Paul-Hermann Balduf

Research

Motivation for Tropical Field Theory

Tropical field theory is a new type of renormalizable model quantum field theory which is at the moment being developed by Michael Borinsky, Erik Panzer, and myself.

There are different perspectives on what tropical field theory is, and each of them gives rise to useful physical and mathematical insights. Below is a short summary (intended to be readable for high energy physicists).

Perspective 1: Extending zero-dimensional QFT

It is well known that the number of Feynman diagrams can be counted with so-called zero-dimensional quantum field theory. As “Feynman rules”, this framework assigns the value unity to every Feynman digram, and the “amplitudes” are therefore the generating functions for the number of graphs, taking into account appropriate symmetry factors. This seems like a poor approximation, but zero-dimensional QFT actually captures a lot of the qualitative behavior of realistic QFTs at large loop order, for example the factorial asymptotic growth rate which can only really be measured from data upwards of 25 loops.

The major shortcoming of zero-dimensional QFT is that it has no meaningful notion of renormalization, and therefore it doesn’t show any of the effects that are (conjecturally) caused through renormalization at large loop order, such as renormalons.

From this perspective, tropical field theory is a natural generalization of zero-dimensional QFT. Its Feynman rules are defined in terms of the superficial degree of divergence \omega_G=E_G-\frac D 2 \ell_G, where E_G is the number of edges and \ell_G the number of loops of a Feynman graph, and D is the spacetime dimension. For concreteness, we consider \phi^4-theory in D=4-2\epsilon dimension. Then, a vertex-type graph (i.e. a connected graph with 4 external legs) has \omega_G=\epsilon \ell_G.

A Feynman integral is in general a complicated function of masses and external momenta. However, it contains a prefactor \Gamma(\omega_G) which represents the superficial divergence (if any). In our setting in dimensional regularization, a vertex-type graph therefore has an overall pole \frac 1 {\omega_G}=\frac 1 {\ell_G \cdot \epsilon} at \epsilon \rightarrow 0. If the graph has subdivergences, this factor is multiplied by an expression that contains additional poles. Conversely, if the graph has no subdivergences, the remaining factor is finite, and is called the Feynman period.

The idea of tropical field theory is to assign a simple Feynman rule to each graph, but instead of just unity as in zero-dimensional QFT, one wants to capture the pole structure. Basically, we would like to assign \frac 1 {\omega_G}, but this by itself is not consistent (because what would then be the Feynman rule of subgraphs of G?). A consistent tropical Feynman rule for a graph G has the form

    \[\theta[G]= \sum_{\gamma_1\subset \gamma_2\cdots G} \frac 1{\omega_1}\frac 1{\omega_2}\cdots \frac 1 {\omega_G},\]

where the sum goes over all possible “flags” of subgraphs \emptyset \subset \gamma_1 \subset \gamma_2 \cdots G, arising from adding one edge at a time. This definition \theta[G] can be reformulated in various ways upon gathering and grouping the terms. For example, one can write it as a recursion formula by removing only one edge:

    \[\theta[G]=\frac 1 {\omega_G} \sum_{e\in E_G} \theta[G\setminus e].\]

The combinatorial definition of tropical field theory has the advantage that it makes obvious the nested divergence structure, which closely resembles that of the full theory: There are all superficial divergences and subdivergences that arise from vertex-type graphs. On the other hand, tropical field theory does not have propagator divergences. This is physically expected from the perspective of long-range theory.

We demonstrate in our article that the limit \epsilon \rightarrow 2 of tropical field theory, i.e. D \rightarrow 0, indeed exactly reproduces the conventional zero-dimensional QFT. It might at first not be obvious, but every tropical Feynman integral goes to unity in this limit, so that the Green functions are once more merely counting the number of graphs when \epsilon=2.

Perspective 2: Scaling long-range field theory

In high energy physics, the momentum-dependence of (scalar) propagators is typically forced to be p^{-2} because of Lorentz-invariance, renormalizability, and unitarity. This amounts, via Fourier transform, to a spacial decay r^{-2}, and with respect to the time coordinate it is related to the Hamiltonian equations of motion being differential equations of first order. In many interesting situations in solid state- and statistical physics, these constraints are weaker and one is naturally led to interactions which decay with different power laws. Under the name long range field theory, such systems have been investigated for many decades, and they often show non-trivial differences compared to the conventional, or short-range, interactions.

From a perspective of statistical physics, it is therefore perfectly natural to consider a theory where the propagator scales like p^{-2\xi}, and which is situated in some ambient dimension D \neq 4. If one has a system with quartic interaction, such as \phi^4 theory, then this interaction stays marginal (i.e. just renormalizable) when the exponent \xi and the dimension D are in a certain relation. We use a setup that reproduces 4-dimensional \phi^4 theory with dimensional regularization, therefore we let D= \xi(4-2\epsilon). For example, the choice \xi = \frac 3 4 amounts to a renormalizable \phi^4 theory in 3 dimensions, with dimensional regularization parameter \epsilon, and with propagator \frac i {(p^2)^{\frac 3 4}}. The non-integer power in the propagator is not as scary as it looks because p^2 is a scalar, not a vector, so that one may take any power of it as long as it is positive. In order to make sense of these definitions, one can always resort to a Fourier transform and imagine e.g. a lattice simulation: On a lattice, one may impose any arbitrary power \alpha in an interaction Hamiltonian

    \[H = \sum_{x,y} \frac{\phi(x) \phi(y)}{|x-y|^\alpha}.\]

For a finite lattice, a computer can evaluate H and will find a finite number. Whether or not a meaningful limit exists for an infinite lattice is another question, but there is no fundamental problem when \alpha is not an integer.

With this setup, tropical field theory emerges in the limit \xi \rightarrow 0, which is therefore called tropical limit. From this perspective, tropical field theory lives in zero spacetime dimensions, but yet it is different from the conventional zero-dimensional field theory. If one carries out the limit naively, all momentum dependence vanishes, and what remains are the correlation functions at zero momentum, that is, the quantum effective potential. From this heuristic argument, it is of course not obvious that the tropical limit exists at all, and that the resulting theory coincides with the purely combinatorial definition above. This can be proved for example from a consideration of parametric Feynman integrals.

Perspective 3: Approximating parametric Feynman integrals

It has been known for decades that Feynman integrals can be represented in a Schwinger-parametric form, based on the identity

    \[\frac 1 {(p^2)^\xi} = \frac 1 {\Gamma(\xi)} \int_0^\infty \d a \; a^{\xi-1} e^{-a p^2}.\]

The parametric Feynman integral is an integral over a projective space of the Schwinger parameters a , one out of many possible ways to write it is

    \[\Gamma(\omega_G) \int \Omega_G \prod_{e\in E_G} \frac{a_e}^{\xi-1}}{\Gamma(\xi)} \frac{\mathcal F^{-\omega_G}}{\mathcal U^{\frac D 2 - \omega_G}}.\]

Here, \Omega_G=\delta(\sum_e a_e-1) \prod_e \text{d} a_e is an integration measure, \omega_G=2|E_G| -\ell \frac D 2 is the superficial degree of convergence, and \mathcal F and \mathcal U are the two Symanzik polynomials which are well understood homogeneous polynomials in the a_e.

The parametric representation has the big advantage that it defines an analytic continuation in propagator powers \xi and spacetime dimension D, that is, both of these parameters appear in the parametric representation as arguments of Euler gamma functions or as powers of positive real numbers, but not as dimensionality of the integration. Apart from particular values where the integral diverges, one may use arbitrary non-integer values for \xi and D. Moreover, the second Symanzik polynomial \mathcal F is the only one that depends on masses and momenta.

If one inserts D=\xi(4-2\epsilon), the dependence on the first Symanzik polynomial \mathcal U degenerates in a particular way: Recall that for some set of numbers \lbrace y_e \rbrace and some finite positive factors \lbrace c_e \rbrace, the maximum norm, or L_\infty norm, is defined as

    \[\lim_{\xi \rightarrow 0} \left( \sum_e c_e \cdot y_e^{\frac 1 \xi} \right)^\xi = \max \lbrace y_e \rbrace.\]

In words: Taking the \xi-th root of all y_e, then adding them up with arbitrary prefactors, and then taking the \xi-th power, selects the maximum of the y_e. The first Symanzik polynomial \mathcal U is a sum of homogeneous terms, each of which corresponds to a spanning tree of the graph. In the tropical limit, a rescaling a_e \rightarrow y_e^{\frac 1 \xi} precisely reproduces the setting of the L_\infty-norm. Effectively, what happens is that \mathcal U, the homogeneous polynomial, gets replaced by \mathcal U^{\text{tr}}, the maximum monomial of \mathcal U. Since the integration domain amounts to positive values of a_e, and the Symanzik polynomial \mathcal U has positive coefficients, all monomials are positive. Therefore, the tropicalization \mathcal U^{\text{tr}} constitutes a bound: The maximum of a sum of positive terms is clearly smaller than the sum itself, but the maximum times the number of summands is larger than the sum. Therefore, the tropicalization produces a bound of Feynman integrals. For integrals without subdivergences, this is called the Hepp bound, as it has been used by Klaus Hepp decades ago to prove convergence.

A similar calculation shows that in the tropical limit, the second Symanzik polynomial \mathcal F vanishes, therefore, the integrals are naturally independent of external momenta in the tropical limit.

Instead of going through all the steps, one can now also view tropical Feynman rules a priori as an approximation: One is interested in Feynman integrals with vanishing momenta, and therefore leaves out \mathcal F. Now, the approximation consists in replacing \mathcal U by its maximum monomial \mathcal U^{\text{tr}}. An integral of a monomial can in principle be solved easily, the difficulty is to figure out where in the integration domain which of the monomials is the maximum. It turns out that this is encoded by the superficial degree of convergence \omega_\gamma of corresponding subgraphs \gamma. Doing all the algebra, one finds that the tropicalized integral evaluates to \theta[G], the formula claimed in the very beginning.