Feynman diagrams and periods of motives

Question

A recent article in the online science magazine Quanta, Strange Numbers Found in Particle Collisions, discusses experimental evidence of a connection between Feynman integrals and periods of motives. Although Quanta usually does a great job with expository articles, in this case so much simplification was necessary for the lay reader that I was left wanting more; a lot more.

Can anyone add more detail to the article, or point me towards another, more in depth exposition?

Answer 1

If I understand correctly, Quantum Field Theory was successful in so far as it was predictive of experimental results. Somehow these badly divergent integrals, when combined correctly product an answer aligning with the result you see in the particle accelerator.

Having gone through a quantum field theory class, I was warned not to worry about being too rigorous as long as I can get the correct result. And the course moved so quickly there wasn't enough time to reflect whether I had seen these tools in my math courses. And I left the course confused and dissatisfied.

There are many types of math and many types of physics. So, I think a great question to ask to is which math and which physics are being related in a given paper.

A few names are mentioned again and again in the paper. Francis Brown, so here is one if his:

• Feynman Amplitudes and Cosmic Galois group

Certainly there is earlier literature. Dirk Kreimer is mentioned so I pull one out of a hat:

• Renormalization and Mellin transforms

None of the Hopf algebra structures or Tannakian categories discussed here ever appear in a QFT course. Real QFT homework consists of pages and pages of integerals followed by more integrals and you are never told which matematical theory can formalize this.

• Peskin Schroeder Quantum Field Theory
• Zuber Quantum Field Theory

Even worse, when you try to formalize these computations, you drown in rigor and entirely lose the spirit of the original computation.

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Using the Feynman Rules we can put together diagrams which contribute to the scattering cross sections of various quantum fields theories. In particular $\phi^3$ and $\phi^4$ theory I am seeing a lot. It doesn't matter because they all use the same diagrams. Here's an integral:

$$ \left[ \prod_{k=1}^L \int \frac{d^4 p_k}{p_k^2 (p_k + p)^2}\right]\prod_{m=0}^L \frac{1}{(p_{m+1}-p_m)^2} $$

and later on in that paper we obtain the value of a slightly different diagram: $$ \int [\dots] \, d^4p_k = p^2 \left( \frac{\pi}{p}\right)^{2L} \binom{2L}{L} \zeta(2L-1) $$ for the ladder with $L$ rungs. And this is nothing short of PHENOMENAL.

I believe part of the miracle is, that although we could write down the contributing integrals, we had no idea what the $\int$ evaluated to, even in these simple cases. In class, I thought we had covered this but I guess not.

And I left out the domain of integration - which loosely involves conservation of momentum - but there may be other factors as well.

And there are many facets to these integrals that we are just beginning to find out.

• Summation of an infinite series of ladder diagrams
• Multiple zeta values and periods of moduli spaces $\mathfrak{M}_{0,n}$

Here are the sources I have looked up. And I recognize that moduli space, it is the moduli space of $n$ marked points on the sphere. Or as I like to think of them as polyhedra (hopefully that is accurate).

The second diagram counts how many Feynman diagrams - which is like a zillion.

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Having explained a tiny bit why there is a question at all, we learn that Feynman diagrams evaluate to multiple zeta values or other periods. As I learn a bit more, maybe I can explain why the first examples of motives. QFT students doing their homework.

Here is theorem 0.1 in Brown - page 1

For any Feynman graph $G$ with generic kinematics $q,m$, there is a canonical way to associate to a convergent integral $$ I_G(q,m) = \int \bigg[ \frac{P(\alpha_E) \Omega_G }{\Psi_G^A \Xi_G(q,m)^B } \bigg] $$

• an object $\text{mot}_G$ in $\mathcal{H}(S)$, where $S$ is a zariski open [set] in the space of kinematics
• a de Rham class $[\omega_G]$ in the generic fiber of $(\text{mot}_G)_{dR} $
• a Betti class $[\sigma_G]$ in a certain (Euclidean) fiber of $(\text{mot}_G)^\hacek_B$

such that the [Feynman diagram] integral is the period $$ \sigma_G\big( c(\omega_G) \big) = I_G(q,m)$$

Unless your Deligne or Kontsevich you probably have no idea what that means. I read it was: in the process of doing Feynman integrals we have done a lot of other things:

a motive is a thing that appears in many cohomology theories

• the space you're integrating over is part of Betti singular cohomology (this is the one from topology)
• the integrand is part of the deRham cohomology (this is the one from calculus)
• By some sort of Poincaré duality there's a way to match the Betti and deRham cohomolog to get a number

I believe Brown's kinematics is what happens when you evaluate the Feynman integral over momentum space $dp$ rather than position space $dx$.

He also does something rather strange getting these cohomology theories over the fractions $\mathbb{Q}$ rather than $\mathbb{C}$. And the space of kinematic data (e.g. from convervation of momentum) he says define a scheme rather than a variety. Complicating things further, but separating it from what a day-to-day high energy physics graduate student might call a "Feynman diagram"

At least now we see what the motives are and why they are appearing. If you are an expert in motives I direct you to the relevant papers and textbooks. Lastly... a great discussion on Motives (in French)

• Le Groupe Fondamental de $\mathbb{P}^1 \backslash \{ 0, 1, \infty\}$
• From Calculus to Cohomology

the last one one of my favorite resources.