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4

For a scalar field $\phi$, the most widely used convention, based on my experience, is to write the Lagrangian with kinetic and potential terms, followed by interactions like so, $$\mathcal{L}=\frac{1}{2}(\partial_\mu\phi)^2 - \frac{1}{2}m^2 \phi^2 - \sum_{n \geq 3} \frac{\lambda^n}{n!}\phi^n$$ where $\lambda_n$ are coupling constants. (We could not have a ...


6

The photon implements the electromagnetic force - it interacts with charged particles. Because the Higgs boson is neutral, it cannot (directly) interact with the photon. Another reason why the Higgs boson and the photon cannot (directly) interact is that interactions with the Higgs boson result in mass (after electroweak symmetry breaking). The photon is ...


2

If you are considering massless neutrinos there is no such a diagram since all interactions would preserve flavor. If you take instead massive neutrinos, you are probing lepton flavor violation within the SM since the new interaction with $\varphi$ respects flavor. It is thus very very small, being controlled by the neutrino masses. In turn, it is therefore ...


4

The $SU(2) \times U(1)$ electroweak gauge theory has 4 symmetry generators, of which the vacuum breaks only three -- corresponding to $W^+$, $W^-$ and $Z$. The vacuum is symmetric under the generator of electromagnetism, hence the photon cannot interact with the vacuum. Think of the vacuum as a medium -- if it was not symmetric under EM, then it could ...


4

The difference between the Higgs boson and the bosons of the three/four fundamental (depending whether you include gravity as a quantized theory or not) actions is that the latter are associated with gauge symmetries, while the Higgs plays a role in spontaneous symmetry breaking. Photons, W- and Z-bosons, gluons and gravitons arise from the requirement that ...


1

The easiest way to form $SU(2)$ singlets in the most general way is to use the techniques of Young Tableau. The method is discussed from a physicists perspective in many lecture notes online. One such example is given here. Using such method its easy to show that 2 lepton doublets make a singlet and a triplet under $SU(2)$, \begin{equation} 2 \otimes 2 = 3 ...


2

The interaction term disappears because when you integrate out the right handed neutrinos you insert in a linear combination of the other fields in their place. For example I expect that we would have something of the form, \begin{equation} G _{ ij} \sigma \bar{\nu} _L ^i \nu _R ^j \rightarrow G _{ij} G _{jk}\sigma ^2 \bar{\nu} _L ^i \nu _L ^k ...


5

Yes, massive particles such as W-bosons, Z-bosons, quarks, and leptons couple to the Higgs field via the cubic (Yukawa) interaction, so they may also exchange the virtual Higgs. Yes, because the virtual particle is massive, one gets the Yukawa potential that includes the exponential dumping with distance. This "Higgs force" is much less fundamental and ...



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