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arXiv:1403.3309v1 [physics.hist-ph] 25 Feb 2014
FERMI AND THE THEORY OF WEAK INTERACTIONS
G. Rajasekaran
Institute of Mathematical Sciences, Chennai 600113 and Chennai Mathematical Institute, Chennai 603103 e-mail: graj@imsc.res.in
Abstract
The history of weak interactions starting with Fermi’s creation of the beta decay the- ory and culminating in its modern avatar in the form of the electroweak gauge theory is described. Discoveries of parity violation, matter-antimatter asymmetry, W and Z bosons and neutrino mass are highlighted.
Introduction
Sun gives us light and heat that makes life possible on the Earth. How do the Sun and stars produce energy and continue to shine for billions of years? Thermonuclear fusion is the answer as Eddington proposed in 1920 and Bethe demonstrated explicitly in 1939. Through a series of nuclear reactions, four protons (which are Hydrogen nuclei) in the core of the Sun combine to form a Helium nucleus emitting two positrons and two neutrinos and releasing 27 MeV of energy:
p + p + p + p → He^4 + e+^ + e+^ + νe + νe + 27M eV This can be regarded as the most important reaction for all life, for without it there can be no life on Earth!
The above reaction is caused by one of the basic forces of Nature, called weak interaction. Beta decays of nuclei and in fact the decays of most of the elementary particles are now known to be due to weak interaction.
Enrico Fermi formulated the theory of weak interactions in 1934 and his theory has stood the ground very successfully with appropriate amendments and generalizations and finally served as a core part of the Standard Model of High Energy Physics, which is now known as the basis of almost ALL of Physics, except for gravitation.
In this article we trace the historical evolution of the theoretical ideas punctuated by the landmark experimental discoveries. This history can be divided into two parts separated by the year 1972 which marks the watershed year since the gauge theoretic revolution that
converted Fermi’s theory into the modern electroweak theory occured roughly around that year. Discovery of P and CP violation as well as the discovery of neutrino oscillation and neutrino mass are weaved into this tapestry as integral partds of weak interaction physics and its history.
Early history: Weak interactions upto 1972
The story of weak interactions starts with Henri Becquerel’s discovery of radioactivity in 1896 and its subsequent classification into alpha, beta and gamma decays of the nucleus by Ernest Rutherford and others. But the real understanding of beta-decay in the sense we know it now came only after Enrico Fermi invented a physical mechanism for the beta-decay process in 1934.
The basic ingredient for Fermi’s theory had been provided by Wolfgang Pauli.To solve the puzzle of the continuous energy spectrum of the electrons emitted in the beta-decay of the nuclei, Pauli had suggested that along with the electron, an almost massless neutral particle also was emitted. Fermi succeeded in incorporating Pauli’s suggestion and thus was born the theory of weak interactions. Fermi also named the particle as neutrino.
Drawing an analogy with electromagnetic interaction which at the quantum level is the emission of a photon by an electron, Fermi pictured the weak interaction responsible for the beta-decay of the neutron as the emission of an electron-neutrino pair, the neutron converting itself into a proton in the process.(Fig 1)
p
p n
p e
Electrodynamics Weak interaction
Figure 1: Fermi’s analogy
By initiating Quantum Electrodynamics Dirac had laid the foundation for Quantum Field Theory (QFT) in 1927. Within a few years Fermi made the first nontrivial application of QFT to weak interactions in which material particles are created.
Either because of the neutrino which most people at that time did not believe in, or because of QFT which most people did not understand at that time or because of both,
(A) currents. Vectors and axial vectors behave differently when we go from left to right- handed coordinate systems and hence the parity violation.This is the V-A form discovered by Sudarshan and Marshak, Feynman and Gell-Mann and Sakurai in 1957.
However there was a period of utter confusion before the above correct form of the inter- action was found. As we saw earlier, Fermi based his intuition on electromagnetism which involves a vector current and we shall see in the context of later developments how sound this intution proved to be. Infact this was a master-stroke of Fermi. However, subsequently, in an attempt at generalization, Fermi’s vector form was replaced by an arbitrary combina- tion S,V,T,A,P (scalar, vector, tensor, axial vector and pseudoscalar) interactions and it led to enormous complication and confusion in the confrontation of experiments with theory. The confusion was resolved and the correct V - A form could be found only because of the additional experimental clues provided by parity violation.
During 1947-55, many new particles such as muons, pions, kaons and hyperons were dis- covered and all of them were found to decay by weak interactions. In fact parity revolution itself was triggered by the famous tau-theta puzzle in the decays of the kaons which was the culmination of the masterly phase-space plot analysis of the three-pion decay mode of the kaon by Richard Dalitz. The field of weak interactions thus got enriched by a multitude of phenomena, of which nuclear beta-decay is just one. Weak interaction is indeed a universal property of all fundamental particles.
Remarkably enough, all the weak phenomena, namely the weak decays of all the particles could be incorporated in a straight-forward generalization of the original Fermi interaction. This was achieved by Feynman and Gell-Mann (1957) in the form of the current x current interaction:
LF G =
GF
(J+J− + J−J+)
J+ = ¯pn + ¯νee + ¯νμμ + ... J− = ¯np + ¯eνe + ¯μνμ + ...
The dots at the end refer to other terms that can be added in order to incorporate the weak decays of other particles such as the strange particles Λ, Σ, and K. A diagrammatic repre- sentation of this is given in Fig 2. The current J+ represents a neutron turning itself into a proton, an electron turning itself into a neutrino or a muon turning itsef into a neutrino - all these transitions result in an increase of electrical charge by one unit. The weak current J+ is called the charge-raising current; J− describes the opposite transition and is called the charge-lowering current.
One can see that Fermi’s original form of the interaction describing the beta decay of the neutron is just one term ¯pn ¯eνe in the product J+J−. The decay of the muon and the absorption of the muon by the proton are described by the terms ¯νμμ ¯eνe and ¯νμμ ¯np respectively. These are illustrated in Fig 3. By turning around the line a particle in the
n (^) e
p (^) e
n
p
e
e
J + (^) J −
Figure 2: Current × current interaction.
initial state can become an antiparticle in the final state. This can be understood from Appendix 1 where it is explained that a field operator can annihilate a particle or create an antiparticle. This happens for the neutrino in n decay and μ decay depicted in Fig 3.
A fundamental experimental discovery - the discovery of CP violation was made by Cronin and Fitch in 1964, in the weak decays of neutral kaons. It is this asymmetry in the basic laws of nature that is presumed to be responsible for the evolution of the original matter-antimatter symmetric Universe into the present-day asymmetric Universe that con- tains only matter.
The story of weak interactions is not complete without due recognition of the neutrino, especially because of more recent developments to be described later.
Pauli proposed the neutrino in 1930. Although because of the success of Fermi’s theory based on neutrino emission in explaining quantitatively all the experimental data on nuclear beta decays, there was hardly any doubt (at least in theorists’ minds) that neutrinos existed, a direct detection of the neutrino came only in 1956. This achievement was due to Cowan and Reines who succeeded in detecting the antineutrinos produced from fission fragments in nuclear reactors.
Subsequently, it became possible to detect the neutrinos from the decays of pions and kaons produced in high-energy accelerators. It is by using the accelerator-produced neutri- nos that the important experiment proving νμ not to be the same as νe was done.
Further, even neutrinos produced by cosmic rays were detected. The underground lab- oratory at the deep mines of the Kolar Gold Fields in South India was one of the first to detect cosmic-ray produced neutrinos called atmospheric neutrinos. This was in 1965.
Electroweak theory: Weak interaction after 1972
p
p n
e p e
e
p
n
e
W
Photon exchange Fermi’s theory^ W−boson exchange
Figure 4: Genesis of W-boson theory.
at the two vertices of the W-exchange diagram in Fig 4. The neutron turns into a proton by emitting a W and so this W should be negatively charged.
(b) Just like the photon, the W particle also has a spin angular momentum of one unit. Both photon and W are bosons.
(c) In contrast to the photon, the W boson has to be a very massive object. For, the weak interaction has a short range unlike the infinite-ranged electromagnetic interaction.
In Fermi’s theory the coupling constant was GF. In the W-boson theory we have a coupling constant g at each vertex and so for the same process GF is replaced by a factor g^2 multiplied by the propagation factor for the W boson. This propagation factor is (^) m^12 W
where mW is the mass of W, for small energy and momentum transfers relevant in beta decay. Thus we have the important relationship:
GF =
2 g^2 8 m^2 W By introducing the field W +^ and W −^ for the positively and negatively charged W bosons, the current x current form of the weak interaction can be split into the form
g(J+W +^ + J−W −).
This form is very similar to the electrodynamic interaction eJE A and so we have achieved a greater degree of symmetry between weak interaction and electrodynamics.(See Fig 5.) The next step of the argument is to realize that the symmetry between the W boson form of weak interaction and electrodynamics noted above is only apparent and does not hold at a deeper level.
Conservation of electric charge is a cornerstone of electrodynamics. The total charge of an isolated system can neither be increased nor decreased and remains constant. A related question concerns gauge invariance which simply means that different electromagnetic po- tentials A lead to the same physical effects as long as the electric field and the magnetic field that are obtained by taking space and time derivatives of A are the same. Are such
n (^) e
p (^) e
n
p
e
e
+...
J + (^) J −
p
p (^) e
e
J E Electrodynamics
Weak dynamics
W^ −^ W +
Figure 5: Symmetry between electrodynamics and weak dynamics.
properties valid for the W boson theory formulated above? The answer is in the negative.
Certain important structural modifications have to be made in the W-boson theory in order to achieve conservation of the generalized charge involved in weak interaction and the corresponding gauge invariance.
The required basic theoretical structure has been known since 1954 when C N Yang and R Mills introduced nonabelian gauge theory which is a generalization of electrodynamics. The gauge invariance of electrodynamics is known as abelian gauge invariance and Yang- Mills theory has nonabelian gauge invariance based on a nonabelian Lie group. But many other ideas had to be discovered before this theory could be tailored to meet the exper- imental facts of weak interactions. The final outcome is the electroweak gauge theory of Glashow, Salam and Weinberg which is the successor to Fermi’s theory.
This theory generalizes the concept of charge. The single electric charge of electrody- namics is replaced in the new theory by four generalized charges. The current corresponding to each charge interacts with its own boson, called gauge boson. An essential point of elec- troweak theory is that the twin requirements of generalized charge conservation and gauge invariance force us to combine weak and electromagnetic interactions dynamically into a single framework. As a consequence of this unification of weak and electromagnetic interac- tions, a new kind of weak interactions is also generated.
n
e
p
J
e
p n
N
Z
Figure 7: Neutral current interaction.
p
p
Z
Figure 8: Neutrino scattering on proton by neutral current interaction.
The discovery of the neutral current weak interaction was made in 1973 at CERN, Geneva. This discovery has its own intrinsic importance because it opened up a whole new class of weak interactions which had remained undetected in all the 70 years’ history of weak interactions. From the point of view of electroweak theory, it has an added significance since the neutral-current (NC) interaction acts as a bridge between electrodynamics and the old charged-current (CC) weak interaction. It is neutral like electromagnetic current, but in- volves a massive boson Z like the W involved in the CC interaction. Hence its discovery with properties identical to those predicted by the electroweak theory was the first great triumph of the theory.
In the history of weak interaction physics the discovery of the V-A structure of the charged current was an important milestone. What is the structure of the neutral current? It is not V-A and so not all of weak interaction is described by V-A theory! The relative amount of V and A in neutral current is specified by an important parameter of the electroweak theory called the weak mixing angle θW and it has been determined experimentally:
sin θW = 0. 23.
Discovery of W and Z
An immediate consequence of the dynamical connection between weak and electromag- netic interactions is that their coupling constants are related:
e = g sin θW ,
gN =
g cos θW
and the masses of W and Z are also related:
mW = mZ cos θW.
The relationship between GF and g^2 derived earlier
GF =
2 g^2 8 m^2 W
now becomes
GF =
2 e^2 8 sin^2 θW m^2 W . This allows us to calculate the masses of W and Z from the known values of GF , e and sin θW. We get mW = 80 GeV.
mZ = 91 GeV.
The discovery of W and Z with these masses at CERN in 1982 was the second great triumph of electroweak theory.
The inverse relationship between GF and and m^2 W given above helps us to answer the question: why is weak interaction weak? It is because the masses of mW (and mZ ) are so large. GF is the effective weak coupling constant at low energies. Once the energy becomes high enough to produce a real W boson, weak interaction attains its real strength g which is comparable to e, the strength of the electromagnetic interaction.
Spontaneous breaking of symmetry and the Higgs boson
An essential ingredient of the electroweak theory described sofar is spontaneous break- down of symmetry, also known as Higgs mechanism. The gauge invariance or gauge symme- try of Yang-Mills theory would lead to massless gauge bosons exactly as the gauge invariance of electrodynamics requires massless photon. But we need massive gauge boson to describe the short-ranged weak interaction. How is this problem solved in electroweak theory? It is solved by the spontaneous breakdown of symmetry engineered by the celebrated Higgs
u
dd
u
u
d
d
u
e
W −
n =
p =
Figure 9: Beta decay of the neutron in terms of quarks.
All composite particles formed out of these quarks are called hadrons. Our familiar proton and neutron are hadrons and many more hadrons are known. Electron and neutrino have remained elementary on par with quarks upto the present. These are called leptons and again 6 types of leptons are known to exist:
(ν 1 , e) (ν 2 , μ) (ν 3 , τ ).
The electroweak interaction in terms of the quarks and leptons is given by
LEW = eJE A + g(J+W +^ + J−W −) + gN JN Z.
The elecromagnetic and neutral currents will contain terms like
uu,¯ dd,¯ ee ...¯
while the charged currents that describe the transitions from one type of quark to another or from one type of lepton to another (as illustrated in Fig 10) are given by
J+ = ¯ud + ¯cs + ¯tb + ¯ν 1 e + ¯ν 2 μ + ¯ν 3 τ J− = du¯ + ¯sc + ¯bt + ¯eν 1 + ¯μν 2 + ¯τ ν 3 The electric charge of the ”up-type” quarks (u, c, t) is + 23 while that of the ”down-type” (d, s, b) is − 13. All the transitions between the up and down type of quarks indicated by the above expression or Fig 10 have a change of charge by one unit and are of the same sign, exactly like the transitions between the charged leptons (e, μ, τ ) and the neutrinos.
e
J
J
+
−
=
=
u (^) c t (^1 2 )
d s (^) b
d s b e
u c t (^1 2 )
Figure 10: The currents in terms of quarks and leptons.
Actually we have to change the down quarks (d, s, b) occuring in J+ and J− by their linear superpositions (d′, s′, b′) defined as follows. Intruducing the notation qi(i = 1, 2 , 3) for (d, s, b), the superposed quarks are given by
q′ i =
j
Uij qj
where U is a 3 × 3 unitary matrix: U †U = 1.
This U is called the CKM matrix and its discovery by Cabibbo, Kobayashi and Maskawa is an important chapter in the history of weak interactions. (For more details on this part of history, see Ref [6]). These superpositions are natural consequences in electroweak the- ory and allow the heavier quarks to decay into all the lighter quarks. The matrix U is parametrized by three angles and one phase that is responsible for the CP violation discov- ered by Cronin and Fitch.
A similar unitary mixing matrix V called PMNS matrix (named after Pontecorvo, Maki, Nakagawa and Sakata) is used in the leptonic part of the currents also and this is what leads to neutrino oscillation and the discovery of neutrino mass. (For more on this, see Ref [7].)
It is important to note that all the 6 quarks and 6 leptons are equally fundamental and all were presumably created in equal numbers in the Big Bang and it is the weak interaction that caused all the heavier particles to decay into the lighter ones u, d, e, ν that make up the
If it is its own antiparticle it is called Majorana particle; otherwise it is a Dirac particle just like the other fermions such as electron or quark. This question can be answered only by the ”neutrinoless double beta decay experiment” which is therefore the most important experiment in all of neutrino physics (See Ref [10] for an account of the Majorana problem).
Neutrino physics is now recognized as one of the most important frontiers in high energy physics and it is vigorously pursued in many underground laboratories around the world. The India-based Neutrino Observatory (INO) that is coming up in South India will be one such [11].
As already mentioned, electroweak theory implies massless neutrinos in a natural way. How is the theory to be extended to incorporate nonzero neutrino masses? Only Future will tell.
Epilogue
We have seen the vast range of phenomena covered by weak interactions: beta decay of nuclei, thermonuclear fusion reactions in the Sun and stars, decays of most of the elementary particles of Nature and removal of antiparticles in the Universe through CP violation. We have touched on the brief history of the important theoretical and experimental discoveries. The milestones in this history are listed in Appendix 3.
Fermi created beta decay theory which was the starting point of all that followed, using the nascent Quantum Field Theory which was perhaps understood by very few physicists at that time. He did this at a time when nuclei were not understood and so nuclear physics did not even exist - not to speak of particle physics (now called high energy physics) which was born only much later. No wonder Fermi responded that it is beta decay theory when asked what he regarded as his most important contribution. There is no doubt that it is not only Fermi’s most important contribution but it is one of the most important contributions made by anybody in that Foundational Epoch of Modern Physics.
Acknowledgements
The author thanks Prof N Mukunda for inviting him to write this article. He thanks Dr Saurabh Gupta for help in drawing the large number of figures that the article contains.
References
- E. Fermi, Attempt at a Theory of β-rays, Il Nuovo Cimento, Vol 11, p 1 (1934); Zeitschrift fur Physik, Vol 88, p 161 (1934)
- For Fermi’s article in English, see The Development of Weak Interaction Theory, Ed: P K Kabir, Gordon and Breach, Science Publishers, (1963). This volume contains most of the original papers in weak interaction physics upto 1962.
- A Pais, Inward Bound, Oxford University Press, New York, 1986, p 417-418.
- S Chaturvedi and Shyamal Biswas, Fermi-Dirac Statistics, Resonance, Jan 2014, p 45.
- G Rajasekaran, Standard Model, Higgs Boson and What Next?, Resonance, Oct 2012, p
- G Rajasekaran, Cabibbo Angle and Universality of Weak Interactions, Physics News, July 2011, p 21.
- G Rajasekaran, An angle to tackle the neutrinos, Current Science,103,No 6 (2012)
- G Rajasekaran, Hans Bethe, the Sun and the Neutrinos, Resonance, Oct 2005, p 49
- D Indumathi, MVN Murthy and G Rajasekaran, Neutrinos and how to catch them, Physics Education, Vol 23, No 2, Aug 2006
- G Rajasekaran, Are neutrinos Majorana particles? arXiv:0803.4387[physics]
- More information on INO can be obtained from www.ino.tifr.res.in
Appendix 1: Quantum Field Theory (QFT)
Quantum Field Theory created by Dirac and used by Fermi to describe weak interactions remains to this day as the correct basic language to understand ALL high energy physics. Here we give an elementary account of its symbolism and interpretation.
We start with Quantum Electrodynamics. The electromagnetic force between two charged particles such as proton and electron can be represented by the diagram in Fig 11. The wavy line denotes the photon which is the quantum of the electromagnetic field and it is exchanged between the proton and the electron. It is this exchange that leads to the electromagnetic force between the charged particles. This is the quantum version of the classical picture where the proton is considered to produce the electromagnetic field which then influences the electron placed in the field.
p
p
e
e
Figure 11: Exchange of photon.
In QFT, the range of a force is inversely proportional to the mass of the quantum that is exchanged. Since photon mass is zero, the electromagnetic force mediated by the exchange of photons is of infinite range.
We shall make a rather liberal use of pictorial representations of interactions and pro- cesses such as in Fig 11. These are called Feynman diagrams, after Feynman whose use of
Figure 12: The basic QED processes.
Particles or quanta of fields come in two varieties, bosons and fermions. Bosons are particles with integral spins, photon with spin 1 being their main representative and they follow Bose-Einstein statistics. Fermions have half-integral spins, electron with spin 12 being a fermionic example and they follow Fermi-Dirac statistics [4]. Since in Fermi’s LF four fermionic lines meet at a point, it is also called the four-fermion interaction.
Appendix 2: Violation of left-right symmetry and CP
Left-right symmetry is also called reflection symmetry or parity symmetry and is denoted by P. Madam Wu’s famous experiment which established P violation was done using the beta decay of Co^60 nuclei. She aligned the spins of Cobalt nuclei by an external magnetic field produced by a circulating current and counted the number of beta electrons emitted in all directions. She found more beta electrons emitted in the direction of the magnetic field as compared to the opposite direction. This was the discovery of P violation.
In the right-handed coordinate system, the directions of the x, y and z axes are such that, if we imagine a screw (actually a right-handed screw which is what we normally use) being rotated from x to y, the screw will advance along z. The left-handed coordinate system is obtained by mirror-reflection. (See Fig 13). Can the laws of physics distinguish between the two coordinate systems? Except for the weak interaction, all other laws of physics are symmetric under mirror reflection and hence cannot be used to distinguish between the left and right coordinate systems.
The significance of this left-right symmetry, as well as its violation can be appreciated better, if we think of the following attempt at intergalactic communication.
Y
Z
X X
Y
Z
Right−handed coordinate system (^) Mirror Left−handed coordinate system
Figure 13: Reflection in a mirror.
Suppose we want to communicate with somebody in a distant galaxy through radio waves. How do we define a right-handed coordinate system for him? Screws will not help here since we do not know whether they use a right- handed screw or a left-handed one in that galaxy! We can use any of the laws of physics for this purpose. If none of the laws dis- tinguishes between the two coordinate systems, we will never be able to convey a definition of the right-handed coordinate system to a being in a distant part of the Universe.
However, thanks to weak interactions, this can be done. The following instruction can be conveyed: ” Take Co^60 nuclei and arrange a sufficient number of electrons to go around these nuclei, thus forming an electric current. If a rotation from the x axis to the y axis is in the direction of the circulating electrons, then the z axis is the direction in which more electrons are emitted.” This would define the right-handed coordinate system for our friend in the distant galaxy.(See Fig 14). Thus weak interaction allows us to define a right-handed coordinate system by using natural physical laws. A word of caution, however. We have to make sure that the planet inhabited by our friend is made of matter and not of antimatter. If it is made of antimatter,he would really take nuclei of anti-Co nuclei and electric current made of positrons and would end up with a left-handed coordinate system by following our instructions!
What is stated in the last para above is the result of CP symmetry, C standing for particle-antiparticle conjugation. In other words, weak interactions violate P symmetry and C symmetry. But if both C and P are applied together weak interactions remain invariant. It was thought until 1964 that CP symmetry remains intact in weak interactions. We now know that even this is not correct, as a consequence of the discovery of CP violation by Cronin and Fitch.
