Ninety years from the origin of the electroweak theory

Riccardo Barbieri

1 An effective theory of the weak interactions

The construction of the theory of the electroweak and strong interactions is an impressive story that took about forty years, made of new concepts introduced to agree with experiments and, in parallel, of the search for logical and mathematical consistency. In this context, one of the most influential papers ever in Particle Physics is that of Fermi, since it is the initiator of this story. Importantly Fermi had previously worked, among other things, in quantum field theory by contributing to Quantum ElectroDynamics.

Being one of the first physicists to believe in the existence of the “neutrino” – as he dubbed the new particle hypothesized by Pauli three years earlier to account for the continuous electron spectrum in $\beta$ decay – Fermi introduced quantized fermion fields for the neutrino, $ \Psi_{\nu} ( x ) $, and for the electron, $ \Psi_{e} ( x ) $, without doing the same for the nucleons due to some confusion existing at the time regarding the magnetic moment of the proton, which made not clear whether the Dirac equation was applicable to them. Fermi bypassed this difficulty by introducing the rising and lowering isospin operators, $\tau_{+}$, $\tau_{-}$, as done earlier by Heisenberg, and he wrote the expression for the $\beta$ decay interaction Hamiltonian as

$ H_{l} = g [ \tau_{-} \overline{\Psi}_{e} ( x ) \Psi_{\nu} ( x ) + \tau_{+} \overline{\Psi}_{\nu} ( x ) \Psi_{e} ( x ) ] $.

The evolution in the understanding of the nuclear particles and forces led already in 1936 to the form under which this Hamiltonian is best known

$ H_{l} = \frac{G_{F}}{\sqrt{2}} \sum_{i} [ \overline{\Psi}_{\rho} ( x ) O_{i} \Psi_{n} ( x ) \overline{\Psi}_{e} O_{i} \Psi_{\nu} ( x ) ]$

where $G_F$ is the Fermi constant and $O_{i} = S, P, V, A, T$ for scalar, pseudo-scalar, vector, axial-vector, tensor Dirac operators, respectively. An important consequence of this Fermi theory was soon realized by Marcus Fierz: it led to a cross section for neutrino-proton scattering increasing with the square of the center-of-mass energy, an unacceptable behavior which haunted for decades the weak interactions and gave rise to the formulation of the gauge theory to which we shall come.

In the following three decades or so, an amazing number of important steps/discoveries were made that progressively gave rise to the current-current form of the weak Hamiltonian

$ H_{l} = \frac{G_{F}}{\sqrt{2}} J_{\mu} J_{\mu}^{+}$,

– with the current being a sum of a leptonic and a hadronic part – and reduced the five operators $O_i$ to the $V–A$ combination, which maximally violates the invariance under space reflections. In a concise description:

i) The discovery of parity violation in the $\beta $ decay of 60Co by Chien-Shiung Wu in 1956.
ii) The identification of the vector component of the hadronic current with the charged component of the SU ( 2 ) isospin current (Conserved Vector Current).
iii) The universality between the $\beta$ decay and the $\mu$ decay.
iv) The extension by Nicola Cabibbo of the CVC hypothesis to include the current with change of strangeness – by the name given to a number of relatively long-lived particles discovered since 1947 – thus making the hadronic part of the weak current a member of an SU ( 3 ) octet.
v) The Partial Conservation of the Axial-vector Current (PCAC): Unlike the case of isospin SU ( 2 ), or of SU ( 3 ), no approximate axial symmetry seems present in the hadron spectrum. No parity doublet is observed, e.g., in the nucleons. Yoshiro Nambu pointed out in 1960 that the strong interactions are nevertheless invariant under chiral transformations, but the symmetry is spontaneously broken, with pions – or the full octet of pseudo-scalar mesons in the case of SU ( 3 ) – being the corresponding Nambu-Goldstone bosons.

2 A gauge theory of leptons

In parallel to this progress in the weak interactions, Quantum ElectroDynamics, i.e. a gauge theory of an Abelian U ( 1 ) group, was being developed and successfully compared with experiments at the level of quantum corrections in the Lamb shift and in the electron anomalous magnetic moment. Not surprisingly the question arose about how to construct a gauge theory for a non-Abelian group. This was indeed the case already after Heisenberg introduced the concept of isospin SU ( 2 ). After some attempts involving a compactified higher-dimensional theory of general relativity, the solution came from Yang and Mills in 1954, who wrote down a gauge theory of an internal SU ( 2 ) symmetry in flat four dimensions.

Whereas the initial motivations for this theory were the strong interactions, the first semirealistic gauge SU ( 2 ) models aimed, following the line of Fermi, at describing the weak and the electromagnetic interactions. The first attempt came from Julian Schwinger in 1957, who assumed the existence of a triplet of vector bosons, called $Z^{\pm , 0}$, the two charged ones mediating the weak interactions à la Fermi and the neutral one being the photon. The most important of these attempts came however in 1961 and is due to Sheldon Glashow. He used an SU ( 2 ) × U ( 1 ) gauge group, thus having two neutral gauge bosons, one associated with U ( 1 ) and the other with the neutral generator of SU ( 2 ). This was the first proposal of a unified description of the weak and the electromagnetic interactions, with the photon being a mixed state, by an angle $\theta$ – now called $\theta_{W}$ – of the two neutral vectors. An unresolved question, however, was pending. The photon is massless and the electromagnetic interactions are long range, whereas the weak interactions are short range, so that the charged vectors had to be massive. How about gauge invariance, which ought to require the conservation of the corresponding currents and a zero mass for the Yang-Mills vector bosons?

The remarkable synthesis of Steven Weinberg came in 1967, followed a year later by Abdus Salam, with the crucial use of the mechanism of spontaneous symmetry breaking of a gauge symmetry, discussed independently a few years earlier by Englert and Brout, by Peter Higgs and by Guralnik, Hagen and Kibble. What became known as the Higgs mechanism allowed to give mass to the $W$ and the $Z$ bosons, setting significant lower bounds (in units where $h=c=1$)

$ M_{W} > \left ( \frac{e^{2}}{4\sqrt{2}G_{F}} \right )^{1/2} \simeq 40$ GeV,

$ M_{Z} > \left ( \frac{e^{2}}{\sqrt{2}G_{F}} \right )^{1/2} \simeq 80$ GeV,

in spite of the ignorance, at that time, of the value of the mixing angle $\theta_{W}$. This was the first case in which spontaneous symmetry breaking was used in the weak interactions. At the same time this same mechanism gives mass to the fermions, which, in the work of Weinberg, were only leptons. In the concluding paragraph of his paper Weinberg argues that his model may be renormalizable, perhaps even when extended to hadrons, thus improving the high-energy behavior of the theory and finally avoiding problems like the one already pointed out by Fierz in 1936, as mentioned above. The formal proof of renormalizability will come a few years later by the works of Gerard ‘t Hooft and Martinus Veltman.

3 The extension to hadrons

What prevented the inclusion of hadrons in the SU ( 2 ) × U ( 1 ) gauge model? Note that already at the beginning of 1964 Murray Gell-Mann, simultaneously with George Zweig, had introduced the notion of quarks, which rapidly emerged as real entities. In any evidence the culprit was the difficulty in reproducing the observed transitions with change of strangeness, the $K_{L} - K_{S}$ mass difference, and the branching ratio $\Gamma (K_{L} \rightarrow \mu\mu) / \Gamma (K_{L} \rightarrow \mathrm{all} )$, both quadratically sensitive to a high energy cutoff. This problem was neatly solved in 1970 by Glashow, Iliopoulos and Maiani. As the 1962 proposal of Cabibbo could be stated by saying that the down and the strange quarks entered the weak current in the combination $ d_{C} \mathrm{cos}\theta d + \mathrm{sin} \theta s$, the so-called GIM hypothesis was that a second up-type quark had to be introduced, the charm quark, coupled in the weak current to the orthogonal combination $s_{C} = -\mathrm{sin} \theta d + \mathrm{cos} \theta s$. In this way the highly suppressed strangeness-changing transitions could be reproduced, provided the charm quark had a mass below a few GeV.

Note a kind of restored symmetry between hadrons and leptons, underlined by the title of the GIM paper: “Weak interactions with lepton-hadron symmetry”, as was the case in the original Fermi theory: proton-neutron versus electron-neutrino. As a matter of fact the discovery of the muon and of its associated neutrino, $\nu_{\mu}$, shown to be different from the electron neutrino, $\nu_e$, in 1962 by Schwartz, Steinberger and Lederman, appeared to break that symmetry. The introduction of charm restored it, however, with two doublets of leptons $( \nu_e , e)$ and $( \nu_{\mu} , \mu)$, corresponding to two doublets of quarks, $( u, d )$ and $ ( c, s )$. Indeed what emerged was a full replica, $c,s,\mu, \nu_{\mu}$, of the constituents of standard matter $u, d, e, \nu_e$. Not the end of the story in this respect, as we shall see. It is worth recalling at this point that the correct interpretation of the muon as the first discovered replica of the electron originated from the experiment of Conversi, Pancini and Piccioni, published in 1947, about which Louis Alvarez in his Nobel lecture of 1968 said: “As a personal opinion, I would suggest that modern particle physics started in the last days of World War II, when a group of young Italians, Conversi, Pancini and Piccioni, who were hiding from the German occupying forces, initiated a remarkable experiment.”

4 The Standard Model of Particle Physics

The GIM hypothesis made it possible to write down a complete gauge theory of the electroweak interactions, paving at the same time the way to a full theory that included the strong interactions as well. The key element that allowed this further step was the emergence in the following two or three years of Quantum ChromoDynamics by several independent contributions: every quark carrying a triplet of colors is glued inside the hadrons by an octet of equally colored gluons, $G$, consistently with a gauged SU ( 3 ) symmetry. Arguably the most important dynamical property of QCD that became clear in those years was Asymptotic Freedom: contrary to the case of the electroweak couplings, the strong gauge coupling gets weaker as the energy increases. This was seen as the way to make sense of the partonic picture of the nucleons – made of quarks and gluons in overall colorless states – shown by the deep inelastic scattering experiments of electrons on nuclei in 1967 by J. Friedman, H. Kendall and R. Taylor. Unlike the electroweak SU ( 2 ) × U ( 1 ) gauge group, the strong SU ( 3 ) is not spontaneously broken, since the Higgs multiplet does not carry color. In the same way the quark masses respect the SU ( 3 ) symmetry although not the electroweak SU ( 2 ) × U ( 1 ), as already stated.

Neglecting for brevity the discussion that went on in the seventies on the precise structure of the model, we are finally in the position, by applying the powerful machinery of gauge theories, to define the SM in a compact but nevertheless fully precise manner. For that:

i) The Lagrangian has to respect the space-time Lorentz symmetry as well as the SU ( 3 ) × SU ( 2 ) × U ( 1 ) gauge symmetry.
ii) The Higgs boson and the fermions have to transform under these symmetries as specified in table 1.
iii) In order for the theory to be renormalizable, only local operators are allowed of dimension $d \le 4$, where d is the dimension of mass in units where $h =c=1$. As a matter of fact, to match observations every operator of dimension $d\le 4 $ has to be included, with one single exception – at least so far, as recalled below – related to the conservation by the strong interactions of CP, the product of charge conservation times parity.

This is the way to describe the behavior of standard matter. As mentioned, one full replica had already emerged, at least accepting the GIM hypothesis. In 1973, however, Kobayashi and Maskawa pointed out the need to introduce a further replica, or generation, in order to describe in the SM, as defined above, the violation of CP in the weak interactions. This means that every fermion field in table 1 has to carry an index $i$ from 1 to 3, called flavor index. A key feature of flavor is the following. As allowed by the gauge symmetry, the couplings of the quarks to the Higgs field are generic 3×3 matrices in flavor space, the so-called Yukawa couplings. To get the quark mass eigenstates these matrices, one for the up-type and one for the down-type quarks, have to be diagonalized by canonical unitary transformations of the quark fields. As a result only the charged weak interactions become non-diagonal in flavor space, with the transition between different flavors described by the Cabibbo-Kobayashi-Maskawa matrix. The expression of $d_C$, $s_C$ in terms of $d$, $s$ given in the previous section is nothing but a manifestation of the CKM matrix restricted to the first two flavor generations. An important difference between quarks and leptons emerges here. Since the SM does not include right-handed neutrinos (see table 1), neutrinos are strictly massless – a fact contradicted by experiments to which we shall have to return – and, when going to the physical mass basis for the charged leptons, no flavor transition arises in the lepton sector, unlike in the quark case. This goes under the name of individual Lepton Flavor Conservation.

The entire particle content expected in the SM is summarized in table 2, with the entries filled by the particles that had already been discovered by the end of 1973. The dates indicate the experimental discoveries in the case of the leptons, whereas for the other particles they correspond to arguable theoretical choices associated with their introduction: the explanation of the photoelectric effect by Einstein for the photon and the already mentioned paper of Gell-Mann for the quarks. The nature of the empty entries will emerge shortly.

5 The extraordinary empirical success of the Standard Model

The empirical adequacy of the SM, remarkable by extension and precision, emerged in the following decades. Here is a concise description:

i) The discovery of weak neutral currents. The phenomenological approach and, in parallel, the search for mathematical consistency produced what is perhaps the most striking consequence of the electroweak theory: the need of two neutral vectors, with the massive $Z$ boson being the combination orthogonal to the photon. The search for the new interaction mediated by the $Z$ exchange emerged as the greatest and, at the beginning, partly unexpected result of the bubble chamber Gargamelle at CERN. After some discussion about the unmistakable events revealing the presence of the so-called neutral current events, the announcement of the discovery was made in 1973, which makes that Gargamelle can be admired still nowadays on CERN ground, see fig. 1.
ii) The progressive discovery of all the expected particles. As shown in table 3, all the particles expected in 1973 were discovered in the following decades. Note the remarkable progression that took place already in the seventies and the full completion achieved in 2012 with the discovery of the Higgs boson at the CERN Large Hadron Collider. It may be curious to note as well that only three of these discoveries were not specifically rewarded with the Nobel prize: the gluon, $G$, the top quark, $t$, and the tau neutrino, $\nu_{\tau}$.
iii) Precision in QCD. The empirical adequacy of the SM emerges not only in the discovery of new phenomena or new particles but, with comparable importance, in the numerical precision of its verified predictions as well. The behavior predicted by QCD of the strong fine-structure constant $\alpha_{S} ( Q^2 )$ is shown in fig. 2, only dependent on the value of $\alpha_S$ at a single point, e.g. $Q^2 = m^{2}_{Z}$. The consequent determination of $\alpha_{S} ( m^{2}_{Z} ) $ at better than 1% precision is not comparable with the ppb uncertainty at which we know the electromagnetic fine-structure constant at $Q^2 = 0$, but is nevertheless remarkable for a theory that becomes non-perturbative at momenta below about one GeV.
iv) Precision in the electroweak sector. The precision tests of the SM in the electroweak sector have been dominated, although not in an exclusive way, by the experiments at the Large Electron Positron collider at CERN in the nineties via numerous observables at the $Z$ pole. The result of an overall fit is illustrated in fig. 3, which shows the pull between the observed experimental values and the SM prediction for the different so-called ElectroWeak Precision Observables. This test has allowed to establish the correct description by the SM of the weak interactions at the level of the quantum corrections, leading as well in this way to the indirect determination, before the corresponding direct discoveries, of the top quark mass and of the Higgs mass, in both cases with a 10-20% precision.

v) Flavor physics. As already mentioned the transition between different flavors, as well as CP violations, are parametrized in the SM by the CKM unitary matrix in terms of four effective parameters, three angles and one phase. The comparison with experiments goes through the observation of the weak decays – i.e. not mediated by the strong or the electromagnetic interactions – and of the mixing of neutral mesons carrying strangeness, $K_0 – \overline{K}_0$, charm, $D_0 – \overline{D}_0$, or beauty, $B_0 – \overline{B}_0$. These flavor tests have a long history, initiated by Cabibbo in 1962 and made complex by the frequent involvement of QCD in the difficult non-perturbative regime. For example in the case of CP violation, discovered in 1964, the emergence of its CKM description, still at qualitative level, had to wait for the observation of the so-called direct CP violation in $K \rightarrow \pi\pi$ decays, first at CERN and then at FermiLab in 1999. Currently no significant deviation has emerged from the SM expectations, although at a level of precision not comparable to the one at which the gauge sector of the theory has been tested. Recent years have seen progress in experiments, in QCD calculations on a lattice and in phenomenological analyses, with the prospect of making flavor physics a fundamental tool in the search for new physics beyond the SM.

6 Questions for (problems of) the Standard Model

As any successful scientific theory, the SM in its maturity leaves a number of open questions and problems, both of observational and of structural origin. The structural problems are of more theoretical nature, but not, in my view, less important.

6.1 Observational problems

Among the observational problems, the most well known are neutrino masses, Dark Matter and the dominance of baryonic matter over antimatter in the universe:

i) Neutrino masses. The oscillations of neutrinos among different flavors, observed in neutrinos coming from the Sun, from collisions of cosmic rays in the atmosphere, from reactors (antineutrinos) and from prepared beams, disprove individual Lepton Flavor Conservation, implied, as already mentioned, by the absence of neutrino masses in the SM. Neutrino masses can be introduced by adding new fermions that do not carry any SM quantum number, which can be called right-handed neutrinos, or by accepting the presence in the SM Lagrangian of an operator of dimension $d=5$. At an effective level these two choices may not be different from each other. A fundamental difference is in the possible nature of the neutrino masses: of Dirac type, i.e. similar to the masses of the charged fermions and, as such, conserving Lepton Number, or of Majorana type – from the work of Ettore Majorana in 1937 – with the neutrino identical to its antineutrino, thus breaking Lepton Number. Oscillations alone cannot distinguish between these two possibilities.
ii) Dark Matter. The observation of matter at all cosmic scales by gravity effects only – hence the name of Dark Matter – is another phenomenon not accounted for in the SM. Unlike the case of neutrino masses, the proposed extensions of the SM to account for Dark Matter are many, currently constrained at very different levels: axions, the Lightest Supersymmetric Particle, Weakly Interacting Massive Particles or other relatively less motivated suggestions. Primordial black holes are considered as well. All this pushes for a synergy between Particle Physics, Astrophysics and Cosmology in the search for signals of Dark Matter and/or in the determination of its properties.
iii) Baryon-antibaryon asymmetry. A third item often mentioned as an observational problem of the SM is the dominance of baryonic matter over antimatter in the universe. In standard cosmology the difference in the number of quarks over the antiquarks, relative to their sum, is in the early universe of order 10–10. There is no logical contradiction in supposing that this tiny excess was built in as an initial condition. This would be contrived, however, and in fact excluded in inflationary cosmology (no quarks nor antiquarks at the beginning during inflation). The problem with the SM is that it cannot give a dynamical explanation for this baryon-antibaryon asymmetry.

6.2 Structural problems

As to the structural problems of the SM, at least three are worth mentioning:

i) Electric charge quantization. The presence of an Abelian U ( 1 ) in the gauge symmetries of the SM leaves in some way unexplained the quantization of electric charge, $Q= T_{L3} +Y$, made of a combination of the (quantized) third component of weak SU ( 2 ), $T_{L3}$, and of the U ( 1 ) hypercharge, $Y$. The preservation, at the quantum level, of the SU ( 3 ) × SU ( 2 ) × U ( 1 ) gauge symmetry – i.e. the absence of gauge anomalies – is not enough to guarantee charge quantization. This is in contrast with the bounds on the neutrality of matter, at the level of 10–21 relative to the electron charge, or on the neutrino charge, of about 10–14, from plasmon decays into neutrinos in stars.
ii) A single lacking operator of dimension $d\le 4$. As already mentioned it is striking that one single such operator does not seem to be present in the SM. The dimensionless coefficient of the operator $G_{\mu\nu} \overline{G}^{\mu\nu}$ – where $G_{\mu\nu}$ is the gluon field strength and $ \overline{G}^{\mu\nu} = \varepsilon^{\mu\nu\rho\sigma} G_{\rho\sigma}$ –, odd under CP, is bound to be less than 10–10 by the absence of any signal, so far, of an electric dipole moment of the neutron, equally odd under CP. The presence of an axion, a very light scalar which might also constitute the Dark Matter in the universe, is a suggestive, although not exclusive, possibility to explain this feature.
iii) A matter of calculability. Of the seventeen particles in table 3 two are massless, the photon and the gluon, due to gauge invariance. Also the three neutrinos are massless, but this is a feature of the SM that needs to be corrected. On the other hand, none of the masses of the remaining twelve particles is predicted by the SM, as are not predicted the four physical parameters of the CKM matrix. The mass of the Higgs boson suffers from its sensitivity to any higher mass scale coupled to the Higgs, the so-called naturalness problem that affects as well the $W$ and $Z$ masses via the related vacuum expectation value of the Higgs field. All the other parameters constitute the flavor puzzle of the SM, itself strongly intertwined with the Higgs boson via the Yukawa couplings.

7 Conclusion

As said, it is proper to the strong scientific theories that they give rise to new unanswered questions. The SM of Particle Physics is no exception in this respect. This hints at the need for its embedding into a suitable more complete extension, thus giving hope for new discoveries. Any such extension, however, will have to be consistent with the enormous success of the SM: not the least reason to celebrate the ninetieth anniversary of the seminal 1933 Fermi work, which originated one of the greatest scientific theories ever.