Abstract
In Chap. 5 we already talked about the phenomenological Fermi theory, as well as the modern fundamental electroweak theory (the Standard Model), and described their salient features in words and pictures. Since then, we have learned a lot of new stuff (what are fermion fields, what is a field theory Lagrangian, how to calculate scattering amplitudes, what is Abelian and non-Abelian gauge invariance, etc.) and are now prepared to give more precise mathematical formulations.
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Notes
- 1.
Fermi wrote down somewhat different formulas, but we modernized his reasoning a little bit. It is worth recalling at this point that the fermion field carries the canonical dimension \(m^{3/2}\) . Bearing in mind that \([\mathcal{L}] = m^4\), this gives the canonical dimension of Fermi’s constant, \([G_F] = m^{-2}\). Its numerical value was given in (5.21).
- 2.
The precise meaning of the latter statement is the following. The current (12.4) enters the classical Lagrangian (12.3). Consider, however, the corresponding quantum Hamiltonian. It is an operator acting in Fock space and involving the product of the current operators. The field operator \(\bar{e}(x)\) in the quantum counterpart of (12.4) may either create an electron state or annihilate a positron one (cf. the discussion on p. 119). In both cases, the electric charge of the state is changed by \(-1\).
- 3.
Actually, one can evaluate this distortion using the ITEP sum rules method, but this is a separate interesting story.
- 4.
This symmetry and its violation was already mentioned in a telegraphic style on p. 78. We ask the reader to forgive us that we decided not go into further details concerning this extremely interesting subject in our book.
- 5.
It is usually called the Pontecorvo–Maki–Nakagawa–Sakata or PMNS matrix.
- 6.
Now I’ve suddenly thought of Sophie who accompanied us during our visit of the Theoretical Physics Museum in Chap. 4. Is she still with us? Is she not too tired?
- 7.
Wishing to stay within the limits of the standard Latin alphabet, we use the same notation e for the charge of the scalar particle as for the electron charge. But they actually need not be the same.
- 8.
We have called it \(B_\mu \) rather than \(A_\mu \) because it is not an electromagnetic field, as we will see very soon.
- 9.
- 10.
The normalization factor \(1/\sqrt{g^2 + g'^2}\) was introduced in (12.34) and (12.36) to ensure the same standard form of the kinetic term for the fields \(Z_\mu \) and \(A_\mu \):
$$\begin{aligned} \mathcal{L}^\mathrm{kin}_{A, Z} \ =\ -\frac{1}{4} (\partial _\mu A_\nu - \partial _\nu A_\mu )^2 - \frac{1}{4} (\partial _\mu Z_\nu - \partial _\nu Z_\mu )^2 \, .{(12.37)} \end{aligned}$$ - 11.
Oops, we are not yet done with our entrées. After few sips of wine then.
- 12.
To be precise, we should have written \(\nu _e\) rather than just \(\nu \) in (12.41) and in all the formulas on the several subsequent pages. But your author is afraid to intoxicate his reader and get intoxicated himself by a plethora of indices. When we bring into consideration other generations, we will restore the accurate notation \(\nu _e, \nu _\mu , \nu _\tau \).
- 13.
Incidentally, the hypercharge of the scalar doublet is \(Y_\phi = 1\), and that means that the upper component of \(\phi \) is positively charged while the lower one is neutral.
- 14.
We now see that, for \(\mathcal{L}\) to carry zero net electric charge and thereby ensure charge conservation, one has to ascribe the positive charge to the combination \(\propto W^1_\mu - i W^2_\mu \), as was done in (12.33)!.
- 15.
As we know from Sect. 5.3, Yukawa’s theory suggested in 1935 was a phenomenological theory of the strong interaction between the nucleons and scalar meson particles (identified later with the \(\pi \) mesons). This seems to have nothing to do with the electroweak theory. But the Lagrangian of this interaction had the form \(\mathcal{L}_\mathrm{Yukawa} \propto \pi \bar{N} N \). Since then, any interaction of the form \( \phi \bar{f}f\) (\(\phi , f\) being scalar and spinor fields of any nature) is called a Yukawa interaction.
- 16.
SPS means Super Proton Synchrotron. It accelerates protons up to \(\sim \)450 GeV. It is still operating, now used as the injector for the Large Hadron Collider.
- 17.
“How come”, you may ask, “the strong interaction is stronger than the electroweak one and the strong widths (decay rates) should be larger than the electroweak ones!”
But everything is perceived in comparison. In our case, the comparison is with the resonance masses. The masses of W and Z are much larger than the characteristic hadron masses, while the ratios \(\Gamma _W/m_W\) and \(\Gamma _Z/m_Z\) are still small. Theoretically, these ratios are of order \(G_F m_{W, Z}^2\).
- 18.
The mass of H is below the threshold for creating two real Z bosons.
- 19.
ATLAS Experiment © 2012 CERN (License: CC-BY-SA-4.0).
- 20.
Then \(^4\)He starts to burn in its turn, but for the Sun at the present stage of its evolution that is not relevant.
- 21.
The cross section of the interaction of a neutrino with energy \(E \sim 1\) MeV with ordinary matter is at the level \(\sigma \sim 10^{-44} \, \mathrm{cm}^2\). The corresponding mean free path of neutrinos is \(\lambda \sim 1/(n \sigma )\), where n is the matter density. It is estimated at the level \(\lambda \sim 10^{20}\) cm \(\approx \) 100 light years.
- 22.
Bruno Pontecorvo, an Italian theorist, who spent the second part of his life in the Soviet Union, predicted this phenomenon back in 1957.
- 23.
Our rabbits, the muon neutrinos generated due to oscillations, would be eager to produce muons, but they do not have enough energy for that.
- 24.
This is the so-called QCD vacuum angle, the constant multiplying a possible structure \(\sim \) \(\varepsilon ^{\mu \nu \alpha \beta } G^a_{\mu \nu } G^a_{\alpha \beta }\) in the QCD Lagrangian. We did not discuss it before and will not do so in future. Experimentally, \(\theta _{QCD}\) is very close to zero (only the upper limit, \(|\theta _{QCD}| \lesssim 10^{-10}\) is known). Nobody understands why.
- 25.
The higgsino is a superpartner of the Higgs particle.
- 26.
From [M. Peskin, hep-ph/9705479, published in the Proceedings of the 1996 European School of Particle Physics.].
- 27.
- 28.
Besides the vector X bosons, the GUT counterparts of W and Z, there are scalar X bosons, the counterparts of the ordinary Higgs boson. We will meet the latter monsters again at the end of Chap. 14.
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Smilga, A. (2017). Theory of the Electroweak Interactions. In: Digestible Quantum Field Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-59922-9_12
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