Detailed Proof of Double Soft Theorems Up to the Sub-Leading Order

Recently collaborated with Freddy Cachazo and Song He, I have been studying the simultaneous emission of two soft particles in several classes of amplitudes, where we found several double-soft theorems which mimic the soft theorems for emission of single soft particles in gravity and Yang-Mills amplitudes.

Specifically, we found two types of double-soft theorems for soft scalars. The first one, up to the sub-sub-leading order, applies to amplitudes in theories without any non-trivial flavor or color groups. Explicit examples include scalar amplitudes in a special Galileon theory with enhanced symmetries (first conjectured in arXiv:1412.4095, and its explicit form found in arXiv:1501.07600), the Dirac-Born-Infeld theory, and the Einstein-Maxwell-Scalar theory obtained from dimensional reduction of gravity. The second one, up to the sub-leading order, applies to theories with a U(N) flavor/color group such that partial amplitudes can be defined. These includes scalars amplitudes in the U(N) non-linear sigma model, as well as the Yang-Mills-Scalar obtained from dimensional reduction of Yang-Mills. There was also evidence that these theorems are valid when external states involve particles other than scalars. Apart from these, we also found a double-soft theorem for two soft photons, which applies to, e.g., photon amplitudes in the Born-Infeld theory. However, this third theorem is only at the leading order.

For all the amplitudes mentioned above, closed formulas using scattering equations are conjectured and checked in our previous work arXiv:1412.3479. A nice feature of this formulation is that it provides a very convenient tool in studying soft limit behaviors. Hence, assuming that these formulas are fully valid, we are able to use them to derive the above-mentioned soft theorems. This was done at the leading order for scalar emissions in our recent paper arXiv:1503.04816. The derivation at the sub-leading order is much more involved and deserves careful explanation. We decided to basically summarize the main results in the paper, while leaving this discussion to a supplementary note only for readers who are interested in hearing more details. The note can be accessed via the following link:

Supplementary Note for New Double Soft Emission Theorems: Proofs Up to the Sub-Leading Order, v1.

This supplementary note includes a detailed derivation of the two types of double-soft scalar theorems up to the sub-leading order, using the formulation based on scattering equations. In particular it provides an explanation to the origin of the sub-leading corrections inside S^{(0)}, which is a new feature of these theorems (as compared to the usual theorem for single soft particle). It also includes an explicit derivation for the double-soft photon theorem mentioned above.


Any comments and remarks are welcome, either by sending us e-mails (preferred) or replying directly to this post. The document here will be updated if we have a better version in future. — Freddy, Song and Ellis.


Last updated on: March 17, 2015

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