or an excited deuteron or triton)
have been found. From a deep phenomenology analysis of the experimental data
and taking into account the already existent related literature, it emerged
that, very likely, one of these two observed peculiar events, say the Event I,
could be due to a certain $\Lambda^{0}$-decay of an unstable hydrogen isotope
rather than a nuclear interaction (which perhaps has would been quite
unusual), whereas, instead, the second one, say the Event II, led to the
conclusion according to which, under certain further hypotheses, it could be a
$\tau$-decay, even supposing that it is not a nuclear interaction.

The second work 2. is a contribution to the experimental evidence for the
existence of the neutral $\tau^{0}$-meson which had been predicted but not
established. To be precise, it is reported and discussed the result of an
experiment made at Jungfraujoch Research Station and consisting of four tracks
originating at a point in the cloud chamber gas which may be interpreted
either as the radiative decay of a $\theta^{0}$-meson or as the decay of a
$\tau^{0}$-meson. Two out of these are tracks of positive and negative
electrons electrons, while the other two are tracks of fast particles
resembling a typical $V$-event which is most readily explained as the decay of
a neutral $\tau$-meson, rather than a $\theta^{0}$-meson, with the subsequent
decay of the secondary $\pi^{0}$-meson into an electron pair and a
$\gamma$-ray, which may be respectively written as
$\tau^{0}\rightarrow\pi^{+}+\pi^{-}+\pi^{0}$ and $\pi^{0}\rightarrow
e^{+}+e^{-}+\gamma$. After having discussed on the various possibilities, this
four-pronged event is deemed to be geometrically associated with a small
nuclear interaction with, in turn, can be interpreted, on the basis of the
experimental data, as a charge exchange reaction of the type
$K^{+}+n\rightarrow\tau^{0}+p$, albeit it is also not excluded a possible
double production according to the scheme
$\pi^{+}+n\rightarrow\tau^{0}+\Sigma^{+}$ in which, in turn, the
$\Sigma^{+}$-particle decays into one proton and one yet undetected
$\pi^{0}$-meson.

Taking into account the previous work 2. discussed above, the work 3. mainly
shows some results coming from certain emulsion experiments which provide the
first two remarkable examples of $K$-meson pairs of the type
$(K^{0},\bar{K}^{0})$ and $(K^{+},\bar{K}^{0})$, produced in elementary
neutron-proton interactions whose production reactions respectively are
$n+p\rightarrow K^{0}+\bar{K}^{0}+n+p$ and $n+p\rightarrow
K^{+}+\bar{K}^{0}+n+n$; these allowed to extend the knowledge on the
phenomenology of heavy mesons, to further confirmation of some of the various
hypotheses suggested by M. Gell-Mann and A. Pais in the years 1952-54 about
associated productions, in particular, the prediction according to which the
$K^{0}$-mesons should exist in two states as particle and antiparticle with
$S=+1$ and $S=-1$. To be precise, from a systematic study of the associated
production of heavy mesons and hyperons in a cosmic-ray cloud chamber, in 3.
examples of very simple nuclear interactions giving rise to pairs of
$K$-mesons have been found. The importance of these observations is that they
provide experimental evidence to support the theoretical prediction that
$K^{0}$-mesons should exist in two states with opposite strangeness $S=+1$ and
$S=-1$, that is to say, these events are evidence that $K^{0}$-mesons with
both positive and negative strangeness exist. In relation to the well-known
$\theta-\tau$ puzzle, the authors also argued on the possible identification
or not of the produced $K^{0}$-mesons with the $\theta^{0}$-particles, but the
experimental measurements weren’t of great usefulness for this.

The work 4. is a brief research note in which it is determined, on the basis
of previous works made by J.A. Newth and M.S. Bartlett, the mean lifetime
estimates of the two decaying particles $\Lambda_{0}$ and $\theta_{0}$,
isolated amongst 115 $V^{0}$-events observed in a multi-plate cloud chamber
triggered for penetrating showers, and respectively interested to the
following main decay reactions $\Lambda^{0}\rightarrow p+\pi^{-}+37$ MeV and
$\theta^{0}\rightarrow\pi^{+}+\pi^{-}+214$ MeV. Moreover, neglecting the
existence of other types of unstable neutral particles with a two-body decay,
it has been possible to classify the above 115 $V^{0}$-particles.

The paper 5. is a research report, presented at CERN Scientific Policy
Committee on 21 October 1957, after the CERN research activity on cosmic rays
taken the decision to stop the so-called Geneva experiment on $K$-meson
decays, whose related motivations were exposed in the subsequent CERN report
No. CERN/SPC/52 (B). In 5., the director-general of the Jungfraujoch Research
Station in detail proposed a new experimental apparatus specifically designed
to prosecute the research activity on high energy interactions with a mountain
experiment based on the nuclear interaction of protons at energies in the
neighborhood of 100 GeV, through a large magnet cloud chamber. The novel
feature of this apparatus was a magnetic spectrometer which measured the
momenta of the primary particles. One of the main aims of this experiment was
also that to fathom new directions on particle physics as, for instance, the
search for new unstable strange particles having very short lifetimes.

As it has been said in section 1, in 1950s gradually started to run the first
particle accelerators of synchrotron type which will be intended to replace
the cosmic-ray researches. But this conclusion did not yet apply to the study
of the production processes of strange particles. Now, the results described
in the work 6. come from an experiment designed to study the production of
strange particles in materials of low and high atomic weight, precisely carbon
and copper consecutively used, through the interaction of energetic
secondaries, sprung out by nuclear interactions in passing through the targets
(of carbon and copper), which produce 79 neutral $V$-particles. The division
of all the $V^{0}$-events into $\Lambda^{0}$\- and $\theta^{0}$\- decays was
made in order to determine their lifetime estimates. If one denotes with
$N(\Lambda^{0})$ and $N(\theta^{0})$ the numbers of $\Lambda^{0}$\- and
$\theta^{0}$-mesons so produced, then a significant difference between the
values of the ratio $N(\Lambda^{0}):N(\theta^{0})$ for their production in
carbon and copper has been found; this asymmetry’s fact occurred in the decay
of the $\Lambda^{0}$ particles respect to the short-lived $\theta^{0}$ ones,
was also explained by H. Blumenfeld, E.T. Booth, L.M. Ledermann and W.
Chinowsky, who conducted similar experiences in 1956 with carbon and lead,
reaching to almost equal results, through the associated production of pairs
of $K$-mesons through which it is possible to increase the number of
$\Lambda^{0}$ particles so slowly produced, with also non-conservation of
strangeness. Thus, the results achieved in 6. as well as by Blumenfeld and co-
workers, may be taken as further evidence for the great importance of the pair
production of $K$-mesons in cosmic-ray experiments.

The decay asymmetry detected in the previous work 6. will be deeper studied in
the next work 7. where many other properties of $\Lambda^{0}$\- and
$\theta^{0}$\- particles, like for example spin, mean lifetime, behavior with
respect to inversion operators and anisotropy effects on geometrical
distributions, have carried out on 107 $\Lambda^{0}$ and $\theta^{0}$
particles produced in iron plates of a multiple cloud chamber exposed to
cosmic radiation at an altitude of 3,500 m.a.s.l. Likewise, the work 8.
reports the first results of an experimental study of the nuclear interaction
of cosmic rays (mainly of the type proton-proton) with a magnet cloud chamber
based at an altitude of 3,500 m.a.s.l. and operating at energy of about 100
GeV, showing that such a type of nuclear interaction study is feasible.

## 3\. Historical introduction: II

In this section, we recall the main events and facts of that historical path
which goes from the introduction of the spin to the notion of anomalous
magnetic moment, with particular attention to the leptonic case. The
necessarily limited historical framework so outlined in this section, covers a
temporal period which roughly goes up from early 1920s to 1960s.

### 3.1 On Landé separation factors

Following (Muirhead 1965, Chapter 2) and (Tomonaga 1997), when a fundamental
interaction is taken into account then the experimental determination of the
basic particle data, like masses, lifetimes, spins and magnetic moments, is
necessarily required. The most accurately known properties of the particles
are those which can be associated with their magnetic moments. Magnetic
properties of elementary particles have been and yet are of paramount
importance both to theoretical and experimental high energy physics. One of
the main intrinsic properties of the elementary particles is the spin, which
can be inferred from the conservation laws for angular momentum. Following
(Landau 1982, Chapter VIII), in both classical and quantum mechanics, the laws
of conservation of angular momentum are a consequence of the isotropy of space
respect to a closed system, so that it depends on the transformation
properties under rotation of the coordinate system. Therefore, all quantum
systems, like atomic nuclei or composite systems of elementary particles,
besides the orbital angular momentum, show to have as well an intrinsic
angular momentum, called spin, which is unconnected with its motion in space
and to which it is also associated a magnetic moment whose strengths are not
quantized and may assume any value. The spin disappears in the classical limit
$\hbar\rightarrow 0$ so that it has no classical counterpart. The spin must be
meant as fully distinct from the angular momentum due to the motion of the
particle in space, that is to say, the orbital angular momentum. The particle
concerned may be either elementary or composite but behaving in some respect
as an elementary particle (e.g. an atomic nucleus). The spin of a particle
(measured, like the orbital angular momentum, in units of $\hbar$) will be
denoted by $\vec{s}$. Following (Rich and Wesley 1972), (Bertolotti 2005,
Chapter 8), (Miller et al. 2007) and (Roberts and Marciano 2010, Chapter 1),
the physical idea that an electron has an intrinsic angular momentum was first
put forward independently of each other by A.H. Compton in 1921 to explain
ferromagnetism181818Furthermore, Compton acknowledges A.L. Parson for having
first proposed the electron as a spinning ring of charge. Compton modified
this idea considering a much smaller distribution of charge mainly
concentrated near the center of the electron. The Compton’s paper is almost
unknown (see (Compton 1921)) albeit it is quoted by the 1926 Uhlenbeck and
Goudsmit paper. Following (Roberts and Marciano 2010, Chapter 3, Section
3.2.1), also R. Kronig proposed, in 1925, the spin as an internal angular
momentum responsible for the electron forth’s quantum number (see (Bertolotti
2005, Chapter 8). and by G. Uhlenbeck and S. Goudsmit in 1925 to explain
spectroscopic observations in relation to the anomalous Zeeman effect, while
spin was introduced into quantum mechanics by W. Pauli in 1927 as an
additional term to the Pauli equation which is obtained by the non-
relativistic representation of the Dirac equation to small velocities (see
(Jegerlehner 2008, Part I, Chapter 3, Section 3.2)) to account for the quantum
mechanical treatment of the spin-orbit coupling of the anomalous Zeeman effect
(see also (Haken & Wolf 2005, Chapter 14, Section 3)). An equation similar to
the Pauli’s one, was also introduced by C.G. Darwin in 1927 (see (Roberts and
Marciano 2010, Chapter 3, Section 3.2.1)).

Following (Jegerlehner 2008, Part I, Chapter 1), (Melnikov and Vainshtein
2006, Chapter 1) and (Shankar 1994, Chapter 14), leptons have interesting
static (classical) electromagnetic and weak properties like the magnetic and
electric dipole moments. Classically, dipole moments may arise either from
electrical charges or currents. In this regards, an important example which
may turns out to be useful to our purposes is the circulating current, due to
an orbiting particle with electric charge $Q$ and mass $m$, which exhibits the
following orbital magnetic dipole moment

(1)
$\vec{\mu}_{L}=\frac{Q}{2c}\vec{r}\wedge\vec{v}=\frac{Q}{2mc}\vec{L}=\Gamma\vec{L}$

where $\Gamma=Q/2mc$ is the classical gyromagnetic ratio191919Usually, the
gyromagnetic ratio is denoted by lower case $\gamma$, but here we prefer to
use the upper case $\Gamma$ to distinguish it by the well-known Lorentz factor
$\gamma=1/\sqrt{1-\beta^{2}}$ with $\beta=v^{2}/c^{2}$. and
$\vec{L}=m\vec{r}\wedge\vec{v}=\vec{r}\wedge\vec{p}$ is the orbital angular
momentum whose corresponding quantum observable is the operator
$-i\hbar\vec{r}\wedge\nabla=\hbar\vec{l}$, so that we have the following
orbital magnetic dipole moment operator (see (Jegerlehner 2008, Part I,
Chapter 3) and (Shankar 1994, Chapter 14))

(2) $\vec{\mu}_{l}=g_{l}\frac{Q\hbar}{2mc}\vec{l}$

where $g_{l}$ is a constant introduced by the usual quantization transcription
rules. For $Q=e$, the quantity $\mu_{0}=e\hbar/2mc$ is normally used as a unit
for the magnetic moments and is called the Bohr magneton. The electric charge
$Q$ is usually measured in units of $e$, so that $Q=-1$ for leptons and $Q=+1$
for antileptons; therefore, we also may rewrite (2) in the following form

(3) $\vec{\mu}_{l}=g_{l}\frac{Qe\hbar}{2mc}\vec{l}=g_{l}Q\mu_{0}\vec{l}.$

Both electric and magnetic properties have their origin in the electrical
charges and their currents, apart from the existence or not of magnetic
charges. Following (Jegerlehner 2008, Part I, Chapter 1) and (Muirhead 1965,
Chapter 9, Section 9.2(d)), whatever the origin of magnetic and electric
moments are, they contribute to the electromagnetic interaction Hamiltonian
(interaction energy) of the particle with magnetic and electric fields which,
in the non-relativistic limit, is given by

(4) $\mathcal{H}_{em}=-(\vec{\mu}_{m}\cdot\vec{B}+\vec{d}_{e}\cdot\vec{E})$

where $\vec{\mu}_{m}$ and $\vec{d}_{e}$ are respectively the magnetic and
electric dipole moments (see (Jegerlehner 2008, Part I, Chapter 1)).

If one replaces the orbital angular momentum $\vec{L}$ with the spin
$\vec{s}$, then we might search for an analogous (classical) magnetic dipole
moment, say $\vec{\mu}_{s}$, associated with it and, therefore, given by
$(Q/2mc)\vec{s}$. Nevertheless, following (Born 1969, Chapter 6, Section 38)
and (Muirhead 1965, Chapter 2, Section 2.5)), to fully account for the
anomalous Zeeman effect, we should consider this last expression multiplied by
a certain scalar factor, say $g_{s}$ (often simply denoted by $g$), so that

(5) $\vec{\mu}_{s}=g_{s}\frac{Q}{2mc}\vec{s}$

which is said to be the spin magnetic moment. Now, introducing, as a
corresponding quantum observable, the spin operator defined by
$\vec{S}=\hbar\vec{s}=\hbar\vec{\sigma}/2$, where $\vec{\sigma}$ is the Pauli
spin operator, it is possible to consider both the spin magnetic moment
operator and the electric dipole moment operator (see (Jegerlehner 2008, Part
I, Chapter 1)), respectively defined as follows

(6) $\vec{\mu}_{s}\doteq g_{s}Q\mu_{0}\frac{\vec{\sigma}}{2},\ \ \ \ \ \ \ \ \
\ \vec{d}_{e}\doteq\eta Q\mu_{0}\frac{\vec{\sigma}}{2},$

where $\eta$ is a constant, the electric counterpart of $g_{s}$. Following
(Caldirola et al. 1982, Chapter XI, Section 3), the attribution of a $s=1/2$
spin value to the electron, led to the formulation of the so-called vectorial
model of the atom. In such a model, amongst other things, the electron orbital
angular moment $\vec{L}$ composes with the spin $\vec{s}$ through well-defined
spin-orbit coupling rules (like the Russell-Saunders ones) to give the
(classical) total angular moment defined to be $\vec{j}\doteq\vec{L}+\vec{s}$,
while the (classical) total magnetic moment is defined to be
$\vec{\mu}_{total}\doteq\vec{\mu}_{L}+\vec{\mu}_{s}$, so that, taking into
account (3) and (6), the corresponding quantum observable counterpart, in this
vectorial model, is

(7)
$\vec{\mu}_{total}\doteq\vec{\mu}_{l}+\vec{\mu}_{s}=g_{l}Q\mu_{0}\vec{l}+g_{s}Q\mu_{0}\frac{\vec{\sigma}}{2}=Q\mu_{0}(g_{l}\vec{l}+g_{s}\vec{S})$

which is said to be the total magnetic moment of the given elementary particle
with charge $Q$ and mass $m$; since $g_{s}\neq 1$, it follows that it is not,
in general, parallel to the total angular moment operator
$\vec{J}\doteq\vec{l}+\vec{S}$, so that it undergoes to precession phenomena
when magnetic fields act.

The existence of the various above constants $g_{l},g_{s}$ and $\eta$ is
mainly due to the fact that, in the vectorial model of anomalous Zeeman
effect, the direction of total angular moment $\vec{j}$ does not coincide with
the direction of total magnetic moment, so that these scalar factors just take
into account the related non-zero angles which are called Landé separation
factors because first introduced by A. Landé (1888-1976) in the early 1920s
(see (Born 1969, Chapter 6, Section 38)). To be precise, only the parallel
component of $\vec{\mu}_{tot}$ to $\vec{j}$, say $\vec{\mu}_{tot}^{\|}$, is
efficacious, so that we should have

(8) $\vec{\mu}_{tot}^{\|}=g_{j}\frac{Q\hbar}{2mc}\vec{j}$

where the scalar factor $g_{j}$ (or simply $g$) takes into account the
difference between the vectorial model of anomalous Zeeman effect and the
theory of the normal one. To may computes this factor, we start from the
relation

(9)
${\mu}_{tot}^{\|}=\mu_{l}\cos(\widehat{\vec{l},\vec{j}})+\mu_{s}\cos(\widehat{\vec{s},\vec{j}})$

with

(10) $\mu_{l}=g_{l}\frac{Q\hbar}{2mc}l,\ \ \ \ \ \ \ \ \ \
\mu_{s}=g_{s}\frac{Q\hbar}{2mc}s$

where $g_{l}$ and $g_{s}$ are known to be respectively the orbital and spin
factors, which, in turn, represent the ratios respectively between the orbital
and spin magnetic and mechanic moments. Replacing (10) into (9), we have

(11)
$g_{j}=g_{l}\frac{l}{j}\cos(\widehat{\vec{l},\vec{j}})+g_{s}\frac{s}{j}\cos(\widehat{\vec{s},\vec{j}})$

from which (see (Born 1969, Chapter 6, Section 38)) it is possible to reach to
the following relation

(12)
$g_{j}=g_{l}\frac{(j^{2}+l^{2}-s^{2})}{2j^{2}}+g_{s}\frac{(j^{2}+s^{2}-l^{2})}{2j^{2}}$

Experimental evidences dating back to 1920s and mainly related to the
anomalous Zeeman effect, seemed suggesting that $g_{l}=1$ and $g_{s}=2$ for
the electron, that is, the atomic vectorial model explains the fine structure
features of alkali metals and the anomalous Zeeman effect if one supposes to
be $g_{s}\neq 1$, that is to say, a spin intrinsic gyromagnetic ratio
anomalous respect to the orbital one ($g_{l}=1$), so speaking of a spin
anomaly. Following (Bohm 1993, Chapter IX, Section 3), the deviations from the
$g_{s}=2$ value for the electron comes from the radiative corrections of
quantum electrodynamics and is of the same order as, and of analogous origin
to, the Lamb shift. The value $g_{s}=2$ was first established as far back as
1915 by a celebrated experiment of A. Einstein and W.J. de Haas which led to
the formulation of the so-called Einstein-de Hass effect and that was also
incorporated in the spin hypothesis put forward in the 1920s (see (S̆polskij
1986, Volume II, Chapter VII, Section 70)). Following (Jegerlehner 2008, Part
I, Chapter 1), the anomalous magnetic moment is an observable which may be
studied through experimental analysis of the motion of leptons. The story
started in 1925 when Uhlenbeck and Goudsmit put forward the hypothesis that an
electron had an intrinsic angular momentum of $\hbar/2$ and that associated
with this there were a magnetic dipole moment equal to $e\hbar/2mc$, i.e. the
Bohr magneton $\mu_{0}$. According to E. Back and A. Landé, the question which
naturally arose was whether the magnetic moment of the electron
$(\mu_{m})_{e}$ is precisely equal to $\mu_{0}$, or else $g_{s}=1$ in
$(10)_{2}$, to which them tried to answer through a detailed study of numerous
experimental investigations on the Zeeman effect made in 1925, reaching to the
conclusion that the Uhlenbeck and Goudsmit hypothesis was consistent although
they did not really determine the value of $g_{s}$. In 1927, Pauli formulated
the quantum mechanical treatment of the electron spin in which $g_{s}$
remained a free parameter, whilst Dirac presented his revolutionary
relativistic theory of electron in 1928, which, instead, unexpectedly
predicted $g_{s}=2$ and $g_{l}=1$ for a free electron. The first experimental
evidences for the Dirac’s theoretical foresights for electrons came from L.E.
Kinster and W.V. Houston in 1934, albeit with large experimental errors at
that time. Following (Kusch 1956), it took many more years of experimental
attempts to descry that the electron magnetic moment could exceed 2 by about
0.12, the first clear indication of the existence of a certain anomalous
contribution to the magnetic moment given by

(13) $a_{i}\doteq\frac{(g_{s})_{i}-2}{2},\ \ \ \ \ \ \ \ \ \ i=e,\mu,\tau.$

With the new results on renormalization of QED by J. Schwinger, S.I. Tomonaga
and R.P. Feynman of 1940s, the notion of anomalous magnetic moment (AMM) falls
into the wider class of QED radiative corrections.

### 3.2 On Field Theory aspects of AMM

Following (Jegerlehner 2008, Part I, Chapter 3), for the measurement of the
anomalous magnetic moment of a lepton, it is necessary to consider the motion
of a relativistic point-particle $i$ (or Dirac particle202020That is to say, a
particle without internal structure.) of charge $Q_{i}e$ and mass $m_{i}$ in
an external electromagnetic field $A_{\mu}^{ext}(x)$. The equations of motion
of a charged Dirac particle in an external field are given by the Dirac
equation

(14)
$\big{(}i\hbar\gamma^{\mu}\partial_{\mu}+Q_{i}\frac{e}{c}\gamma^{\mu}(A_{\mu}+A_{\mu}^{ext}(x))-m_{i}c\big{)}\psi_{i}(x)=0,$

and by the second order wave equation

(15) $\big{(}\Box
g^{\mu\nu}-(1-\xi^{-1}\big{)}\partial^{\mu}\partial^{\nu})A_{\nu}(x)=-Q_{i}e\bar{\psi}_{i}(x)\gamma^{\mu}\psi_{i}(x).$

The first step is now to find a solution to the relativistic one-particle
problem given by the Dirac equation (14) in the presence of an external field,
neglecting the radiation field in first approximation. In such a case, the
equation (14) reduces to

(16) $i\hbar\frac{\partial\psi_{i}}{\partial
t}=\big{(}-c{\vec{\alpha}}(i\hbar{\vec{\nabla}}-Q_{i}\frac{e}{c}{\vec{A}})-Q_{i}e\Phi+\beta
m_{i}c^{2}\big{)}\psi_{i}$

where

(17) $\beta=\gamma^{0}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right),\ \ \
\vec{\alpha}=\gamma^{0}\vec{\gamma}=\left(\begin{array}[]{cc}0&{\vec{\sigma}}\\\
\vec{\sigma}&0\\\ \end{array}\right)$

are the Dirac matrices, $A^{\mu\ ext}=(\Phi,\vec{A})$ is the electromagnetic
four-potential with scalar and vector potential respectively given by $\Phi$
and $\vec{A}$ (of the external electromagnetic field) and $i=e,\mu,\tau$. For
the interpretation of the solution to the last Dirac equation (16), the non-
relativistic limit plays an important role because many relativistic QFT
problems may be most easily understood and solved in terms of the non-
relativistic problem as a starting point. To this end, it is helpful and more
transparent to work in natural units, the general rules of transcription being
the following: $p^{\mu}\rightarrow
p^{\mu},d\mu(p)\rightarrow\hbar^{-3}d\mu(p),m\rightarrow mc,e\rightarrow
e/(\hbar c),\exp(ipx)\rightarrow\exp(ipx/\hbar)$ and, for spinors,
${}^{t}(u,v)\rightarrow{{}^{t}(u/\sqrt{c},v/\sqrt{c})}$; furthermore, we shall
consider a generic lepton $e^{-},\mu^{-},\tau^{-}$ with charge $Q_{i}$,
dropping the index $i$. Moreover, to get, from the Dirac spinor $\psi$, the
two-component Pauli spinors $\varphi$ and $\chi$ in the non-relativistic
limit, one has to perform an appropriate unitary transformation, the so-called
Foldy-Wouthuysen transformation212121It is a unitary transformation introduced
around the late 1940s by L.L. Foldy and S.A. Wouthuysen to study the non-
relativistic limits of Dirac equation as well as to overcome certain
conceptual and theoretical problems arising from the relativistic
interpretations of position and momentum operators. Following (Foldy and
Wouthuysen 1950), in the non-relativistic limit, where the momentum of the
particle is small compared to $m$, it is well known that a Dirac particle
(that is, one with spin 1/2) can be described by a two-component wave function
in the Pauli theory. The usual method of demonstrating that the Dirac theory
goes into the Pauli theory in this limit makes use of the fact that two of the
four Dirac-function components become small when the momentum is small. One
then writes out the equations satisfied by the four components and solves,
approximately, two of the equations for the small components. By substituting
these solutions in the remaining two equations, one obtains a pair of
equations for the large components which are essentially the Pauli spin
equations. See (Bjorken and Drell 1964, Chapter 4)., upon the Dirac equation
(16) rewritten as follows

(18) $i\hbar\frac{\partial\psi}{\partial t}=\vec{H}\psi,\ \ \ \ \
\vec{H}=c\vec{\alpha}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}+\beta
mc^{2}+Q\Phi,$

with $\vec{\alpha}$ and $\beta$ given by (17) (see (Bjorken and Drell 1964,
Chapter 1, Section 4, Formula (1.26)).

Then, following (Bjorken and Drell 1964, Chapter 1, Section 4) and
(Jegerlehner 2008, Part I, Chapter 3), in order to obtain the non-relativistic
representation for small velocities, we should split off the phase of the
Dirac field $\psi$, which is due to the rest energy of the lepton

(19) $\psi=\tilde{\psi}\exp\big{(}-i\frac{mc^{2}}{\hbar}t\big{)},\ \ \ \ \
\tilde{\psi}=\left(\begin{array}[]{c}\tilde{\varphi}\\\
\tilde{\chi}\end{array}\right)$

so that the Dirac equation takes the form

(20) $i\hbar\frac{\partial\tilde{\psi}}{\partial
t}=(\vec{H}-mc^{2})\tilde{\psi}$

and describes the following coupled system of equations

(21) $\big{(}i\hbar\frac{\partial}{\partial
t}-Q\Phi\big{)}\tilde{\varphi}=c\vec{\sigma}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}\tilde{\chi},$
(22) $\big{(}i\hbar\frac{\partial}{\partial
t}-Q\Phi+2mc^{2}\big{)}\tilde{\chi}=c\vec{\sigma}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}\tilde{\varphi}$

which, respectively, provide the Pauli description in the non-relativistic
limit and the one of the negative-energy states. As $c\rightarrow\infty$, it
is possible to prove that

(23)
$\tilde{\chi}\cong\frac{1}{2mc}\vec{\sigma}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}\tilde{\varphi}+O(1/c^{2}),$

by which we have

(24) $\big{(}i\hbar\frac{\partial}{\partial
t}-Q\Phi\big{)}\tilde{\varphi}\cong\frac{1}{2m}\big{(}\vec{\sigma}(\vec{p}-\frac{Q}{c}\vec{A})\big{)}^{2}\tilde{\varphi}$

and since $\vec{p}$ does not commute with $\vec{A}$, we may use the relation

(25)
$(\vec{\sigma}\vec{a})(\vec{\sigma}\vec{b})=\vec{a}\vec{b}+i\vec{\sigma}(\vec{a}\wedge\vec{b})$

to obtain

(26)
$\big{(}\vec{\sigma}(\vec{p}-\frac{Q}{c}\vec{A})\big{)}^{2}=\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}^{2}-\frac{Q\hbar}{c}\vec{\sigma}\cdot\vec{B}$

where $\vec{B}=rot\ \vec{A}$, so finally reaching to the following 1927 Pauli
equation

(27) $i\hbar\frac{\partial\tilde{\varphi}}{\partial
t}=\tilde{H}\tilde{\varphi}=\Big{(}\frac{1}{2m}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}^{2}+Q\Phi-\frac{Q\hbar}{2mc}\vec{\sigma}\cdot\vec{B}\Big{)}$

which, up to the spin term, is nothing but the non-relativistic Schrödinger
equation. Following too (Muirhead 1965, Chapter 3, Section 3.3(f)), the last
term of (27) has the form of an additional potential energy. Now, by (4),
since the potential energy of a magnet of moment $\vec{\mu}_{m}$, in a field
of strength $B$, is $-\vec{\mu}_{m}\cdot\vec{B}$, equation (27) shows that a
Dirac particle with electric charge $Q$ should possess a magnetic moment equal
to $(Q\hbar/2mc)\vec{\sigma}=2Q\mu_{0}\vec{\sigma}/2$ that, compared with
$(6)_{1}$, would imply $g_{s}=2$. This is what Dirac theory historically
provided for an electron. Later, Pauli showed as the Dirac equation could be
little modified to account for leptons of arbitrary magnetic moment by adding
a suitable term.

Indeed, in222222See also (Pauli 1973, Chapter 6, Section 29). (Pauli 1941,
Section 5)), the author concludes his report with some simple applications of
the theories discussed in (Pauli 1941, Part II, Sections 1, 2(d) and 3(a)),
concerning the interaction of particles of spin 0, 1, and 1/2 with the
electromagnetic field. In the last two cases we denote the value $e\hbar/2mc$
of the magnetic moment as the normal one, where $m$ is the rest mass of the
particle. The assumption of a more general value $g_{s}(e\hbar/2mc)$ for the
magnetic moment demands the introduction of additional terms, proportional to
$g_{s}-1$, into the Lagrangian or Hamiltonian. Pauli concludes his report with
some simple applications of the theories discussed in (Pauli 1941, Part II,
Sections 1, 2(d) and 3(a)) concerning the interaction of particles of spin 0,
1, and 1/2 with the electromagnetic field. In the last two cases, Pauli
denotes the value $e\hbar/2mc$ of the magnetic moment as the normal one, where
$m$ is the rest mass of the particle. The assumption of a more general value
$g(e\hbar/2mc)$ for the magnetic moment demands the introduction of additional
terms, proportional to $g-1$, in the Lagrangian or Hamiltonian. To be precise,
following (Dirac 1958, Chapter 11, Section 70), (Corinaldesi and Strocchi
1963, Chapter VII, Section 4), (Muirhead 1965, Chapter 3, Section 3.3(f)) and
(Levich et al. 1973, Chapter 8, Section 63 and Chapter 13, Section 118), Pauli
modified the basic Dirac equation, written in scalar form as follows

(28) $i\hbar\gamma_{\mu}\frac{\partial}{\partial
x_{\mu}}\psi+mc^{2}\psi-i\hbar\frac{Q}{c}\gamma_{\mu}A_{\mu}\psi=0,$

to get the following Lorentz invariant Dirac-Pauli equation

$\displaystyle i\hbar\gamma_{\mu}\frac{\partial}{\partial
x_{\mu}}\psi+mc^{2}\psi-i\hbar\frac{Q}{c}\gamma_{\mu}A_{\mu}\psi-i\hbar
a_{\mu}\gamma_{\mu}\gamma_{\nu}(A_{\mu,\nu}-A_{\nu,\mu})$ (29)
$\displaystyle=i\hbar\gamma_{\mu}\frac{\partial}{\partial
x_{\mu}}\psi+mc^{2}\psi-i\hbar\frac{Q}{c}\gamma_{\mu}A_{\mu}\psi-i\hbar
a_{\mu}\sigma_{\mu\nu}q_{\nu}A_{\mu}=0$

replacing the gauge invariant interaction term
$-i\hbar\sigma_{\mu\nu}q_{\nu}A_{\mu}$ with the following phenomenological
term (see also (Sakurai 1967, Chapter 3, Section 3-5) $-i\hbar
a_{\mu}\sigma_{\mu\nu}q_{\nu}A_{\mu}$ called an anomalous moment interaction
(or Pauli moment), where $a_{\mu}$ represents the anomalous part of the
magnetic moment of the particle, $q$ is the momentum transfer and
$\hat{\sigma}=-(i/2)[\vec{\gamma},\vec{\gamma}]$ is the spin $1/2$ momentum
tensor. In the non-relativistic limit, this last expression reduces to the
following equation (compare with (27))

(30) $i\hbar\frac{\partial\psi}{\partial
t}=\Big{(}\frac{1}{2m}\big{(}\vec{p}-\frac{Q}{c}\vec{A}\big{)}^{2}+Q\psi-\big{(}a_{\mu}+\frac{Q\hbar}{2mc}\big{)}\vec{\sigma}\cdot\vec{B}\Big{)}$

so justifying the use of the term ’anomalous’ to denote a deviation from the
classical results. Thus, the transition from the non-relativistic
approximation of the Dirac equation goes over into the Pauli equation;
furthermore, from this reduction there results not only the existence of the
spin of particles but also the existence of the intrinsic magnetic moment of
particle and its anomalous part. Namely, we should have $g_{s}=2(1+a_{\mu})$,
where its higher order part $a_{\mu}=(g_{s}-2)/2\geq 0$ just measures the
deviation’s degree respect to the value $g_{s}=2$ (Dirac moment) as predicted
by the 1928 Dirac theory for electron232323Following (Roberts and Marciano
2010) and (Miller et al. 2007, Section 1), the non-relativistic reduction of
the Dirac equation for an electron in a weak magnetic field $\vec{B}$, is as
follows $i\hbar(\partial\psi/\partial
t)=[(p^{2}/2m)-(e/2m)(\vec{L}+2\vec{S})\cdot\vec{B}]\psi$, by which it follows
that $g_{s}=2$. as well as by H.A. Kramers in 1934 (see (Farley and
Semertzidis 2004, Section 1)) developing Lorentz covariant equations for spin
motion in a moving system. Later, this Pauli ansatz was formally improved and
generalized by L.L. Foldy and S.A. Wouthuysen in the forties to obtain a
generalized Pauli equation which will be the theoretical underpinning of
further experiments. Indeed, at the first order in $1/c$, the lepton behaves
as a particle which has, other than a charge, also a magnetic moment given by
$\mu_{m}=(Q\hbar/2mc)\vec{\sigma}=(Q/mc)\vec{S}$, as said above. Following
(Corinaldesi and Strocchi 1963, Chapter VII, Section 5), (Bjorken and Drell
1964, Chapter 4, Section 3) and (Jegerlehner 2008, Part I, Chapter 3), from an
expansion in $1/c$ of the Dirac Hamiltonian given by $(18)_{2}$, we have the
following effective third order Hamiltonian obtained applying a third
canonical Foldy-Wouthuysen transformation to $(18)_{2}$

(31) $\displaystyle\vec{H}^{\prime\prime\prime}_{FW}$ $\displaystyle=$
$\displaystyle\beta\Big{(}mc^{2}+\frac{\big{(}\vec{p}-(Q/c)\vec{A}\big{)}^{2}}{2m}-\frac{\vec{p}^{4}}{8m^{3}c^{2}}\Big{)}+Q\Phi-\beta\frac{Q\hbar}{2mc}\vec{\sigma}\cdot\vec{B}+$
$\displaystyle-\frac{Q\hbar^{2}}{8m^{2}c^{2}}div\vec{E}-\frac{Q\hbar}{4m^{2}c^{2}}\vec{\sigma}\cdot\big{[}(\vec{E}\wedge\vec{p}+\frac{i}{2}rot\vec{E})\big{]}+O(1/c^{3})$

where each term of it, has a direct physical meaning: see (Bjorken and Drell
1964, Chapter 4, Section 3) for more details. In particular, the last term
takes into account the spin-orbit coupling interaction energy and will play a
fundamental role in setting up the experimental apparatus of many $g-2$ later
experiments. The last Hamiltonian, to the third order, gives rise to the
following generalized Pauli equation $i\hbar(\partial\tilde{\varphi}/\partial
t)=\vec{H}^{\prime\prime\prime}_{FW}\tilde{\varphi}$, which is a generalized
version, including high relativistic terms via the application of a Foldy-
Wouthuysen transformation, of the first form proposed by Pauli in 1941 (see
(Pauli 1941)) and that leads to the second approximation Schrödinger-Pauli
equation as a non-relativistic limit of the Dirac equation (see (Corinaldesi
and Strocchi 1963, Chapter VIII, Section 1)).

Our particular interest is the motion of a lepton in an external field under
consideration of the full relativistic quantum behavior which is ruled by the
QED equations of motions (14) and (15) that, in turn, under the action of an
external field, reduce to (16). For slowly varying field, the motion is
essentially determined by the generalized Pauli equation which besides also
serves as a basis for understanding the role of the magnetic moment of a
lepton at the classical level. The anomalous magnetic moment roughly estimates
the deviations from the exact value $g_{s}=2$, because of certain relativistic
quantum fluctuations in the electromagnetic field (initially called
Zitterbewegung) around the leptons and mainly due, besides weak and strong
interaction effects, to QED higher order effects as a consequence of the
interaction of the lepton with the external (electromagnetic) field and which
are usually eliminated through the so-called radiative corrections. At
present, we are interested to QED contributions only. Following (Muirhead
1965, Chapter 11, Section 11.4), (Jegerlehner 2008, Part I, Chapter 3) and
(Melnikov and Vainshtein 2006, Chapter 2), the QED Lagrangian of interaction
of leptons and photons is (see also (Muirhead 1965, Chapter 8, Section
8.3(a)))

(32)
$\mathcal{L}^{QED}_{int}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}(i{\gamma_{\mu}\partial_{\mu}}-m)\psi-
QJ^{\mu}A_{\mu}$

where $\psi$ is the lepton field, $A^{\mu}=(\Phi,\vec{A})$ is the vector
potential of the electromagnetic field,
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ is the field-strength
tensor of the electromagnetic field,
$J^{\mu}(x)=\bar{\psi}(x)\gamma^{\mu}\psi(x)$ is the electric current and $Q$
is the lepton charge. Let us consider an incoming lepton
$l(p_{1}^{\mu},r_{1})$, with 4-momentum $p_{1}^{\mu}$, rest mass $m$, charge
$Q$ and $r_{1}$ as third component of spin, which scatters off the external
electromagnetic potential $A_{\mu}$ towards a lepton $l(p_{2}^{\mu},r_{2})$ of
4-momentum $p_{2}^{\mu}$ and third component of spin $r_{2}$. To the first
order in the external field and in the classical limit of
$q^{2}=p_{2}^{2}-p_{1}^{2}\rightarrow 0$, the interaction is described by the
following scattering amplitude

(33) $\mathcal{M}(x;p)=\langle
l(p_{2}^{\mu},r_{2})|J^{\mu}(x)|l(p_{1}^{\mu},r_{1})\rangle$

where $\vec{q}=\vec{p}_{2}-\vec{p}_{1}$ is the momentum transfer. In practice,
it will be more convenient to work, through Fourier transforms, with invariant
momentum transfers rather than spatial functions. So, in momentum space, due
to space-time translation invariance for which
$J^{\mu}(x)=\exp(iPx)J^{\mu}(0)\exp(-iPx)$, and to the fact that the lepton
states are eigenstates of 4-momentum, that is to say
$\exp(-iPx)|l(p_{i},r_{i})\rangle=\exp(-ip_{i}x)|l(p_{i};r_{i})\rangle,i=1,2$,
we find the following Fourier transform of the scattering matrix

(34) $\displaystyle\tilde{\mathcal{M}}(q;p)$ $\displaystyle=$
$\displaystyle\int\exp(iqx)\langle
l(p_{2},r_{2})|J^{\mu}(x)|l(p_{1},r_{1})\rangle d^{4}x=$ $\displaystyle=$
$\displaystyle\int\exp[i(p_{2}-p_{1}-q)x]\langle
l(p_{2},r_{2})|J^{\mu}(0)|l(p_{1},r_{1})\rangle d^{4}x=$ $\displaystyle=$
$\displaystyle(2\pi)^{4}\delta^{(4)}(q-p_{2}+p_{1})\langle
l(p_{2},r_{2})|J^{\mu}(0)|l(p_{1},r_{1})\rangle$

which is proportional to the Dirac $\delta$-function of 4-momentum
conservation. Therefore, the $T$-matrix element is given by

(35) $\langle l(p_{2},r_{2})|J^{\mu}(0)|l(p_{1},r_{1})\rangle.$

Via the current conservation law $\partial_{\mu}J^{\mu}(\vec{x})=0$ and the
parity conservation in QED, the most general parametrization of the $T$-matrix
element has the following QED relativistically covariant decomposition

(36) $\langle
l(p_{2})|J^{\mu}(0)|l(p_{1})\rangle=\bar{u}(p_{2})\Gamma^{\mu}(p_{2},p_{1})u(p_{1})$

where $\Gamma^{\mu}$, called lepton-photon vertex function, is any expression
(or group of expression) which has the transformation properties of a 4-vector
and is also a $4\times 4$ matrix in the spin space of the lepton. Following
(Muirhead 1965, Chapter 11, Section 11.4(c)) and (Roberts and Marciano 2010,
Chapter 2, Section 2.2; Chapter 3, Section 3.2.2), we shall have the following
Lorentz structure for the scattering amplitude

(37)
$\bar{u}(p_{2})\Gamma^{\mu}(p_{2},p_{1})u(p_{1})=-iQ\bar{u}(p_{2})\Big{(}F_{D}(q^{2})\gamma^{\mu}+F_{P}(q^{2})\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}\Big{)}u(p_{1})$

where $u(p)$ denotes the Dirac spinors, while
$\sigma^{\mu\nu}=(i/2)(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})=(i/2)[\gamma^{\mu},\gamma^{\nu}]$
are the components of the Dirac spin operator
$\hat{\sigma}=-(i/2)\vec{\gamma}\wedge\vec{\gamma}$ or else the spin $1/2$
angular momentum tensor. $F_{D}(q^{2})$ (or $F_{E}(q^{2})$) is the Dirac (or
electric charge) form factor, while $F_{P}(q^{2})$ (or $F_{M}(q^{2})$) is the
Pauli (or magnetic) form factor, which roughly are connected respectively with
the distribution of charge over the lepton and with the anomalous magnetic
moment to the interaction lepton-electromagnetic field. We now need to know
the relationships between these form factors and the anomalous part of the
lepton magnetic moment.

In the non-relativistic quantum mechanics, a lepton interacting with an
electromagnetic field is described by the Hamiltonian

(38) $H=\frac{(\vec{p}-Q\vec{A})^{2}}{2m}-\vec{\mu}_{s}\cdot\vec{B}+Q\Phi,\ \
\ \ \ \ \vec{B}=rot\vec{A}$

which is nothing that $\tilde{H}$ of (27). To find the relations between the
lepton magnetic moment $\mu_{s}$ and the Dirac and Pauli form factors, we
consider the scattering of the lepton off the external vector potential
$A_{\mu}$ in the non-relativistic approximation, using the Hamiltonian (38)
and comparing the results with (33). Following (Melnikov and Vainshtein 2006,
Chapter 2), the non-relativistic scattering amplitude in the first order Born
approximation is given by

(39)
$\Omega=-\frac{m}{2\pi}\int{\bar{\psi}}(\vec{p}_{2})V\psi(\vec{p}_{1})d^{3}\vec{r}$

where $\psi(\vec{p}_{1})=\tilde{\varphi}\exp(i\vec{p}_{1}\cdot\vec{r})$ and
$\psi(\vec{p}_{2})=\tilde{\chi}\exp(i\vec{p}_{2}\cdot\vec{r})$ are the wave
functions of the lepton described by the two components of Pauli spinors (see
(19)) $\tilde{\varphi}$ and $\tilde{\chi}$, and

(40)
$V=-\frac{Q}{2m}(\vec{p}\cdot\vec{A}+\vec{A}\cdot\vec{p})-\mu_{s}\vec{\sigma}\cdot\vec{B}+Q\Phi.$

By a Fourier transform, we have

(41)
$\Omega=-\frac{m}{2\pi}\tilde{\chi}\Big{(}-\frac{Q}{2m}\vec{A}_{q}\cdot(\vec{p}_{2}+\vec{p}_{1})+Q\Phi_{q}-i\mu_{s}\vec{\sigma}\cdot(\vec{q}\wedge\vec{A}_{q})\Big{)}\tilde{\varphi}$

where $\Phi_{q}$ and $\vec{A}_{q}$ stands for the Fourier transforms of the
electric potential $\Phi$ and of the vector potential $\vec{A}$. Therefore, we
will derive (41) starting from the relativistic expression for the scattering
amplitude (33) and taking then the non-relativistic limit. If the Dirac
spinors are normalized to $2m$, the relation between the two oscillating
amplitudes in the non-relativistic limit, is given by

(42) $-i\lim_{|\vec{p}|\ll m}\mathcal{M}(x;p)=4\pi\Omega.$

To derive the non-relativistic limit of the scattering amplitude
$\mathcal{M}$, we use the explicit representation of the Dirac matrices, given
by

(43) $\gamma^{0}=\left(\begin{array}[]{cc}I&0\\\ 0&-I\\\ \end{array}\right),\
\ \ \ \ \gamma^{i}=\left(\begin{array}[]{cc}0&{{\sigma}_{i}}\\\
-{\sigma}_{i}&0\\\ \end{array}\right)\ \ \ i=1,2,3,$

and the Dirac spinors $u(p)$. Using these expressions in $\mathcal{M}$ and
working at first order in $|\vec{p}_{i}|/m\ i=1,2$, we obtain

(44) $\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle-2iem\tilde{\chi}\Big{[}F_{D}(0)\Big{(}\Phi_{q}-\frac{\vec{A}_{q}\cdot(\vec{p}_{1}+\vec{p}_{2})}{2m}\Big{)}+$
$\displaystyle-i\frac{F_{D}(0)+F_{P}(0)}{2m}\vec{\sigma}\cdot(\vec{q}\wedge\vec{A}_{q})\Big{]}\tilde{\varphi}.$

Using (41), (42) and (44), we find

(45) $F_{D}(0)=1,\ \ \ \ \ \ \ \ \ \
\mu_{s}=\frac{Q}{2m}\big{(}F_{D}(0)+F_{P}(0)\big{)}$

which compared with (5) and (6), give

(46) $g_{s}=2(1+F_{P}(0))$

so that, if the Pauli form factor $F_{P}(q^{2})$ does not vanish for $q=0$,
then $g_{s}$ is different from 2, the value predicted by Dirac theory of
electron. It is conventional to call this difference the muon anomalous
magnetic moment and write it as

(47) $a_{\mu}=F_{P}(0)=\frac{g_{s}-2}{2}$

so that, in the static (classical) limit we have too

(48) $F_{D}(0)=1,\ \ \ \ \ \ \ \ \ \ F_{P}(0)=a_{\mu}$

where the first relation is the so-called charge renormalization condition (in
units of $Q$), while the second relation is the finite prediction for
$a_{\mu}$ in terms of the pauli form factor. In QED, $a_{\mu}$ may be computed
in the perturbative expansion in the fine structure constant242424Following
(Muirhead 1965, Chapter 1, Section 1.3(b)), the interaction of the elementary
particles with each other can be separated into three main classes, each with
its own coupling strength. To be precise, the common parameter appearing in
the electromagnetic processes is the fine structure constant
$\alpha=e^{2}/4\pi\hbar c$; the strength of strong interactions is
characterized by the dimensionless coupling term $g^{2}/4\pi\hbar c$, while
the weak interactions are ruled by the Fermi coupling constant $G_{F}$.
$\alpha=Q^{2}/4\pi$ as follows

(49)
$a_{\mu}^{QED}=\sum_{i=1}^{\infty}a_{\mu}^{(i)}=\sum_{i=1}^{\infty}c_{i}\Big{(}\frac{\alpha}{\pi}\Big{)}^{i}.$

The first term in the series is $O(\alpha)$ since, when radiative corrections
are neglected, the Pauli form factor vanishes. This is easily seen from the
QED Lagrangian $\mathcal{L}_{int}^{QED}$ given by (32), which implies that,
through leading order in $\alpha$, the interaction between the external
electromagnetic field and the lepton, is given by
$-iQ\bar{u}(p_{2})\gamma^{\mu}u(p_{1})A_{\mu}$. A consequence of the current
conservation, is the fact that the Dirac form factor satisfies the condition
$F_{D}(0)=1$ to all orders in the perturbation expansion. The renormalization
constants influence the Pauli form factor only indirectly, through the mass,
the charge and the fermion wave function renormalization, because there is no
corresponding tree-level operator in QED Lagrangian. Therefore, the anomalous
magnetic moment is the unique prediction of QED; moreover, the $O(\alpha)$
contribution to $a_{\mu}$ has to be finite without any renormalization. The
QED radiative corrections provide the largest contribution to the lepton
anomalous magnetic moment. The one-loop result was computed by J. Schwinger in
1948 (see (Schwinger 1948)), who found the following lowest-order radiative
(or one-loop) correction to the electron anomaly (see (Rich and Wesley 1972)
and (Roberts and Marciano 2010, Chapter 3, Section 3.2.2.1))

(50) $a_{e}^{(2)}=F_{P}(0)=\alpha/2\pi\cong 0.00116.$

In 1949, F.J. Dyson showed that Schwinger’s theory could be extended to allow
calculation of higher-order corrections to the properties of quantum systems.
Since Dyson showed too that the one-loop QED contribution to the anomalous
magnetic moment did not depend on the mass of the fermion, the Schwinger’s
result turned out to be valid for all leptons, so that we have
$a_{i}^{(2)}=F_{P}(0)=\alpha/2\pi,\ i=e,\mu,\tau$. Currently, QED calculations
have been extended to the four-loop order and even some estimates of the five-
loop contribution exist. It is interesting however to remark that Schwinger’s
calculation was performed before the renormalizability of QED were understood
in details; historically, this provided a first interesting example of a
fundamental physics result derived from a theory that was considered to be
quite ambiguous at that time. Therefore, the anomalous magnetic moment of a
lepton is a dimensionless quantity which may be computed order by order as a
perturbative expansion in the fine structure constant $\alpha$ in QED and
beyond this, in the Standard Model (SM) of elementary particles or extensions
of it. As an effective interaction term, the anomalous magnetic moment is
mainly induced by the interaction of the lepton with photons or other
particles, so that it has a pure QED origin. It corresponds to a dimension 5
operator (see (51)) and since any renormalizable theory is constrained to
exhibit terms of dimension 4 or less only, it follows that such a term must be
absent for any fermion in any renormalizable theory at tree (or zero-loop)
level. It is the absence of such a Pauli term that leads to the prediction
$g_{s}=2+O(\alpha)$. Therefore, at that time, it was necessary looking for
other theoretical tools and techniques to experimentally approach the
determination of the anomalous magnetic moment of leptons. Following
(Jegerlehner 2008, Part I, Chapter 3), in higher orders the form factors for
the muon in general acquires an imaginary part. Indeed, if one considers the
following effective dipole moment Lagrangian with complex coupling

(51)
$\mathcal{L}_{eff}^{DM}=-\frac{1}{2}\Big{[}\bar{\psi}\sigma^{\mu\nu}\Big{(}D_{\mu}\frac{1+\gamma_{5}}{2}+\bar{D}_{\mu}\frac{1-\gamma_{5}}{2}\Big{)}\psi\Big{]}F_{\mu\nu}$

with $\psi$ the muon field, we have

(52) $\Re D_{\mu}=a_{\mu}\frac{Q}{2m_{\mu}},\ \ \ \ \ \Im
D_{\mu}=d_{\mu}=\frac{\eta}{2}\frac{Q}{2m_{\mu}},$

so that the imaginary part of $F_{P}(0)$ corresponds to an electric dipole
moment (EDM) which is non-vanishing only if we have $T$ violation. The
equation (51) provides as well the connection between the magnetic and
electric dipole moments through the dipole operator $D$. As we will see later,
the incoming new ideas on symmetry in QFT will turn out to be of extreme
usefulness to approach and to analyze the problem of determination of the
anomalous magnetic moment of the leptons, the equation (51) being just one of
these important results.

### 3.3 Experimental determinations of the lepton AMM: a brief historical
sketch

#### 3.3.1 On the early 1940s experiences

Following (Kusch 1956), (Rich and Wesley 1972), (Farley and Picasso 1979),
(Hughes 2003) and (Jegerlehner 2008, Part I, Chapter 1), in the same period in
which appeared the famous 1948 Schwinger seminal research note, thanks to the
new molecular-beams magnetic resonance spectroscopy methods mainly worked out
by the research group leaded by I.I. Rabi in the late of 1930s, P. Kusch and
H.M. Foley detected, in 1947, a small anomalous $g_{L}$-value for the electron
within a 4% accuracy (see also (Weisskopf 1949)), analyzing the
${}^{2}P_{3/2}$ and ${}^{2}P_{1/2}$ state transition of Gallium: to be precise
they found the values $g_{s}=2.00229\pm 0.00008$ and $g_{l}=0.99886\pm
0.00004$; later, J.E. Nafe, E.B. Nelson and Rabi himself were able, in May
1947, to detect a discrepancy between theoretical and predicted values of
about 0.26% by the measurements of the hyperfine structure level splitting of
hydrogen and deuterium in the ground state on the accepted Dirac $g$-factor of
2, which was quickly confirmed in the same year by D.E. Nagle, R.S. Julian and
J.R. Zacharias (see also (Schweber 1961, Chapter 15, Section d)). In this
regards, in September 1947, G. Breit (1947a,b) suggested that such
discordances between theoretical expectations and experimental evidences could
be overcome if one had supposed $g\neq 2$. Independently by Breit, also J.M.
Luttinger (1948) (as well as T.A. Welton and Z. Koba - see (Rich and Wesley
1972) and references therein - between 1948 and 1949) stated that some
experiments of then, seemed to require a modification in the $g$-factor of the
electron. In this regards, Schwinger suggested that the coupling between the
electron and the radiation field could be the responsible of this, calculating
the effect on the basis of a general subtraction formalism for the infinities
of quantum electrodynamics. Luttinger, instead, shown that the possible change
in the electron magnetic moment could be derived very simply without any
reference to an elaborate subtraction formalism. Soon after, P. Kusch, E.B.
Nelson and H.M. Foley presented, in 1948, another precision measurement of the
magnetic moment of the electron, just before Schwinger’s theoretical result
whose 1948 paper besides quotes them, which together the discovery of the fine
structure of hydrogen spectrum (Lamb shift) by W.E. Lamb Jr. and R.C.
Retherford in 1947, as well as the corresponding calculations by H.A. Bethe,
N.M. Kroll, V. Weisskopf, J.B. French and W.E. Lamb Jr. in the same period,
were the main triumphs of testing the new level of QED theoretical
understanding with precision experiments. All that was therefore a stimulus
for the development of modern QED. These successes had a strong impact in
establishing the QFT as a general formal framework for the theory of
elementary particles and for our understanding of fundamental interactions.
The late 1940s were characterized by a close intertwinement between theory and
experiment which greatly stimulated the rise of the new QED. On the
theoretical side, a prominent role was gradually undertaken by the new non-
Abelian gauge theory proposed by C.N. Yang and R.L. Mills in 1954 as well as
by the various relativistic local QFT symmetries amongst which the discrete
ones of charge conjugation $(C)$, parity $(P)$ and time-reversal $(T)$
reflection which are related amongst them by the well-known $CPT$ theorem,
according to which the product of the these three discrete transformations,
taken in any order, is a symmetry of any relativistic QFT (see (Streater and
Wightman 1964)). Actually, in contrast to the single transformations $C$, $P$
and $T$, which are symmetries of the electromagnetic and strong interactions
only (d’après T.D. Lee and C.N. Yang celebrated work), $CPT$ is a universal
symmetry and it is this symmetry which warrants that particles and
antiparticles have identical masses as well as equal lifetimes; but also the
dipole moments are very interesting quantities for the study of the discrete
symmetries mentioned above.

#### 3.3.2 Some previous theoretical issues

The celebrated 1956 paper of T.D. Lee and C.N. Yang (see (Lee and Yang 1956))
on parity violation, has been an invaluable source of theoretical insights.
The paper discusses the question of the possible failure of parity
conservation in weak interactions taking into account what experimental
evidences existed then as well as possible proposal of experiments for testing
this hypothesis. Amongst these last, they discuss, since the beginning, on
some experiments concerning polarized proton beams which would have led to an
electric dipole moment if the parity violation were occurred. The related
important consequences were too discussed, like the proton and neutron EDM,
taking into consideration the previous early 1950s experiences made by E.M.
Purcell, N.F. Ramsey and J.H. Smith for the proton who made an experimental
measurement of the electric dipole moment of the neutron by a neutron-beam
magnetic resonance method, finding a value less than $10^{-20}$ $e$-cm ca. in
agreement with parity conservation for strong and electromagnetic
interactions. Nevertheless, Lee and Yang argued that yet lacked valid
experimental confirmations of parity conservation for weak interactions
suggesting, to this end, to consider the measure of the angular distribution
of the electrons coming from $\beta$ decays of oriented nuclei like those of
$Co^{60}$, thing that will be immediately done, with success, by C.S. Wu and
co-workers, furnishing a first experimental evidence for a lack of parity
conservation in $\beta$ decays. Subsequently, Lee and Yang also argue on the
question of parity conservation in meson and hyperon decays, as well as in
those strange particle decays having the following features: 1) the strange
particle involved has a non-vanishing spin and (2) it decays into two
particles at least one of which has a non-vanishing spin or rather it decays
into three or more particles. Thus, what conjectured by Lee and Yang could be
also applied to the decay processes a) $\pi\rightarrow\mu+\nu$ and b)
$\mu\rightarrow e+2\nu$. So, in the sequential decay
$\pi\rightarrow\mu\rightarrow e$, starting from a $\pi$ meson at rest, one
might study the distribution of the angle $\theta$ between the $\mu$-meson
momentum and the electron momentum, the latter being in the center-of-mass
system of the $\mu$ meson. The decay b) is then a pure leptonic one, so no
hadronic phenomenon is involved, this making easier the related calculations
(see (Okun 1986, Chapter 3)). Lee and Yang then argue that, if parity is
conserved in neither a) nor b), then the distribution will not in general be
identical for $\theta$ and $\pi-\theta$ directions. To understand this, one
may consider first the orientation of the muon spin. If a) violates parity
conservation, then the muon would be in general polarized along its direction
of motion. In the subsequent decay b), the angular distribution problem with
respect to $\theta$ is therefore closely similar to the angular distribution
problem of $\beta$ rays from oriented nuclei, as discussed before, so that, in
this way, it will be also possible to detect possible parity violations in
this type of decays. These last remarks on $\pi\mu e$ sequence will be
immediately put in practice in the celebrated 1956 experiences pursued by R.L.
Garwin, L.M. Lederman with M. Weinrich and by J.L. Friedman with V.L. Telegdi,
which will further confirm Lee and Yang hypothesis of parity violation in weak
interactions. Following (Sakurai 1964, Chapter 7, Section 2) and (Schwartz
1972, Chapter 4, Section 11), polarized muons slow down and stop before they
decay, but depending on the material (graphite, aluminium, etc.) the muon spin
direction is still preserved, so we have a source of polarized muons. Negative
muons are emitted with their angular momenta pointing along their directions
of motion, whereas positive muons are emitted with their angular momenta
pointing opposite to their directions of motion. Furthermore, if these
positive muons were stopped in matter and allowed to decay, then the direction
of this angular momentum (or spin) at the moment of decay could be determined
by the distribution in directions of the emitted decay electron which follow
the former. If parity is not conserved in muon decay either, then there will
be a forward-backward asymmetry in the positron distribution with respect to
the original $\mu^{+}$ direction. The just above mentioned experiences showed
more positrons emitted backward with respect to the $\mu^{+}$ direction,
showing that parity is not conserved in both $\pi$ and $\mu$ decays.

As it has said above, Lee and Yang already argued on electric dipole moments
in relation to parity conservation law for fundamental interactions, in some
respects enlarging the discussion to the general framework of discrete
symmetry transformations. To understand about the properties of the dipole
moments under the action of such transformations, in particular the behavior
under parity and time-reversal, we have to look at the interaction Hamiltonian
(4) and, above all, at the equations (6) which both depend on the axial vector
$\vec{\sigma}$, so that also $\vec{\mu}_{m}$ and $\vec{d}_{e}$ will be also
axial vectors. On the other hand, the electric field $\vec{E}$ and the
magnetic one $\vec{B}$ transform respectively as a (polar) vector and as an
axial vector. Then, an axial vector changes sign under $T$ but not under $P$,
while a (polar) vector changes sign under $P$ but not under $T$. Furthermore,
since electromagnetic and strong interactions are the two dominant
contributions to the dipole moments, and since both preserve $P$ and $T$, it
follows that the corresponding contributions to (4) must conserve these
symmetries as well. Indeed, following (Muirhead 1965, Chapter 9, Section
9.2(d)), we have

(53) $\displaystyle P\vec{\sigma}P^{-1}=\sigma,\ \ \ \ \
T\vec{\sigma}T^{-1}=-\vec{\sigma},\ \ \ \ \ P\vec{H}P^{-1}=\vec{H},$
$\displaystyle T\vec{H}T^{-1}=-\vec{H},\ \ \ \ \ P\vec{E}P^{-1}=-\vec{E},\ \ \
\ \ T\vec{E}T^{-1}=\vec{E},$

whence it follows that

(54) $\displaystyle
P(\vec{\sigma}\cdot\vec{H})P^{-1}=\vec{\sigma}\cdot\vec{H},\ \ \ \ \ \ \ \ \ \
T(\vec{\sigma}\cdot\vec{H})T^{-1}=\vec{\sigma}\cdot\vec{H},$ $\displaystyle
P(\vec{\sigma}\cdot\vec{E})P^{-1}=-\vec{\sigma}\cdot\vec{E},\ \ \ \ \ \ \ \ \
\ T(\vec{\sigma}\cdot\vec{E})T^{-1}=-\vec{\sigma}\cdot\vec{E}.$

Therefore, as L.D. Landau and Ya.B. Zel’dovich pointed out (see (Landau 1957)
and (Zel’dovich 1961)), due to these symmetry rules on $P$ and $T$, the
magnetic term $-\vec{\mu}_{m}\cdot\vec{B}$ is allowed, while an electric
dipole term $-\vec{d}_{e}\cdot\vec{E}$ is forbidden so that we should have
$\eta=0$ in $(6)_{2}$. Now, $T$ invariance (that, by $CPT$ theorem, is
equivalent to $CP$ invariance) is also violated by weak interactions, which
however are very small for light leptons. Nevertheless, for non-negligible
second order weak interactions (as for heavier leptons \- see (Chanowitz et
al. 1978) and (Tsai 1981)), an approximate $T$ invariance will require the
suppression of electric dipole moments, i.e. $d_{e}\rightarrow 0$. Thus,
electric dipole interaction cannot occur unless both $P$ and $T$ invariance
breaks down in electrodynamics. Following (Roberts and Marciano 2010, Chapter
1, Section 1.3), P.A.M. Dirac discovered, in 1928, an electric dipole moment
term in the relativistic equations involved in his electron theory. Like the
magnetic dipole moment, the electric dipole moment had to be aligned with
spin, so that we have an expression of the type
$\vec{d}=\eta(Q\hbar/2mc)\vec{s}$ (see $(6)_{2}$) where, as already said,
$\eta$ is a dimensionless constant which is the analogous to $g_{s}$. Whilst
the magnetic dipole moment is a natural property of charged particles with
spin, electric dipole moment are forbidden both by parity and time reversal
symmetries as said above. Nevertheless, from a historical viewpoint, the
search for an EDM dates back to suggestions due to E.M. Purcell and N.F.
Ramsey since 1950 who however pointed out that the usual parity arguments for
the non-existence of electric dipole moments for nuclei and elementary
particles, albeit appealing from the standpoint of symmetry, weren’t
necessarily valid. They questioned about these arguments based on parity and
tried, in 1957, to experimentally measure the EDM of the neutron through a
neutron-beam magnetic resonance method, finding a value for $d$ of about
$(-0.1\pm 2.4)\cdot 10^{-20}$ $e$-cm. This result was published only after the
discovery of parity violation although their arguments were provided in
advance of the celebrated 1956 T.D. Lee and C.N. Yang paper on parity