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The Higgs Mechanism with Diagrams: a didactic approach
by Jochem Kip, Ronald Kleiss
Submission summary
Authors (as registered SciPost users):  Jochem Kip 
Submission information  

Preprint Link:  https://arxiv.org/abs/2404.08329v2 (pdf) 
Date submitted:  20240709 15:32 
Submitted by:  Kip, Jochem 
Submitted to:  SciPost Physics Lecture Notes 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We present a pedagogical treatment of the electroweak Higgs mechanism based solely on Feynman diagrams and Smatrix elements, without recourse to (gauge) symmetry arguments. Throughout, the emphasis is on Feynman rules and the SchwingerDyson 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.
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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 "selfinteracting tachyons are physically acceptable [$\cdots$]", but that "the eventual particle spectrum is perfectly nontachyonic". 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 reenter 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 reenter 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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Ask for minor revision
Author: Jochem Kip on 20240726 [id 4661]
(in reply to Report 1 on 20240724)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 selfinteraction. 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 apriori. In particular, we do not introduce 'tachyons' as particles but rather as defined by propagators with the `wrong' mass. Since the SchwingerDyson equations are satisfied, this is perfectly allowed; and at the end a physical, nontachyonic 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.