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A standard argument that the Higgs boson must exist is that without it, amplitudes in the Standard Model at the TeV scale violate unitarity. This is explained in section 21.2 of Peskin and Schroeder and also described in many sets of lecture notes.

However, when I look at these arguments, I don't see any apparent unitarity violation at all. Usually, they show that tree-level amplitudes become large at the TeV scale, so that the tree-level amplitude alone would violate unitarity bounds. But that's no big deal; who cares if the first term in a Taylor series is large, if the whole series sums to something reasonable? Some sources instead say that unitarity is not being violated but "tree-level" unitarity is, but it's unclear to me why that's an important principle that must be preserved.

What am I missing about this argument?

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The standard argument goes something like this:

  1. The tree-level amplitude alone without a Higgs boson violates unitarity.
  2. If you believe that your theory should be valid for all choices of your couplings (or at least for a neighborhood around zero) then different orders of perturbation theory should be independent and unitarity bounds should hold at every order.
  3. Hence, the violation of the unitarity bound at tree-level implies either a) you need to include extra degrees of freedom to restore unitarity (eg a Higgs boson) or b) perturbation theory is not valid for your theory, you have strongly-coupled interactions.

Note that this does not imply a Higgs boson, per se. If you take route (b) you can argue that the spontaneous symmetry breaking happens through strongly-coupled interactions in the UV, as do the Technicolor family of models. The apparent unitarity violation is just a sign that you are reaching scales where the strong interactions become important and you need to include them in your theory.

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  • $\begingroup$ Thanks for the answer! I guess step (2) is exactly what I'm stuck on. Can you spell it out in extreme detail? $\endgroup$ – knzhou Mar 23 '18 at 22:32
  • $\begingroup$ 1. If you believe the theory does not need a special value of the coupling constant to be well-defined, you can make the loop corrections arbitrarily small than the tree level answer by considering small coupling constant. This is a generic statement regarding tree-level truncations in QFT. 2. Besides, assuming your theory is well defined at TeV energies, you can consider an effective field theory for those scales i.e. the coupling is "renormalized" at that scale. This would mean that loop corrections are negligible, and the tree amplitude captures the full answer. $\endgroup$ – Siva Mar 24 '18 at 6:55
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    $\begingroup$ @Siva Sorry, I still don't follow. I agree that for tiny coupling constant we should have tree-level unitarity. And indeed we do! The smaller you make the coupling, the further away you push tree-level unitarity violation; it doesn't happen at all in the limit $g \to 0$. I don't see how that says anything about $g$ finite. $\endgroup$ – knzhou Mar 24 '18 at 10:09
  • $\begingroup$ @Siva Similarly I don't follow argument (2). The point is that we know the full theory is unitary. Basically, if I already know a Taylor series sums to $1$, so what if I can redefine the expansion variable so the first term is $2$? Even if it intuitively looks like the rest of the terms will be small, I already know that they sum to $-1$. $\endgroup$ – knzhou Mar 24 '18 at 10:12
  • $\begingroup$ @knzhou Imagine that the amplitude is a polynomial in the coupling constant. The only way for a polynomial to be zero for all values of its variable is for all of its coefficients to be zero. Likewise, the only way for the unitarity-violating amplitude to be zero for all values of the coupling is for the contributions at every order to be zero. The amplitude is not exactly like a polynomial, but the argument is pretty much the same. $\endgroup$ – Luke Pritchett Mar 24 '18 at 13:14
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User Luke Pritchett has already given a good answer. For completeness, I want to mention that there is an alternative way to think about this, one that I learnt very recently and that I found to be fascinating. I cannot help but recommend the book Quantum Gauge Theories: A True Ghost Story, by G. Scharf. It is short, concise and to the point. I read it a couple of days ago, and I loved every page of it.

In its first chapter, the book introduces free fields. Here, the author argues that the unphysical (longitudinal) polarisation of a spin $j=1$ field is in fact a gradient: $A_\mu=A_\mu^\mathrm{physical}+\partial_\mu \Lambda$. This determines the gauge transformation for free fields to be $A_\mu\to A_\mu+\partial_\mu \lambda$. So far, so good: this is just standard gauge theory.

The key point is that, as the author shows, the gauge invariance of free fields is in fact restrictive enough to determine the gauge transformation of interacting fields. For example, the author does not introduce the (ad-hoc) postulate that gauge fields are to transform according a Lie algebra: this is in fact a conclusion rather than an axiom.

Furthermore, the author does not introduce the (ad-hoc) Higgs mechanism, but rather derives it from the gauge invariance for free fields. All in all, in this book there are (almost) no unjustified ingredients: no covariant derivatives, no Lie Groups, no spontaneous symmetry breaking, etc. The only working principle is the gauge invariance of free fields, $A_\mu\to A_\mu+\partial_\mu \lambda$, which is perfectly well-motivated. Everything else is derived as a consequence of this simple principle.

Finally, and concerning OP's main question, the author argues that the theory is unitary if and only if it is gauge invariant, so this constitutes a proof that unitarity requires the Higgs field to exist.

If this is not enough for me to convince the reader to read the book, let me mention that the author does not introduce negative norm states (which is also a rather unconvincing aspect of gauge theories), but he doesn't introduce non-covariant (Coulomb, axial) gauges either. Moreover, the author explains from first principles how General Relativity emerges from a spin $j=2$ field, using only the gauge transformation for free fields (which is, as before, completely natural from the point of view of unphysical polarisations). Finally, the book follows the Epstein-Glaser formulation of QFT, so there are no divergences nor counter-terms anywhere.

Needless to say, it is impossible for me to explain how this works in practice: doing so would require for me to rewrite the whole book here. Let me nevertheless quote a paragraph from the introduction that I hope will pique the reader's interest.

In Chapter 4 the same method is applied to massive gauge fields. These are the incoming and outgoing free fields which appear in the expansion of the $S$-matrix (corresponding to the $W^\pm$- and $Z$-bosons in the electroweak theory). We have no generation of mass by spontaneous symmetry breaking; instead, perturbative gauge invariance does the job. It forces us to introduce an unphysical (Goldstone-like) and physical (Higgs) scalar fields and determines their coupling. For example, the so-called Higgs potential need not be put in by hand but follows naturally from third-order gauge invariance.

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  • $\begingroup$ Yes, causal perturbation theory (the subject of Scharf's book) is very nice, and derives much of the older perturbative renormalization stuff on a sound basis. I have written a short introduction to causal perturbation theory, to motivate reading more. See physicsforums.com/insights/causal-perturbation-theory $\endgroup$ – Arnold Neumaier Mar 28 '18 at 17:35

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