#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 is "Lips: p-adic and singular phase space" from 2023. This article is quite different from the ones I usually read, in that it is not about a computation, but rather it describes a software. It is normal for research projects in theoretical #physics to involve large amounts of programming and computer use. Almost all of these operations rely on purpose-built #scientificSoftware , and there is a great number of open source libraries for all kinds of specialist physics computations.
The present article describes the package "Lips" (short for "Lorentz invariant phase space"), whose primary purpose is to generate valid sets of momenta for scattering amplitudes: A scattering amplitude is a function of masses and momenta of a set of particles, and it usually implies various constraints to these (e.g. masses should be positive, momenta should be conserved, individual momenta should square to given values, etc.). This makes it a non-trivial task to produce concrete numerical values of momenta that satisfy these constraints. Beyond that, one might also require specific types of numbers (rational, complex, p-adic, etc), or represent the momenta as spinors. The package can also do further calculations that arise in this context, such as evaluating spin-helicity expressions.
https://arxiv.org/abs/2305.14075

#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

Monday's #paperOfTheDay : "Resistance Minimum in Dilute Magnetic Alloys" by Jun Kondo from 1964. This paper concerns the #electric properties in solid state #physics , more precisely the electrical resistance at low temperatures (around 10K, or -265̀C) in materials that have only a few, and far apart (hence "dilute") impurities. The assumption is made that these impurities are spacially fixed, and have a spin, while the conduction occurs with free moving electrons. Now these electrons scatter with the spins, and, under the given assumptions, this scattering process can occur in a number of ways. For concrete calculations, Kondo uses a Hamiltonian that had been known before, and computes the individual probabilities to second order perturbatively. Taking them all together, the terms combine nicely and produce an effective resistance that depends logarithmically on the temperature. If the constant prefactor has the correct sign, this implies that the resistance -- which normally decreases as the temperature is lowered -- obtains a minimum, and then shows a sharp increase at even lower temperatures. This is nowadays known as the Kondo effect. https://academic.oup.com/ptp/article/32/1/37/1834632

For Sunday, my #paperOfTheDay is "On spin three self interactions" from 1984. It had been known since Fierz and Pauli in the 1930s what the Lagrangian density for a massless free field of arbitrary integer spin s looks like: The field variable is a rank-s tensor, and its kinetic term involves certain second derivatives and tensor contractions.
The present paper is a very clear and instructive exposition of how this fact restricts the form of possible interactions: 1. From the known free-field Lagrangian, one concludes that not all entries of the field tensor are independent, but instead, they satisfy certain conditions, expressible by a differential operator (think spin 1: del_mu A^mu=0). 2. These identities imply that the field has a gauge invariance. 3. Expand an interaction in powers of the field. The lowest order involves three fields. Gauge invariance completely fixes its functional form, up to redefinitions of the field variable. In particular, a 3-valent interaction vertex for spin-s fields contains s derivatives. 4. This can be iterated for interaction terms with more than 3 field variables. 5. Additionally, one obtains a commutator relation for the gauge transformation, i.e. its Lie algebra structure. In principle, this determines what the gauge symmetry is, macroscopically, if one can recognize the commutator as the Lie bracket of some known group.
The authors apply this programme to the spin-3 field. The resulting expressions are somewhat large, and it's not entirely clear how to interpret them intuitively, but still I like the clarity of their exposition. https://link.springer.com/article/10.1007/BF01410362

#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

#paperOfTheDay for Friday was "Tree hook length formulae, Feynman rules and B-series" from 2014. This is a #mathematics paper, more specifically #combinatorics . It deals with rooted trees, that is, connected graphs without cycles and with one distinguished vertex. For a given vertex, the subtree is the unique tree that is obtained by taking the vertex as a root, and discarding everything that was above the new root in the original tree. One then defines a "hook length formula" to be a mapping from rooted trees to some ring, which is given by evaluating some function on each subtree and multiplying the result. The classical example is the "tree factorial", where the function on the subtree is the number of vertices, so that the entire tree evaluates to the product of the number of vertices of all subtrees (which equals the ordinary factorial if the tree is a path). This construction might seem obscure, but it is widely used, and the present paper makes an effort to unify these results. For example, Runge-Kutta schemes for numerical integration of differential equations have an algebraic form called B-series, which essentially is a hook length formula. Also, renormalization of divergent subdiagrams in #quantumFieldTheory has this structure. The present paper discovers various new closed-form expressions for hook length formulae. From the perspective of QFT, what they do is invent new toy model Feynman rules that give rise to nice closed-form Green functions. I find this quite useful for a systematic qualitative understanding of QFT, even if these particular Feynman rules don't have an immediate physical interpretation. https://arxiv.org/abs/1412.6053

Thursday's #paperOfTheDay: "Almost zero-dimensional quantum field theories" from 1992. That paper considers the behaviour of #quantumMechanics and #quantumFieldTheory close to zero spacetime dimensions. The actual limit D=0, the zero-dimensional field theory, is well understood. The authors now study a (radially symmetric) Schrödinger equation, and then a free field theory, close to D=0. They find that this limit exists (i.e. there is a continuous family of theories for real parameters D which interpolates between the physical and the 0-dimensional theory), and the linear approximation in D already gives numerically meaningful estimates of the physical theory.
This setup is in the same spirit as our #tropicalFieldTheory , but the difference is that the older paper varies D alone, whereas the tropical limit arises when one reduces D and the power of the kinetic term (i.e. spacial decay rate of propagators) simultaneously. Secondly, we now have a much better understanding of analytical properties of #FeynmanIntegrals than 30 years ago, so that we can perform this limit in a mathematical clean way for all graphs of an interacting field theory. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.46.5557

#paperOfTheDay is "PT-symmetric -g phi^4 theory" from 2022. The phi^4 theory is a model #quantumFieldTheory where the interaction term of the field variable phi is of the form phi^4. A plot of this function shows that it is bounded from below when its coefficient, the coupling g, is positive. Conversely, if one chooses an interaction term -g phi^4 (for positive g), the potential is unbounded. Heuristically, this means that very large values of the field variables are energetically favored, and this theory is not stable: No matter what field configuration one starts in, there will always be a configuration with larger values of the field, which has lower energy.
This distinction becomes more subtle if one considers the coupling as a complex variable, and allows the field to take complex values. In the present article, this is investigated in detail for the zero-dimensional theory. There, the "field" phi is merely a (globally constant) number, and the "path integral" is an ordinary integral, which manifestly diverges if the interaction term has negative sign. However, if one integrates over a complex contour (that is motivated by PT symmetry), one can obtain a meaningful result even for negative g. The authors show that the PT-symmetric integral equals the imaginary part of an analytic continuation of the ordinary partition function, and they conjecture that the same relation holds also beyond the zero-dimensional case. This would mean that a theory with negative coupling could indeed give rise to physically meaningful observables, if interpreted correctly.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.106.125016