The $\lambda \phi^4$ quantum filed theory is the textbook example (which probably cannot be constructed nonperturbatively; I'm purely interested in perturbation theory). However, usually one treats massive case. I suppose that, in the massless case 1PI and Green function should be also well defined at least outside the mass shell (?). The infrared problems should appear on the level of the S-matrix elements and cross sections. The standard way to solve those problems in case of QED is to consider only inclusive cross-section. They are analysed thoroughly in the context of QED in several textbooks.

Firstly, I would like to make sure whether similar (infrared) problems as in QED appear also in $\lambda \phi^4$ theory. Can I find the analysis of these problems in some book or paper? I'm also interested in actual computations which shows how to deal with those problems. I suppose that one has to consider at lease the second order perturbation theory to encounter IR problems. Am I right?


1 Answer 1


The QFT for the scalar is considered to be massive for a very good reason: it is infinitely unlikely for the mass to vanish.

There is no symmetry principle that would protect the scalar field from acquiring a generic mass. (The gauge symmetry is the principle that protects the masslessness of the photon but the scalar fields can't sacrifice to lose components by a gauge symmetry because nothing would be left.) This is also the reason behind the "hierarchy problem", i.e. why the Higgs is so light. It could be expected to have mass comparable to the highest scales in physics, like the GUT scale or the Planck scale. The real-world Higgs is arguably lighter than most particle physicists would "predict" based on pure thought but it's surely not massless.

This non-protection isn't just a matter of philosophy or shaky speculations about naturalness. The mass is actually a parameter that depends on the RG scale, too. So if it's zero at one scale, it doesn't imply that it's zero at another scale! And indeed, even the real-world Higgs boson may converge to the zero mass at some high enough energy scale. In all these cases, the mathematical machinery that you need is pretty much equivalent to the mathematics needed to analyze the general scalar with the general mass and quartic interactions.

One may construct theories where the vanishing mass is protected by some principle, e.g. supersymmetry. In unbroken $N=1$ SUSY, the mass of the scalar is linked to the mass of the partner fermion. There can still be a quartic (and by SUSY related, also Yukawa) interaction. However, the masslessness will only be preserved if the field (both boson and its fermionic superpartner) will carry a nonzero conserved charge analogous to the electric charge. Then the fermion is a 2-component charged Weyl fermion that must stay massless. But if these particles carry a conserved charge, then they cannot be easily produced as a "bonus" to processes, in analogy with the soft photons. So the directly analogous problem of the IR divergences that we know from photons is absent.

You may also consider scalar fields whose whole potential vanishes, $V=0$, including the quartic term. In $N=2$ supersymmetry and higher, there are symmetry reasons why the scalar fields may be potential-less. In that case, they are moduli and "all" values of the vacuum expectation value are as good as any other. That's on top of the new long-range forces that these exactly massless fields mediate.

But the "moduli" character of these fields makes the situation different from photons because with photons, there is a preferred "photon-less" vacuum state. If there are exactly massless scalar fields i.e. moduli, the vacuum isn't unique but labeled by the vev.

There are many issues but your general expectation that all these things for the scalar field are closely analogous to the IR problems with the photons is an incorrect expectation.

  • $\begingroup$ I agree that there is no reason why $\lambda \phi^4$ theory should be massless and certainly, even in massless case, there should be added a mass counterterm to the lagrangian. However, in principle, it should be possible to impose renormalization condition corresponding to zero mass of particle. Does it lead to serious IR problems? Do the problems appear on the level of 1PI functions or only S-matrix elements. $\endgroup$
    – user72829
    Commented Jul 1, 2014 at 19:53
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    $\begingroup$ Hi, I already wrote it, but if $m=0$ is imposed at one scale, it will be violated at all other scales, anyway. The $m=0$ condition randomly occurs at one scale where the $\phi$ starts to get a vev because there is one minimum on one side and two minima on the other side. Otherwise, nothing special happens at that scale and there are no IR divergences analogous to those with the soft photons. Have you read my answer at all? $\endgroup$ Commented Jul 1, 2014 at 20:14
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    $\begingroup$ The renormalization group equations act on the parameters of the theory (mass, coupling constants,...) defined in some energy scale dependent renormalization scheme (for example MS). In such schemes mass is scale dependent but this is not the physical mass of the particle (the pole of the propagator). Is it possible to use on shell renormalization scheme and impose condition m=0 in $\lambda \phi^4$ theory. $\endgroup$
    – user72829
    Commented Jul 1, 2014 at 20:23

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