CU theory group connects previously unseen dots with new proof of principle

At the heart of so many investigations in theoretical physics is how to reconcile the physics principle with the mathematical proof. Occasionally, there is an even more fundamental concern: recognizing that there is a relationship to be reconciled at all.

In a recently published paper in the Physical Review Letters, three Columbia Physics PhD students -- Ameya Chavda, Daniel McLoughlin, and John Staunton -- collaborated with Assistant Professor Sebastian Mizera to do just this. In their “Unitarity Flow Conjecture: An On-Shell Approach to the Renormalization Group,” they posit through a new concept of unitarity flow conjecture that renormalization group flow is tied to unitarity. 

To set the stage, McLoughlin explained that in traditional quantum field theory (QFT), which predicts what happens when particles collide by adding up Feynman diagrams, “the final answer is often much simpler than the machinery used to obtain it.” So Professor Mizera’s team decided to use the modern “on-shell” approach, which “works directly with measurable scattering processes and the consistency conditions they must satisfy,” and the results unquestionably paid off.

While renormalization group flow and unitarity are not conflicting concepts in QFT, they’re also not considered related. In the renormalization group framework, the properties of physical systems change relative to the scale used to measure them. It's a way of making unwieldy infinite quantities manageable, and as the authors note, it has sweeping applications that "range from statistical and condensed-matter systems, through fluid mechanics and cosmology, to nuclear and particle physics." 

Unitarity, meanwhile, is a condition that brings a tidiness to the unruly probabilities of quantum mechanics. Chavda, McLoughlin, and Staunton used the example of an electron to explain it this way: "although we cannot know exactly where a particle is quantum mechanically-speaking, we do know there is a 100% probability that it is somewhere." In other words, the 100% certainty of unitarity means that the sum of all probable outcomes of a quantum event must add up to one.

With their Unitarity Flow Conjecture, the team not only suggests that these concepts are related, but more specifically that renormalization group flow is, at least in part, a consequence of unitarity.

The genesis of this project dates back to the summer of 2024, when the three students attended a summer school that Professor Mizera had organized, which focused on scattering amplitudes. They began working on Unitarity Flow Conjecture shortly afterwards, and as with any research project, they tried various approaches before finding the right fit by computing a quantity that was true for all powers of a variable λ. As Staunton recalls, they'd demonstrated that it worked through the powers of 1, 2, 3, and 4, but he decided on a whim to try to the power of 5. And? (cue drumroll)…it didn't work. This forced them to backtrack and spend another month scrutinizing their work until they found a way forward. Although Staunton described the setback as “a bit of a disaster,” it actually served to make them more confident in the form their solution ultimately took because they could now demonstrate it worked for powers significantly higher than 5.

The potential applications for their project are wide-ranging. The authors raise the possibility that Unitarity Flow Conjecture could be extended to mass renormalization, soft collinear effective theory, and potentially to streamline computations in operator mixing. They even suggest that their results could be viewed as a version of the S-matrix bootstrap, which raises the prospect of combining it with the nonperturbative bootstrap and positivity programs.

Of course, the nature of quantum mechanics means the concrete precision that one finds in classical mechanics will always be elusive. But as Staunton points out, advancements in QFT can help simplify calculations and make higher precision predictions easier. This is something that McLoughlin considers to be one of the most interesting aspects of modern theoretical physics: “by rethinking well-established concepts in a new way with new tools, we can uncover structure and simplicity that would’ve remained opaque with traditional methods.”