Mastodawn

#paperOfTheDay is "Complex poles and spectral functions of Landau gauge #QCD and QCD-like theories" from 2020.
In #QuantumFieldTheory , one generally wants to compute n-point correlation functions of fields. Of particular interest is the 2-point function, which can be interpreted as describing how a "particle" of that theory moves. If one uses perturbation theory, the leading order of the 2-point function is the "propagator", the quantity that represents the edges in a #FeynmanIntegral . A typical expectation is that a general, non-perturbative 2-point function should still be similar in nature, namely admit a Källen-Lehmann representation, which is qualitatively "an integral over propagators with different mass, weighted by some spectral density function". The density is supposed to describe density of states of the theory. Conversely, the 2-point function as a function of complex energy should have a branch cut on the positive real line.
The present article considers the 2-point functions in theories roughly resembling QCD, and studies their form for complex energy. They find that in addition to the branch cut, these propagators can have pairs of poles in the complex plane off the real axis. In addition, the spectral "density" function can be negative for low energies (and hence can not actually be interpreted as a density).
These findings are based on various approximations and probably not precisely correct, but similar effects have been seen in other work, too. The interpretation, basically, is "confinement": In low-energy QCD, individual quarks or gluons are not meaningful "particles", so their 2-point function is different from that of a typical particle. #physics https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.074044

Paul Balduf 4d ago

#paperOfTheDay is "Use of analyticity in the calculation of nonrelativistic scattering amplitudes" from 1968. In theoretical #physics , scattering amplitudes arise in various contexts, and their calculation is usually quite hard because they involve all sorts of "oscillations": For example, a real-time path integral in #quantumFieldTheory has an oscillatory Boltzmann factor exp(i S), which is numerically unstable. Or, more intuitively: When waves "collide", a tiny inaccuracy in the relative phase has big influence on the outcome of the process. Conversely, these calculations become quite easy in imaginary time ("Wick rotation"): QFT turns into statistical physics, the path integral gets the weighting factor exp(-S), which strongly suppresses fluctuations, and the wave equation turns into the heat equation, which is numerically well behaved and stable.
This intuition is the basis of the present paper: Formally, the difference between the two (physically very different) situations is whether certain quantities are real or imaginary. One of the settings is easily computable. But the answer is a function of this complex input variable, so one can get to the "hard" situation by analytic continuation. Concretely, the paper studies non-relativistic scattering as a function of complex energy, thereby interpolating between bound states and scattering.
On the technical side, the article is noteworthy because it introduces numerical algorithms to compute rational function approximations for a function where only a finite set of evaluations f(x_i) is given (if, conversely, a finite number of Taylor coefficients of f are given, one would use a Padé approximant). https://journals.aps.org/pr/abstract/10.1103/PhysRev.167.1411

Paul Balduf 6d ago

#paperOfTheDay is another classic from the early days of #quantumFieldTheory : "A relativistic equation for bound-state problems" by Bethe and Salpeter in 1951. In this article, they derive the equation that now bears their name, an integral equation describing bound states.
The starting point is to use perturbative relativistic quantum field theory in the form of #FeynmanIntegral s (which was still a novelty in #physics at that time). But a Feynman integral describes essentially an "instantaneous" interaction, whereas a bound state between two particles means that these interact for a long time (e.g. to make an atom, nucleus and electrons need to attract each other permanently, as opposed to a scattering process, where they merely interact for a short moment). In principle, the sum over all (infinitely many) Feynman diagrams should give the full solution of the theory, including bound states, but in practice this is impossible to compute.
The Bethe-Salpeter equation is essentially a rearrangement of Feynman diagrams: One introduces an "interaction kernel" (which can be determined e.g. from solving Feynman integrals), and this kernel is then used "infinitely often". Of course, the exact kernel can not be computed with reasonable effort, but the approximation is still much better than using only a few Feynman diagrams. One can also view the Bethe-Salpeter equation as a rearranged version of a Dyson-Schwinger equation: An integral equation whose self-consistent solution represents an infinite sum of Feynman diagrams.
https://journals.aps.org/pr/abstract/10.1103/PhysRev.84.1232

Paul Balduf 6d ago

#paperOfTheDay is "Analytic structure of three-point functions from countour deformations" from 2022. This article concerns #FeynmanIntegral s, which are the coefficients of the perturbation series for e.g. scattering amplitudes in #quantumFieldTheory . These integrals are functions of momenta and masses, and for specific values of these arguments, they show non-trivial analytic properties. The basic example is the 1-loop massive propagator correction, which starts having an imaginary part when p^2 >= (2*m)^2. Physically, this threshold is the point where the two "virtual" particles in the loop can actually be physical, i.e. there is enough energy to make their rest masses.
Today, there is an active research community in mathematical physics dealing with such questions from the perspective of algebraic geometry: One uses a representation of Feynman integrals in terms of Schwinger parameters, where the integrand is a rational function, so the singularities are a zero locus of the denominator polynomial (and these types of objects -- algebraic varieties -- are well studied in #mathematics ).
The present article, instead, uses the momentum representation. They discuss the 1-loop 2-point and 3-point scalar diagrams in great detail, and comment on generalizations. The momentum representation is more complicated than the parametric one because the integrand is less structured (the integration variables are vector-valued, and can be negative). On the other hand, it has the advantage that it makes very clear the physical reason for analytic properties, so that this article is also a nice pedagogic introduction to the basic mechanism.
https://arxiv.org/abs/2212.02515

The analytic structure of three-point functions from contour deformations

We explore the analytic structure of three-point functions using contour deformations. This method allows continuing calculations analytically from the spacelike to the timelike regime. We first elucidate the case of two-point functions with explicit explanations how to deform the integration contour and the cuts in the integrand to obtain the known cut structure of the integral. This is then applied to one-loop three-point integrals. We explicate individual conditions of the corresponding Landau analysis in terms of contour deformations. In particular, the emergence and position of singular points in the complex integration plane are relevant to determine the physical thresholds. As an exploratory demonstration of this method's numerical implementation we apply it to a coupled system of functional equations for the propagator and the three-point vertex of $ϕ^3$ theory. We demonstrate that under generic circumstances the three-point vertex function displays cuts which can be determined from modified Landau conditions.

arXiv.org

Paul Balduf Jun 9

#paperOfTheDay is "Exact evolution equation for the effective potential" from 1993 by Christof Wetterich. This is the paper where the Wetterich equation is first introduced in its modern form.
In #quantumFieldTheory , there are "quantum fluctuations" which lead to the full theory being different from the Lagrangian or action one starts with. A famous example is light by light scattering in QED: The Lagrangian in quantum electrodynamics contains basically one type of allowed interaction, namely matter (such as electrons) emitting or absorbing one photon. This can take many different forms in practice, for example Bremsstrahlung (electron being accelerated and producing photon), or electrostatic repulsion (photons being exchanged between two electrons, accelerating them away from each other). QED does not allow for an elementary interaction between photons. But such interaction does in fact take place due to #quantum fluctuations with "virtual" intermediate electrons. A central task for theoretical #physics is to compute such effects.
The effective action contains all such quantum effects, that is, if one uses the effective action as a "classical" one (without adding further quantum corrections), one obtains the full quantum answer. Many different methods are known to compute the effective action, several of them based on the renormalization group. Compared to the previous work by Wilson, Wegner, and Polchinski, the Wetterich equation is somewhat more explicit in terms of interpretation, and it has the advantage of directly giving the quantum effective action and not some proxy quantity. It has by now become the cornerstone of functional renormalization group ( #FRG ) methods. https://www.sciencedirect.com/science/article/pii/037026939390726X?via%3Dihub

Paul Balduf Jun 8

#paperOfTheDay is "Bounding scalar operator dimensions in 4D CFT" from 2008.
A conformal field theory (#CFT ) is a #quantumFieldTheory that, in addition to the usual Lorentz symmetry, also has conformal #symmetry (a stronger version of scale invariance). This implies that the 2-point correlation function is a power law, i.e. instead of an arbitrary complicated function of distance, it is fully specified by knowing a single number, the operator's scaling dimension. Moreover, even the 3-point function is fixed once one knows the scaling dimensions of all three operators (where an operator is a polynomial in field variables and derivatives, evaluated at one point) involved in it. This illustrates the enormous importance that scaling dimensions of operators have in a CFT.
Secondly, a CFT allows for an operator product expansion (OPE), where one can rewrite a product of two operators at distinct spacetime points as an infinite sum of operators at only one of the points (morally analogous to a Taylor series for an ordinary function).
The most straightforward operator is the field variable itself. The present paper asks the question: When we know that the field has scaling dimension d, then what can we say about the scaling dimension of the leading operator that appears in the OPE of field x field? By combining the various structural features of the CFT, it turns out that one can find a strict upper bound to the possible scaling dimension of this operator.
The paper is relatively long, but very readable, since it includes detailed reviews and examples of how the various CFT constructions work. Beyond the actual result, it serves as a good introduction to CFT. #physics
https://iopscience.iop.org/article/10.1088/1126-6708/2008/12/031

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Paul Balduf Jun 4

#paperOfTheDay is "Accurate critical exponents for Ising like systems in non-integer dimensions" from 1987. This is one out of a sequence of articles with very similar titles and similar sets of authors, appearing over several years with updated numerical values.
The basic scheme is to use perturbative #quatumFieldTheory in 4-epsilon dimensions to compute critical exponents of the Ising model. This produces a power series in epsilon, which is then, in principle, evaluated at epsilon=1 in order to compute values for the physically interesting dimension 3 (computing directly in D=3 dimensions is less accurate because there, the quartic coupling is relevant, and therefore the theory is strongly interacting, whereas the expansion around 4 dimensions describes a weakly interacting system).
Of course, the power series in epsilon is divergent, therefore it would be inaccurate to simply substitute epsilon=1 (in the truncated series). Instead, one computes a Borel transform of the series data, conformally maps to the complex unit disk, and computes a Laplace integral from 0 to 1 (this is a widely known standard method for resummation of this type of divergent series). Here, the authors additionally introduce three free parameters, which they tune to improve numerical accuracy. The output is a table of resummed critical exponents between 1.5 and 4 dimensions in small steps.
Today, we know 8-loop data instead of 4, which gives rise to much more accurate values, but the basic picture has not changed much.
#physics
https://jphys.journaldephysique.org/articles/jphys/abs/1987/01/jphys_1987__48_1_19_0/jphys_1987__48_1_19_0.html

Accurate critical exponents for Ising like systems in non-integer dimensions | Journal de Physique

Journal de Physique, Journal de Physique Archives représente une mine d informations facile à consulter sur la manière dont la physique a été publiée depuis 1872.

Paul Balduf Jun 3

#paperOfTheDay is "On the Numerical calculation of a class of definite integrals and infinite series" by G. Stokes from 1850.
This article is a follow up on the work of Airy I mentioned a few days ago, where the Airy function was introduced through its convergent integral representation. This function gives the intensity in interference patterns of light (e.g. the location of "higher order" rainbows). In #physics experiments, what one actually measures is the location of maxima or minima of intensity, which would correspond to the zeros of the Airy function. With the technology of 1850 it was hard to numerically compute integrals. Airy derived a convergent Taylor series expansion about zero, but from that, it is still hard to find higher order zeros.
The present paper introduces a new approach to such calculations. The idea is to first derive a differential equation for the Airy function, and then use it to obtain a series expansion around infinity. This expansion is divergent (which is clear since the Airy function qualitatively changes character around zero), but it gives very high numerical accuracy for not-too-small arguments. In particular, one can obtain the locations of all zeros.
The plot shows the true Airy function in black, and the first 9 terms of its convergent power series expansion around zero in red. We see that they reproduce the first zero, but completely miss all other zeros. The green curve is the first (!) term of the divergent expansion around -infinity. It is visually indistinguishable from the true function for all zeros. This high numerical accuracy of divergent series expansions is a typical feature seen in #resurgence computations.
#mathematics

https://www.cambridge.org/core/books/abs/mathematical-and-physical-papers/on-the-numerical-calculation-of-a-class-of-definite-integrals-and-infinite-series/1DA9818868338CC0E684A6CB63B2503D

Paul Balduf Jun 2

#paperOfTheDay is "Dirac traces and the Tutte polynomial" from 2025. A Fermion is a particle that is subject to the Pauli exclusion principle, namely at most one Fermion can occupy any one position at a given time. All constituents of matter, such as electrons or quarks, are Fermions.
Mathematically, Fermions and their interactions are being described by certain matrices, the Dirac matrices (recall that for two matrices, A*B can be distinct from B*A, therefore matrices allow for effects such as flipping sign when being encountered in reverse direction).
In calculations, one then encounters products of such Dirac matrices. In particular, one often wants to compute the trace of them. In principle, this could be done by multiplying the matrices and computing the trace of the result matrix, however, there are 2 disadvantages: Firstly, matrix multiplication is slow, and secondly, in #quantumFieldTheory one wants to compute in a setting where the spacetime dimension is a symbolic parameter, and it is impossible to compute a matrix product when the number of rows in the matrix is an undetermined symbolic value. Hence, instead, one uses the defining properties of Dirac matrices: How they flip signs when the order of their product is changed. In this way, one can "order" an arbitrary product, so that adjacent factors are the same matrix, in which case they become a unit matrix.
The present paper shows that this combinatorial operation is encoded in a specific type of graph, and that the value of the trace is given by the Tutte polynomial of that graph.
https://link.springer.com/article/10.1007/JHEP05(2025)235 #physics

Dirac traces and the Tutte polynomial - Journal of High Energy Physics

Perturbative calculations involving fermion loops in quantum field theories require tracing over Dirac matrices. A simple way to regulate the divergences that generically appear in these calculations is dimensional regularisation, which has the consequence of replacing 4-dimensional Dirac matrices with d-dimensional counterparts for arbitrary complex values of d. In this work, a connection between traces of d-dimensional Dirac matrices and computations of the Tutte polynomial of associated graphs is proven. The time complexity of computing Dirac traces is analysed by this connection, and improvements to algorithms for computing Dirac traces are proposed.

SpringerLink

Paul Balduf May 30

The #paperOfTheDay is "On the intensity of light in the neighbourhood of a caustic" by George Biddell Airy, 1838.
This is a paper from the time long before anything quantum or relativity, where elementary questions of #optics were at the forefront of theoretical #physics . Since light has the character of both a wave and a ray, it is important not only where a ray can point, but also "how long" it is; one will find destructive or constructive interference depending on the difference of path length (in relation to the wave length). In the framework of geometric (=light ray) optics, this amounts to the fact that when light is reflected, a small change of the point of reflection does not change the total length of the path travelled to first order (i.e. light is reflected at such angle that the path length is stationary).
For a point of focus, this relation is true not only infinitesimally, but for the full reflecting surface: All incoming rays sent to a focal point have the same length. Airy studies the intermediate case: Not ALL rays have the same length, but still, the difference vanishes to higher than first order locally. Geometrically, this gives rise to a "caustic", basically a focal point deformed to a focal line. The existence of such line amounts to the reflecting surface satisfying a certain differential equation. Most notably, for the total intensity (in the easiest case), Airy finds the expression
integral from z=0 to z=infinity of cos(z^3 + t*z) dz , where t is a parameter. This function Ai(t) is nowadays known as the Airy function, and it constitutes the basic example of many effects in the theory of #resurgence and special functions in #mathematics .
https://www.semanticscholar.org/paper/ON-the-Intensity-of-Light-in-the-neighbourhood-of-a-Airy/8d6349998be20279220e8a3e033de1679a58a916