# Interactions with high helicities particles

As it can be shown, there are no interacting helicity-3 (and higher) particles (i.e., massless spin-3 or higher particles) in soft limit (small momentums of emitting particles of given helicity). Сan this result tell us something about high-momentum limit (i.e., the principal possibility of interaction of corresponding particles with other fields in high-energy limit)?

The proof of the statement above can be done in the following way.

First we assume an arbitrary process with an external lines which correspond to particles which can interact with our massless particles ("charged" particles). Then we modify one of the external lines by adding an external line of massless particle, after which we take the soft limit and sum over all possible ways of emission of this particle (external line of this particle may begin from each external line of charged particles). As it can be shown, the summary amplitude is given in a form $$M = M_{0} \sum_{n} M^{\mu_{1}...\mu_{2m}}_{n}(p_{n}, q)\varepsilon_{\mu_{1}...\mu_{2m}}(q).$$ Here $M_{0}$ refers to the amplitude without emission of our massless particles, while the other corresponds to the "emission" part: $$M^{\mu_{1}...\mu_{2m}}(p_{n},q) = f_{n}\eta_{n}\frac{p_{n}^{\mu_{1}}...p_{n}^{\mu_{2m}}}{(p_{n} \cdot q) - i\varepsilon},$$ $\eta_{n} = \pm 1$ ($+1$ for emission from outgoing particle and $-1$ for emission from ingoing particle), $f_{n}$ is the coupling constant of interaction of $n$-th charged particle with our massless particle, $\varepsilon_{\mu_{1}...\mu_{2m}}(q)$ is only the polarization tensor of massless particle with momentum $q$.

The requirement of lorentz-invariance of process leads us to the statement that $$\tag 1 M^{\mu_{1}...\mu_{2m} }q_{\mu_{1}} = 0 \Rightarrow \sum_{n} f_{n}\eta_{n}p^{\mu_{1}}...p^{\mu_{2m - 1}} = 0.$$ there isn't non-trivial conserved (like summary momentum or summary charge) object rank $l \geqslant 3$ built only from momentums, so the only possible way to satisfy $(1)$ without prohibiting on all non-trivial processes is to set all of $f_{n}$ to zero.

Analogous thinking for helicity 1, 2 cases leads us to the proof of charge conservation and the equivalence principle without an additional words about gauge invariances of correspond interaction theories.

The details can be found in Weinberg QFT (chapter about infrared photons).

• How can you show that such particles cannot exist under these conditions? – HDE 226868 Aug 4 '14 at 22:27
• @HDE226868 : I have given proof and the link into the question body. – Andrew McAddams Aug 4 '14 at 23:08
• You have the Coleman-Mandula theorem too. – Trimok Aug 5 '14 at 9:47