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

Talks at Oxford & Cambridge

In the past couple of days I have been visiting Oxford and then Cambridge, during which I gave one seminar in each place, and an extra talk in an informal journal club in Oxford which was meant to be a pedagogical introduction to the formulation of tree-level amplitudes based on scattering equations. Both seminar talks focus on our recent two papers (arXiv:1409.8256, arXiv:1412.3479) which extended the application of the above mentioned formulation to various theories of massless particles.

Some highlights are that, starting with only two very simple objects (functions of kinematics data and a set of variables parametrizing an auxiliary Riemann sphere with marked points) together with three operations on them, we managed to find out closed formulas for amplitudes in Yang-Mills coupled to gravity (including the well-studied pure gluon amplitudes and pure graviton amplitudes), Einstein-Maxwell, Dirac-Born-Infeld, U(N) non-linear sigma model, and a very special type of the Galileon theory. Furthermore, these new formulas indicates very amusing similarity and connections among amplitudes in these different theories, some of them relatively well-understood from the normal point of view from Kaluza-Klein while others remaining more or less unclear (as of the current status).

The slides for the Oxford talk can be found here.

Due to some change of plan, I switched to a blackboard talk in Cambridge. Most of the contents are similar to that in Oxford, except for a slightly extended review on the general formulation, which can be found here.

In the same file, there is also an additional page related to an update on the Galileon theory that I discussed in the Cambridge talk. In short, the Galileon theory that we identified with one of the new formulas turns out to be a special class of Galileon theories that possess an enhanced symmetry. More discussions on these can be found in paper by Hinterbichler and Joyce (arXiv:1501.07600) late last week.

Soft Limits and Factorizations of the Pfaffian Formula

Very recently F. Cachazo, S. He and I have proposed a new formula (arXiv:1307.2199) for the complete tree-level S-matrix of Yang-Mills and gravity in any dimension, which is based on an integration of the Pfaffian of a skew-symmetric matrix depending on momentum and polarization vectors, over the moduli space of n punctures on a Riemann sphere.

This post is to supplement the paper by providing a detailed proof that our formula satisfies correct soft limits and factorizations both for gluons and gravitons, the most up-to-date notes of which can be found in the following link:

Scattering of Massless Particles in Arbitrary Dimension: Soft Limits and Factorization, v2

Notice: 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.

Updates in Version 2:

  1. Minor mistakes corrected;
  2. A slightly different convention for the formula has been chosen (which however agrees with that in our previous work arXiv:1306.6575), so that it exactly matches the results from the standard spin-helicity formalism when restricting to 4-dimensions;
  3. Detailed discussions on the behavior of the integration measure by Faddeev-Popov method has been added.

Previous versions can still be found here:

Scattering of Massless Particles in Arbitrary Dimension: Soft Limits and Factorization, v1

Last updated on: 29/08/2013.


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