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The Higgs Mechanism with Diagrams: a didactic approach
by Jochem Kip, Ronald Kleiss
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Submission summary
| Authors (as registered SciPost users): | Jochem Kip |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2404.08329v2 (pdf) |
| Date submitted: | 2024-07-09 15:32 |
| Submitted by: | Kip, Jochem |
| Submitted to: | SciPost Physics Lecture Notes |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We present a pedagogical treatment of the electroweak Higgs mechanism based solely on Feynman diagrams and S-matrix elements, without recourse to (gauge) symmetry arguments. Throughout, the emphasis is on Feynman rules and the Schwinger-Dyson equations; it is pointed out that particular care is needed in the treatment of tadpole diagrams and their symmetry factors.
List of changes
Added a paragraph in Section 1.1 and a comment in Section 2.1 in order to clarify the nature of tachyonic instability.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2024-7-24 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2404.08329v2, delivered 2024-07-24, doi: 10.21468/SciPost.Report.9456
Report
Thanks to the authors for the clarifying remarks. I'm not sure that I have entirely grasped what they intended to say however. For instance, in sct. 2.1 they now state that "self-interacting tachyons are physically acceptable [$\cdots$]", but that "the eventual particle spectrum is perfectly non-tachyonic". I take it that this means that being "physically acceptable" is not the same as being "physically real" (i.e., that which makes up the eventual particle spectrum), although it is unclear in what precise sense physical acceptability should then be viewed. To demand that this point be cleared up would be a little unfair on the part of the authors however, since similar issues crop up in the standard approach (where massless goldstone bosons aren't present in the eventual particle spectrum, but do appear at intermediate stages and are often argued to leave their mark on physical observables). Yet, there is a crucial difference here, which goes back to the main point earlier. In the usual approach, the "equivalence" at high energies between longitudinallly polarized gauge bosons and massless goldstone bosons - which re-enter the particle spectrum upon turning off the gauge couplings - refers (in the standard language) to perturbative expansions around two different ground states. In the current approach, the equivalence (established in the article) between longitudinally polarized gauge bosons and scalar tachyons however does $not$ refer (in the standard language) to perturbative expansions around different $ground$ states, for the reason stated earlier. It is therefore not clear in what sense there can be an equivalence (of a different kind) between the two $approaches$ (i.e., diagrammatic and canonical). Put somewhat differently: in the usual formulation it is reasonable to expect Goldstone modes to re-enter the spectrum at high energies - corresponding to a different kind of SSB - $i\!f$ it can be justified that the gauge couplings tend to zero in this regime. It seems that the analogous statement for scalar tachyons, however, cannot be made, since, as acknowledged by the authors, the vanishing tadpole corresponds to a physically unacceptable situation.
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Author: Jochem Kip on 2024-07-26 [id 4661]
(in reply to Report 1 on 2024-07-24)We thank the referee for their comments. We would like to hopefully clarify a few points in our reply before submitting a new version. We first would like to emphasise that our diagrammatic approach is equivalent to the Lagrangian (or canonical) approach in the sense that it deals with the exact same physics and energy scales and results in the same physical spectrum, it is only the method in which it is obtained that differs. In short, the Mexican hat potential as used in the Lagrangian approach SSB is constructed via a negative mass term and quartic self-interaction. We take these two interactions as a starting point and interpret them as Feynman rules. Since the mass term is negative in order to construct the Mexican hat potential we are, for now, forced to view the particles partaking in these interactions as tachyons. However, these vanish from the spectrum when we perform SSB. In the Lagrangian approach this is of course done by expanding around the new minimum by substituting $\phi \to v + h$. In the diagrammatic approach we cannot substitute terms in the Lagrangian, and thus perform a Dyson summation in which we include the vev as a tadpole interaction. Thus the Dyson summation is the diagrammatic way of performing a field substitution and expanding around the new minimum. We thus expand around the exact same field minimum as in the Lagrangian approach.
We would moreover like to clarify that our text explicitly does not deal with energy scales or the energy dependence of various parameters, simply because it is not needed. Just as many introductory courses and books on the SM simply write down the relevant Lagrangian and derive the resulting spectrum, we do the same but via diagrammatics. It is in this sense the approaches are equivalent and must therefore not be confused with equivalences such as the Goldstone equivalence theorem, which deals with specific energy regimes.
We want to stress that, in our treatment, particle spectra are derived from scattering Green's functions, rather than being posited a-priori. In particular, we do not introduce 'tachyons' as particles but rather as defined by propagators with the `wrong' mass. Since the Schwinger-Dyson equations are satisfied, this is perfectly allowed; and at the end a physical, non-tachyonic spectrum is obtained.
Lastly, we would like to clarify that we use ‘physically acceptable’ in the sense that such parameters result in an acceptable spectrum; just as in the Lagrangian approach where one chooses an a priori negative mass term for the Higgs doublet, which would make it tachyonic. However, because SSB results in either massless or massive scalars, these parameter choices are ‘physically acceptable’ because the resulting spectrum is free of any tachyonic issues.
Carlo Beenakker on 2024-08-13 [id 4693]
(in reply to Jochem Kip on 2024-07-26 [id 4661])Response by the referee [inserted by the editor for administrative reasons]
Unfortunately the second response of the authors has only added to the confusion and I do feel some further clarification is therefore indeed needed.
In the second part of the response it is stated that the text ``explicitly does not deal with energy scales or energy dependence of various parameters''. The equivalence theorem proven in the article (the counterpart in the diagrammatic approach of the so-called Goldstone boson equivalence theorem of the Lagrangian based approach) obviously does however!
That is, the equivalence between longitudinally polarized massive vector modes and scalar tachyons so needed in order to prove unitarity of the model (last paragraphs sections 1.2 and 3.4) only applies asymptotically, i.e. at large energy scales. Otherwise one would be forced to conclude that scalar tachyons are part of the physical spectrum - as obviously longitudinally polarized massive vector modes are.
The statement that the two approaches (Lagrangian and diagrammatic) deal with the exact same physics also cannot be correct, as one approach relies on massless Goldstone modes appearing at high energies, while the other relies on tachyons appearing in this regime.
But above all, and as already mentioned before, the two approaches are inequivalent in the sense that the Lagrangian-based approach relies on expanding around a vacuum state at high energies, whereas the tachyon-based approach does not (tachyonic instability caused by expanding around a local energy maximum). And, as mentioned, perturbative expansions are needed in both cases for the respective equivalence theorems (required for unitarity) to work.
Anonymous on 2024-09-03 [id 4733]
(in reply to Carlo Beenakker on 2024-08-13 [id 4693])Dear referee,
We would first like to emphasise that the statement of identity (53) is not the Goldstone equivalence theorem. The Goldstone equivalence theorem indeed states that in the asymptotic limit of increasing energy the longitudinal polarisation of a massive vector boson is equivalent to a scalar goldstone boson. However, identity (53) states not that, but rather that the scalar field provides an additional degree of freedom to the (now) massive vector boson, which is true at all energy scales (see e.g. Peskin and Schroeder Ch 20.1). This is markedly different from the Goldstone equivalence theorem, since this additional degree of freedom need not be the longitudinal degree of freedom. In the Lagrangian approach the scalar degrees of freedom are removed (or rather hidden) by choosing the unitary gauge. The removal of the scalar degrees of freedom can now of course explicitly be seen in identity (53). The fact that this is equivalent to using the unitary gauge can be seen by e.g identity (58), which is exactly the unitary-gauge propagator.
Regarding tachyonic instability, we would like to emphasise that we do not treat tachyons as particles, but only as a bare propagator, as stated in the beginning of Section 2.1. This is of course markedly different to the Lagrangian approach, where one considers a free particle that is subsequently dressed with interactions. In such a scenario tachyons of course cannot be used as a starting point for a perturbative expansion. However, in the diagrammatic approach we first consider the two-point Greens’ function, which has an a-priori tachyonic propagator, but after evaluating the Schwinger-Dyson equation the resulting spectrum consists of a single massive scalar boson and a number of massless scalar fields. It is only at this point that we can perform a particle interpretation. This is then equivalent to an expansion around the local energy minimum in the Lagrangian approach, seen by the fact that both approaches result in the exact same spectrum.