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This enabled theorists to describe everything in terms of quantum field theory. According to QED, a vacuum is filled with electron-positron fields. Real electron-positron pairs are created when energetic photons, represented by the electromagnetic field, interact with these fields. During the s it became clear that, as it stood, QED gave the wrong answers for quite simple problems. For example, the theory said that the emission and reabsorption of the same photon would occur with an infinite probability.

This led in turn to infinities occurring in many situations; even the mass of a single electron was infinite according to QED because, on the timescales of the uncertainty principle, the electron could continuously emit and absorb virtual photons.


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It was not until the late s that a number of theorists working independently resolved the problems with QED. Basically, renormalization acknowledges all possible infinities and then allows the positive infinities to cancel the negative ones; the mass and charge of the electron, which are infinite in theory, are then defined to be their measured values. Once these steps have been taken, QED works beautifully. It is the most accurate quantum field theory scientists have at their disposal.

Figure 4. Basic vertices of the electromagnetic interaction a , strong interaction b , and weak interaction c. From left to right: examples of annihilation, pair production, absorption, and emission. If we now look at the basic vertex of the strong interaction see figure 4 b , we notice that it looks very similar to the vertex of the electromagnetic interaction.

For example, an anti-quark and a corresponding quark transforming into a gluon can be described as an annihilation process. In the reverse reading direction, this process can also be interpreted as pair creation, where a gluon transforms into a quark and an associated anti-quark. Additionally, by rotating the vertex further, we obtain the Feynman diagrams for gluon absorption and gluon emission. Last but not least, the transformation processes of the weak interaction can be illustrated in a similar way as well figure 4 c. The weak interaction differs from the electromagnetic and the strong interactions in that it transforms one matter particle into another, for example an electron into an electron-neutrino and vice versa.

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We consider processes of the weak interaction involving a W boson to be particularly interesting for introduction in the classroom. In many physics textbooks this process is called 'beta-minus decay' or in the case of : 'beta-plus decay'. The emitted electron or positron is then introduced as 'beta radiation'.

Here, we recommend using the term 'transformation' instead of 'decay', as this more accurately describes the physical process. In addition, doing so can prevent the triggering of misconceptions of the electron or positron as 'fragments' of the original neutron or proton.

Instead of using the word 'beta radiation', we also recommend referring directly to emitted electrons or positrons to focus more strongly on the particle aspect of the transformation process. Figure 5.

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Note: the weak interaction allows only particle transformations between two specific elementary particles, so-called 'weak isospin doublets'. Prominent examples are the electron-neutrino and electron doublet , and the up-quark and down-quark doublet. Overall, this second term of the Lagrangian is of special importance for our everyday life, and therefore merits discussion in the physics classroom.

Indeed, apart from gravity, all physical phenomena can be described on a particle level by the basic vertices of the strong, weak, and electromagnetic interaction. Furthermore, given that the strong and weak interactions play a minor role in high-school curricula, almost all physical phenomena can be described using the basic vertex of the electromagnetic interaction figure 4 a. However, as discussed above, once this basic vertex is introduced, it is possible to draw connections to the basic vertices of the strong interaction figure 4 b and the weak interaction figure 4 c as well.

This term represents the 'hermitian conjugate' of term 2. The hermitian conjugate is necessary if arithmetic operations on matrices produce complex-valued 'disturbances'.

By adding , such disturbances cancel each other out, thus the Lagrangian remains a real-valued function. Actually, the addition of is not required for term 2, since term 2 is self-adjoint. Therefore, this term is often omitted. Anyway, should not be taken literally. Theorists often use it as a reminder: 'If a term changes when conjugating it, then add! If nothing changes because it is self-adjoint , then add nothing'. This term does not have a physical meaning, but it ensures that the theory is sound.

Tip: we recommend the CERN-wide interpretation of term After all, the Lagrangian is printed on a coffee mug for a good reason. It is therefore advisable to take a break at half time with a mug of coffee. Afterwards, it will be easier to enjoy the full beauty of terms 4 to 7, which we explain next. This term describes how matter particles couple to the Brout—Englert—Higgs field and thereby obtain mass.

The entries of the Yukawa matrix y ij represent the coupling parameters to the Brout—Englert—Higgs field, and hence are directly related to the mass of the particle in question. These parameters are not predicted by theory, but have been determined experimentally. Parts of this term still cause physicists headaches: it is still not clear why neutrinos are so much lighter than other elementary particles, in other words, why they couple only very weakly to the BEH field. In addition, it is still not possible to derive the entries of the Yukawa matrix in a theoretically predictive way.

It is known that particles with high mass, in other words with a strong coupling to the Brout—Englert—Higgs field, also couple strongly to the Higgs boson. This is currently being verified experimentally at the LHC, where Higgs bosons are produced in particle collisions. Depending on their mass, i.

Particle physics

This is because the coupling parameter, which describes the coupling to the Higgs boson, is simply the mass of the particle itself. The Higgs boson is thus more likely to be transformed into pairs of relatively more massive particles and anti-particles.

Measurements by the ATLAS detector show, for example, evidence of the direct coupling of the Higgs boson to tauons [ 5 ], see figure 6. See term 3, but here this term is really necessary, since term 4 is not self-adjoint. While term 4 describes the interaction between a Higgs particle and matter particles, term 5, the hermitian conjugate of term 4, describes the same interaction, but with antimatter particles.

Depending on the interpretation, however, we recommend at least one more mug of hot coffee. This term describes how the interaction particles couple to the BEH field. This applies only to the interaction particles of the weak interaction, which thereby obtain their mass. This has been proven experimentally, because couplings of W bosons to Higgs bosons figure 7 have already been verified. Photons do not obtain mass by the Higgs mechanism, whereas gluons are massless because they do not couple to the Brout—Englert—Higgs field.

Figure 7. One example of a Feynman diagram that is encoded in. Figure 8.

Standard model key points

Possible vector-boson fusion process from two colliding protons. The two W bosons transform into an electrically neutral Higgs boson. This term describes the potential of the BEH field. Contrary to the other quantum fields, this potential does not have a single minimum at zero but has an infinite set of different minima. This makes the Brout—Englert—Higgs field fundamentally different and leads to spontaneous symmetry-breaking when choosing one of the minima.

As discussed for terms 4 and 6, matter particles and interaction particles couple differently to this 'background field' and thus obtain their respective masses. Term 7 also describes how Higgs bosons couple to each other see figure 9. Figure 9. Diagrams of Higgs self-interaction 3-Higgs vertex and 4-Higgs vertex that originate from. Our experience at CERN is that both high school students and teachers are greatly fascinated by the Lagrangian.

Hence, introducing it in the classroom can contribute positively when discussing particle physics.


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However, due to the complex level of mathematical formalism used in the Lagrangian, it is probably not favourable to aim for a complete, in-depth discussion. Some of these theories lean against the Standard Model's solution for adding mass to otherwise mass-less particles. Others might use entirely different schemes.

One way to tell different models apart is by looking at how the models vary in their predictions. Common I think to all Higgs look-a-likes is that they are unstable and decay fast in various ways. We call the probability for a specific decay the decay-mode branching-ratio.

For example, the results shown from ATLAS and CMS both rely on two main decay-modes: one in which the Higgs decays into two photons and another in which the Higgs decays to two Z bosons that each again decay to either two muons or two electrons. For the standard model and supersymmetry SUSY some of my friends and I published a small report with the predicted branching ratios. Let's compare:. For a few reasons, the figures are not directly comparable. In the plot, one of them tan beta is fixed at 10, but this value could be something else in reality. Luckily just the mass of one of the Higgs together with tan beta is enough to estimate the mass of the rest.

So, as we see our measured particle decay to two photons and two Zs, it is already in better agreement with the SM Higgs than the SUSY one in the figure. Another way to measure the difference is simply to look at the overall production cross-section, or how many times per proton-proton collision do we find a Higgs particle with any decay-mode?