Width of mass spread for electron infraparticles

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According to the work of Buchholz and others, charged particles fit conceptually closer to the infraparticle notions than to the traditional Wigner classification of particles with well-defined mass eigenvalues and gaps.

But wouldn't such a mass spread of the 511 KeV line have been detected already? What is the experimental width expected from the electron if it was indeed an infraparticle, as classically understood?

asked Jun 20, 2014

The e-p annihilation line is broadened by soft-photons, when an electron and positron come to annihilate in a collision, they always emit radiation along the way in the form of soft-photons, and these make the definition of mass for an electron difficult. The infraparticle business refers to the soft-photons. Once you form positronium, it's net neutral, and it's just an unstable particle, with a well-defined photon annihilation spectrum. I assume the lines you are talking about are positronium lines. I am interested in the technical details of Buchholz, I don't understand the theory, it seems interesting, but your criticism seems to be naive.

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I believe that a honest answer to your (final) question is "nobody knows". I'll explain why.

One way to understand the notion of "infraparticle" is to look back at the derivation of the LSZ reduction formula for the S-matrix elements. Strictly speaking, it needs the assumption of a mass gap in the energy-momentum spectrum - namely, the lowest part of the energy-momentum spectrum away from zero is an isolated mass hyperboloid. This enters the derivation of the formula through the spacelike decay of truncated correlation functions, which in the presence of a mass gap are exponential (i.e. Yukawa-like). The LSZ reduction formula (together with Haag's no-go theorem about the non-existence of the interaction picture in QFT) also implies a very important fact: elementary particles are only an asymptotic concept, they only appear at the large-time limit which defines the S-matrix.

If there is no mass gap, one only gets a power-law (i.e. Coulomb-type) decay, which is insufficient to get a definite limit as the "in" time and the "out" time go respectively to $-/+\infty$. In other words, the S-matrix does not exist in the usual sense in the presence of long-range interactions, for they do not approach zero sufficiently fast as time goes to infinity. Recall that this is also true in non-relativistic quantum mechanics - the Coulomb potential doesn't have a S-matrix in the usual sense. Therefore, in both cases one has to modify the asymptotic limit used to define scattering. In non-relativistic quantum mechanics, this has been done by Dollard for the Coulomb potential and by other people (Alsholm-Kato, Bulsaev-Matveev, Hörmander, etc.) for general long-range potentials. In QFT, there is the heuristic "zero-recoil" recipe proposed by Bloch and Nordsieck for QED: "hard" particles, i.e. with energy above a certain "detector threshold", do not "recoil" when emitting any given number of "soft" (massless) particles with total energy below that threshold. In particular, the energy of massive particles is always bounded below by their (positive) rest mass, so they are usually considered "hard". The effect of this approximation is that "soft" particles see "hard" ones as classical sources, hence they behave as in a coherent state as time evolves. The limit of zero threshold can be performed in the formula for the resulting "inclusive" cross-section. So far, so good - the detector we use to determine the cross-section of th process cannot see the "soft" particles, which are emitted at an indefinite "number" - forming the so-called "infrared clouds" of the "hard" particles emitting them, which are therefore called "infraparticles". Scattering processes which are "essentially insensitive" to the additional emission / absorption of any (finite but undetermined) number of "soft" particles are called "infrared safe". The impossibility of detecting a mass spread for a realistic electron around 511 keV is related to the fact that the energy threshold for "infrared clouds" can be made arbitrarily small in the course of the Bloch-Nordsieck approximation.

Seems good, right? The problem is that nobody knows which is the exact effect of the Bloch-Nordsieck approximation at finite threshold on the energy-momentum spectrum of the theory, let alone of the zero-threshold limit. Put differently, what are we really calculating when we follow this recipe? Does it really have anything to do with a notion (however asymptotic it may be) of particles? Therefore, from a more rigorous viewpoint, one should rethink the notion of "particle" one wants to extract from and compare with collision experiments. A more "infrared safe" notion of a particle is related to what it means to "detect a particle", hence to what is the mathematical definition of a "particle detector". Such a notion enters implicitly in the very definition of a collision cross-section.

An abstract way to define a detector is any observation device that removes some energy of the state being observed (since "detected particles" are absorbed by it) in some Lorentz frame. As already implicit in the reasoning behind the Bloch-Nordsieck approximation, realistic detectors should have a lower energy threshold in this frame and are therefore insensitive to "infrared clouds". Moreover, ideally a detector as defined above should also have an upper energy threshhold in any Lorentz frame, but the time-energy uncertainty principle prevents us to achieve that with a finitely extended detector during an experiment of finite laboratory lifetime. Phrased differently, we also cannot distinguish in practice "true" mass eigenstates from mass resonances. What we do is to ascertain "particles" to sufficiently sharp resonance peaks (the "clicks" in the detectors).

If the theory has a mass gap, one recovers the usual LSZ notion of particle states through this reasoning in the (suitably defined) limit of infinite observation time intervals. If not, one cannot really say one ends up with "particles" (in the LSZ sense) in this limit, since in this case the LSZ S-matrix does not exist. The inclusive cross-sections defined with the aid of our detectors, on the other hand, do exist. The corresponding "clicks" in our detector arrangement are then interpreted as "infraparticles". The relation of infraparticles to features of the energy-momentum spectrum, however, is much more elusive than in the case of LSZ particles. Another problem is how to relate this framework to the (so far quite successful) Bloch-Nordsieck-type approximations - can the former explain why the latter work? More precisely, can they explain what we are really doing with such approximations? Such questions remain so far unanswered.

Just a small but important proviso to the last paragraph: for processes where the energy of "hard" particles is below the pair creation threshold, these particles can be treated as non-relativistic. In this regime, the Bloch-Nordsieck approximation can be brought into rigorous control and shown to be correct. This was done by T. Chen, J. Fröhlich and A. Pizzo in the following papers:

• T. Chen, J. Fröhlich, A. Pizzo, "Infraparticle Scattering States in Non-Relativistic QED: I. The Bloch-Nordsieck Paradigm". Commun. Math. Phys. 294 (2010) 761-825, http://arxiv.org/abs/0709.2493;
• T. Chen, J. Fröhlich, A. Pizzo, "Infraparticle Scattering States in Non-Relativistic QED: II. Mass Shell Properties". J. Math. Phys. 50 (2009) 012103, http://arxiv.org/abs/0709.2812.
answered Jun 21, 2014 by (580 points)
edited Jun 22, 2014

Really nice summary, I appreciate it. From the experimental side of things, what would you say is the roadblock to detecting these soft photons? I know that there has been some work on single photon detectors in the GHz range, but how 'soft' could be the energies that could be detected on, say an electron collider?

I'm not a particle physics experimentalist, so I have no idea of the current state of the art of photon detectors used in collider experiments with respect to their lower energy thresholds. My guess is that it's a matter of interest, not a technical one - since the recoil due to soft photon emission is essentially negligible in infrared safe processes, people doing collider experiments simply don't care about it, specially in the very high energy processes which are the main focus of elementary particle physics - for those, infrared safety is a key feature that makes the comparison of theory with experiment practically feasible, since multiple emission of soft particles is too complicated a process to be fully taken into account in calculations. That's an important advantage of the Bloch-Nordsieck approximation, in spite of its conceptually unsatisfying status as exposed in my answer.

So arguably, we observe this soft photon recoil more easily in a lower energy collider - say, one that make electron-electron collisions of just a few KeV, rather than on a LEC

Possibly, yes. However, I believe that, even in that energy range, the cross-section for the production of (say) bremsstrahlung photons with $~10^{-5}$ eV should be extremely small.

@Ribeiro: My electronium model answers these questions. If you do not like it, look at the atomic case, it is as well instructive.

@Vladimir: it seems to me that it's not a matter of model building. Charles's questions concern the experimental verification of the concept of infraparticle. My answer was aimed at how to properly define this concept starting from first principles, so we know better what we are really computing in practice and, therefore, what we are comparing to experiment. For electrons in the non-relativistic regime (i.e. with energies below the pair creation threshold), the Bloch-Nordsieck approximation is actually much better understood - I'll add references to that in my answer.

@Ribeiro: On the contrary, it is a matter of a model building. The current model has severe drawbacks, which have provoked so many researches and studies.

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