#paperOfTheDay for Friday is "Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions" from 2016.
Statistical #physics and #quantumFieldTheory usually involve two parameters that physically only allow integer values: The dimension of space(time), and the dimensions of internal symmetry groups (such as the SU(3) in the standard model QCD, or O(N) symmetry in scalar fields). On the other hand, it is routine to formally assign non-integer values to them. Dimensional regularization sets D=4-epsilon, where epsilon is not assumed to be integer, and the #FeynmanDiagram s of O(N) symmetric theories are polynomials in N, hence allowing any value.
The present article points out that even a free, 1-component scalar field theory contains states with negative norm if one lets D be non-integer. The argument is surprisingly simple: Consider operators which are built from spacetime-derivatives d_mu acting on fields. In particular, we are interested in those operators which are antisymmetric in their indices. However, in D integer dimensions, there are D coordinate directions, and hence an operator with n>D derivatives can not be fully antisymmetric. Hence, the antisymmetric operators vanish when n>D for integer D. This does not hold for non-integer D, so that the operator actually has zeros at all the integer D. One can then see, by explicit calculation, that the 2-point functions of such operators (in the free theory!) flip sign at integer D, hence they are sometimes negative, hence the theory is not unitary.
This shows that the extension to non-integer D is very subtle; similar trouble exists for the Dirac matrices gamma_mu.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.93.125025

Yesterday's #paperOfTheDay was "Critical Equation of State from the Average Action" from 1996. This paper is one of the first applications of the Wetterich equation: A numerical solution of the local potential approximation for the vector model. The vector model is physically interesting in 3 dimensions.
In conventional perturbative #QuantumFieldTheory, one would probably start in 4-2epsilon dimensions, and compute a power series in epsilon with #FeynmanDiagram s, to then arrive at a 3-dimensional theory with epsilon=1/2. The Feynman diagrams have 4-valent vertices, which is thought of as a microscopic point-like "collision" between 4 "particles".
In the functional/statistical #physics perspective on field theory, the same situation is interpreted quite differently. One is in 3 dimensions throughout, and considers a system (e.g. lattice) where the constituents do not move. A phi^4-term is then a potential, i.e. every individual particle oscillates in its own local quartic potential. Additionally, the particles are coupled to neighbours. This is the "microscopic" theory, represented by the classical action. At longer distances, one effectively merges many of the lattice sites, and the so-obtained average quantities have all sorts of complicated interactions. The present paper uses the "local potential" approximation, which says that the only coupling to neighbours is still an elastic next-neighbour interaction (i.e. standard kinetic term p^2 in momentum space), and all that changes is that the particles are now in a more complicated potential. In particular, this might have non-trivial minima, which reflects a broken symmetry. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.77.873

#paperOfTheDay "Non-Wilsonian ultraviolet completion via transseries" from 2021. A #quantumFieldTheory with marginally renormalizable coupling, such as the standard model of particle #physics usually leads to a power series solution that is divergent in two different ways: The number of #FeynmanDiagram s grows too fast, and the renormalized value of individual diagrams grows too fast. The latter is called #renormalon , and it can also be found in many other frameworks of QFT.
The present paper uses an analysis based on the renormalization group equation, truncated to low-order terms, to argue that the presence of the renormalon implies an ambiguity in the resummation. The technical machinery for this is called "resurgence", the basic mechanism is really intuitive: The Borel resummation is an integral along the positive real line, the renormalon is an algebraic singularity on that line, hence there is an ambiguity of which side (and how often, etc) one wants to pass the singularity. The paper arrives at two possible, mutually exclusive, interpretations of this finding.
I find these considerations really exciting, and they are closely related to my own work. However, I think it is fair to say that the many papers that have been written about renormalon chain resummation often raise more new questions than they answer, and at least to me the "big picture" of how this is supposed to work beyond leading-order is largely unclear. https://doi.org/10.1142/S0217751X21500160

#paperOfTheDay is "Effective field equations for expectation values" from 1986. Scattering in #quantumFieldTheory is usually defined as a time evolution from some infinite-past to infinite-future state, both of which are plane waves (=#particles on straight trajectories). In this setup, a natural definition for an "expectation value" of a field variable is the expectation between these states, i.e. <in | phi(x) | out> . The present paper introduces another class of expectation values, of the form <in | phi(x) | in>, and derive various equations for them. These,equations are different, but structurally equivalent, to the usual ones (e.g. if one uses perturbation theory, there still are the usual #FeynmanDiagram s, but one uses a retarded propagator in place of a Feynman propagator). The crucial difference is the interpretation of these quantities: The paper contains a nice example for a system where the |in> and |out> states have different plane wave states (i.e. a harmonic oscillator where the spring constant changes over time). In that case, it can be hard to interpret the conventional expectation values, whereas the new ones always refer to the |in> plane wave basis. Also, the conventional setup requires boundary conditions at the past and the future to compute the evaluation of the mean field, while the new setup can compute time evolution from just initial conditions in the past, which is more natural in certain quasi-classical setups. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.33.444

Many computations in #quantum field theory are "perturbative", that means that one computes a series expansion in powers of some parameter, called g. The usual intuition is that the parameter is small, so that higher powers of the parameter are negligible. However, in most cases, these series are divergent so that their sum is infinite for every non-zero value of the parameter g. This sometimes leads to confusion. The solution is to realize that there exist perfectly fine functions where a power series expansion is divergent. Not every function has a convergent series expansion. In the context of quantum field theory, one can see this explicitly in the case of a zero-dimensional theory. In that case, every #FeynmanDiagram is assigned the value 1 (which is a huge simplification). One can then solve the entire perturbation series. It is divergent, but it can equivalently be represented as a convergent integral. The conclusion is that quantum field theory delivers a finite, smooth function, but its perturbation series diverges since that function is imaginary for certain parameter ranges (which has physical interpretation as decay of particles). The mathematics is fine, the problem is that physics education is very much focused on polynomials and Taylor series, which leads to the wrong expectation that every physically sensible function should have a convergent Taylor series expansion. I recently gave a talk about this at PI, slides are available on my homepage https://paulbalduf.com/research/vector-asymptotics/
#physics #math

At the @EPSHEP2025 #physics #conference, I gave a talk about behaviour of #feynmandiagram s at large loop order, and that at 18 loops we are still not observing the leading asymptotic growth rate. The slides and various data sets are available from my website as always. https://paulbalduf.com/research/vector-asymptotics/