Double copy

The {\it double copy} is a remarkable relationship between our theories of particle physics --- gauge theories --- and our theory of gravity. When computing so-called {\it scattering amplitudes}, describing the probabilities of different outcomes when particles of the theory are collided, in certain variables one can obtain the result in the gravity theory by merely squaring that of the gauge theory. This relationship is important for a number of reasons. First, the notorious non-linearity of gravity makes any computation on the gravity side extremely complicated. If the double copy relation always holds, we can simplify the computation on the gravity side by instead performing the gauge theory computation, which is well established in particle physics theory. Second, the relationship is telling us something fundamental about two theories that are, on the face of it, describing completely different physical phenomena. Hence, this may contain a clue to the unified description of all physics.

It is therefore important to understand the key ingredients of the double copy relation and their relations to each other. This is a highly technical project, involving three different constraints on the theories; the so-called {\it color-kinematic duality}, the {\it Bern-Carrasco- Johansson(BCJ) relation} \cite{Bern:2008qj} and the {\it Kleiss-Kuijf(KK) relation} \cite{Kleiss:1988ne}.

In recent years, various modifications of the double copy have been of increasing interest, both because they demonstrate ever increasing relationships among seemingly different theories, and also because they may shed light on the origin of the deep structure underlying the double copy relationship.

In my dissertation, I will discuss my recent paper (with collaborators)\cite{Gonzalez:2022mpa} describing several important technical issues in generalizing the double copy relation to theories in which the particles have mass. We began from something called the {\it mass spectral condition}, which has been proposed \cite{Johnson:2020pny} as a way to ensure that such theories remain {\it local} --- an important principle that physical objects can only be directly impacted by their immediate surroundings, rather than by events at arbitrary distances. We have shown that it is not difficult to achieve this mass spectral condition if we construct a massive theory in a very specific way, by starting with a higher dimensional theory without mass (where we know a double copy holds) and then compactifying the extra spatial dimension to obtain a massive theory in our four dimensions of space plus time. We have explored the double copy relationship among these lower-dimensional massive theories and have found that the massive extension of the original double copy relationship still holds. Fascinatingly though, we were able to show that in more general theories with mass, the interactions are much more constrained if one requires that the theory obey a double copy relationship.