Landau Singularities Research Articles

Holonomy Structure of Landau Singularities and Feynman Integrals

Since the pioneering work of Landau and Nakanishi on general Feynman integrals, it has been known that the singularities of the S-matrix, causal Green's functions and related functions are described by the so called Landau equations. These Landau singularities were physically interpreted as the macroscopic causality by the theoretical physicists working in 5-matrix theory, and the notion of essential support was obtained (Chandler-Stapp [3], lagolnitzer-Stapp [5]). In the branch of mathematics, on the other side, the theory of microfunction has evolved and has been applied powerfully to the general theory of partial differential equations (Sato-Kawai-Kashiwara [11]). It contained the essential support theory as the singularity spectrum of a function (i.e. its support viewed as a microfunction), and was far-reaching because of its close connection with the theory of differential equations. Namely the method of microlocal analysis, based on the theory of holonomic systems, has provided a systematic way of handling functions with natural background and found most effective applications to various problems of mathematics, such as the theory of ^-functions and Fourier transformations (Sato [10], Kashiwara [6]). It was then recognized that the Landau equations give holonomic varieties, which led one to the holonomicity postulate of ^-matrix and related quantities (Sato [10]). In the present paper we shall study the holonomy structure of Landau singularities and Feynman integrals from this standpoint. First we review the notion of Landau varieties. In contrast with

Generalized Euler-Pochhammer integral representation for single-loop Feynman amplitudes

Kershaw's power-series representation for the single-loop Feynman amplitudes is cast into a new integral representation which, as a generalization of the Euler-Pochhammer type, is closer in spirit to the Veneziano representation than the conventional Feynman-parameter representation. Landau singularities, which are obscure in the power-series expansion, are recovered in the new integral representation. Explicit calculation is carried out for the cases $N=3$ and $N=4$.

Algebraic Factorization of Scattering Amplitudes at Physical Landau Singularities

The behavior of the scattering amplitude in the vicinity of a physical Landau singularity is considered. It is shown that its singular part may be written as an algebraic product of the scattering amplitudes for each vertex of the corresponding Landau graph times a certain explicitly determined singularity factor which depends only on the type of singularity (triangle graph, square graph, etc.) and on the masses and spins of the internal particles. Thus the well-known result for single-particle-exchange poles is generalized to arbitrary physical Landau singularities. Also, it is shown that for any Landau singularity there exists a finite polynomial in the scalar products of the external four-momenta whose vanishing gives the Landau singularity curve. A general, purely algebraic, method is given for constructing this polynomial.

Elementary Discontinuity Formulas and Mueller's Regge Hypothesis

We propose to interpret Mueller's recent work on inclusive cross sections in terms of the elementary discontinuity equations for multiparticle scattering amplitudes. The paths of continuations for obtaining these discontinuities are given. We find that, in order to make our identification meaningful, the emergence of anomalous thresholds is necessary to "shield" the effects of box-diagram Landau singularities. We also discuss how Mueller's Regge hypothesis can be derived from the multi-Regge model, provided singularities other than normal thresholds are absent in the asymptotic expansion.

Forward Compton Scattering Amplitude as a Simultaneous Analytic Function of Complex Photon Mass and Energy

The analytic structure of the virtual forward Compton scattering amplitude as a function of one and two complex variables is investigated for various combinations of variables which involve the virtual photon mass. This is done both by using the Deser-Gilbert-Sudarshan representation and by using the Feynman perturbation theory. The role and significance of complex Landau singularities is discussed. In general these are branch points, but it is found that at $t=0$ some of these become poles. The effect of these complex singularities and of overlapping cuts on the ordinary single-variable dispersion relations and the mass extrapolations in the vector-meson-dominance model is explained. Using the Cutkosky discontinuity formula for Feynman graphs, the analytic structure of the discontinuities across various normal threshold cuts of the non-Born-term part of this amplitude is deduced. One finds that the two-variable analyticity of the amplitude implies a single-variable analyticity for these discontinuities. It is shown that in general the inelastic structure functions $W$ and $\overline{W}$ cannot be expected to have a simply determinable analytic structure. But under certain conditions $W$ and $\overline{W}$ can be identified with boundary values of the discontinuities across the $s$-and $u$-channel normal threshold cuts in the virtual forward Compton scattering amplitude, respectively. This is used to show that the contribution to $W$ and $\overline{W}$ from certain types of Feynman graphs under certain conditions are analytic functions of one complex variable, and only for such cases can one use crossing symmetry to relate the inelastic-electron-scattering structure functions and the annihilation structure functions. The motion of the Landau singularities of $\ensuremath{\nu}{W}_{2}$ is shown to provide a possible explanation for its observed rapid approach to "scaling" for finite but large final-state masses.

Perturbative unitarity of dual loops

TheN-loop dual graphs are classified and enumerated in terms of integration contours on a sphere withN handles. Perturbative unitarity at the leading Landau singularity dictates that all inequivalentN-loop diagrams (unrelated by duality) be added with equal weight. A proof of perturbative unitarity for the resulting expansion is hindered by the presence of complex anomalous thresholds.

Landau singularities in multi-Regge theory and fixed-angel behaviour

A study of the implications of the Landau singularity structure in multi-Regge theory is begun. The incorporation of a physical-region pole in the three-Reggeon description of the six-particle amplitude is considered. Triple-Regge limits are found to involve fixed-angle limits of the four-particle amplitudes appearing in the residue of the pole. It is shown that under certain conditions the presence of the physical-region pole necessitates an « undesirable » set of angular-nmmeutum singularities of the forml1+l2+l3 = const. Exponential decrease of thc four-particle amplitude at fixed angle is then considered and it is shown that some examples of such behaviour including those most favoured by experimental results, do not necessitate these singularities.

Possible Explanation for the Rapid Approach to "Universality" of the Inelastic Electron-Scattering Structure Functions

We have determined the single-variable analyticity of the inelastic electron-scattering structure functions in the complex $x$ plane, with $s$ kept fixed and real, to all orders of Feynman perturbation theory. We find that its Landau singularities, which move as a function of $s$, rapidly approach their asymptotic $s$-independent position once $s$ is large. We discuss how this observation offers a possible explanation for a rapid approach to universality of the inelastic electron-scattering structure functions and shows that $s{W}_{2}(s,\frac{{q}^{2}}{s})$ should scale faster than $\ensuremath{\nu}{W}_{2}(s,x)$.

The Landau singularities for a double-stream instability plasma

Dispersion relation roots in a double stream instability plasma are given: the pole responsible for this instability is not the Maxwellian Landau pole. Then, we determine the analytic value of the Landau pole in a quasi-linear theory.

The structure of singularities of Feynman integrals

We study in detail the structure of the pinch manifold of second-type singularities of Feynman integrals. Further, we argue that the strength of second-type singularities, in contrast to the strength of Landau singularities. does not in general decrease with the order of the graph. Finally, we give a simple analysis of mixed-type singularities using a modified α-representation for the Feynman integral and a prescription for determining the strength of these singularities.

A homological approach to parametric Feynman integrals

The methods of homology theory are applied to Feynman integrals in α-space. In the case of single-loop graphs the relevant relative homology groups are computed and the results compared with those obtained byk-space methods. For graphs with more than one loop the permanent pinch difficulty is overcome by modifying the ambient manifold and applying Thom's isotopy theorem. Pinching conditions are then found for Landau singularities in which complete circuits are contracted out and for mixed second-type singularities. The breakdown, of the hierarchical principle in perturbation theory is also explained from this point of view.

Observable Effects of the Leading Landau Singularity of the Box Graph

Observable Effects of the Leading Landau Singularity of the Box Graph

Normal Thresholds in Subenergy Variables

The discontinuity across the normal threshold in a subenergy variable is evaluated by S-matrix methods. The effect on the discontinuity of continuing past higher Landau singularities is discussed.

Analytic Properties of the Elastic Unitarity Integral

In this paper, certain results concerning the singularities of holomorphic functions with integral representations are used to investigate the analytic properties of the elastic unitarity integral of the quantum theory of fields. In particular, it is shown that each of the Landau singularities of the scattering amplitude is an actual singularity. It is also shown that if the scattering amplitude is not identically zero it must have a natural boundary on the unphysical sheet.

Proof that Scattering Implies Production in Quantum Field Theory

A proof of the necessity of production processes in quantum field theory is carried out in the axiomatic framework. It is shown that a field theory that is assumed to have a nontrivial scattering amplitude violates crossing symmetry, if production processes are absent. The proof is based on the rigorous analytic properties of the scattering amplitude, particularly the analyticity in the invariant-scattering variables s, t, and u. In the case of a scalar theory with pairing symmetry, the scattering amplitude is known to be analytic within the domain |stu| < 7168m6, except for the usual cuts. Under the working assumption that production processes are null, it is shown that this domain can be enlarged by applying the elastic unitarity conditions beyond the (usual) elastic region. The domain is enlarged sufficiently to include the first Landau singularity of the absorptive part of the scattering amplitude. This singularity is not symmetric in s and t within the extended domain, and this is incompatible with the crossing symmetry of the scattering amplitude. In order to avoid a contradiction, the discontinuity across this Landau singularity must be null. It follows that the scattering amplitude must itself be null. In the course of the proof it is shown that the conclusion is valid for a scattering amplitude satisfying the requirements of an S-matrix theory embodied in the Mandelstam representation.

The singularities of asymptotic quantum field theory

It has been shown recently that the L.S.Z. formulation of quantum field theory can produce finite solutions for theS-matrix elements to all orders in perturbation theory. In this paper these solutions for the scattering of an arbitrary number of scalar bosons are examined in momentum space and their singularity structure is found. It is shown that the extension off the mass shell, as carried out byPugh (1), of the fundamental L.S.Z. equations, yields solutions which are the continuation of the right-hand side of the unitarity relation,R+R †=R † R to unphysical off-mass-shell regions. The analytic mass-shell amplitudes are constructed using the unitarity condition and this continuation. By induction it is shown that the perturbation solutions of asymptotic quantum field theory possess in any order only the singularities which are required by unitarity. These have been shown byStapp (3) to correspond to the Landau singularities of ordinary quantum field theory.

Locations of Landau Singularities

It is shown for the third-order triangle and the fourth-order box Feynman diagrams that singularities exist on the physical boundaries of the amplitudes if, and only if, the external momenta are such as to allow the reactions to actually proceed with real intermediate particles. If, furthermore, the information concerning the locations and appropriate branches of singularities implied by this principle is combined with the requirement that discontinuities be given by Cutkosky's rules, the amplitudes are uniquely determined.

Perturbation Analyticity and Axiomatic Analyticity. I. Connection of the Landau Singularity Manifold with KällénΞn(t)Manifold and Jost DANAD Manifold

For a class of Feynman graphs ${\mathcal{G}}_{n}$ (single-loop diagrams with all internal diagonals), the $p$-space perturbation Landau singularity manifolds are shown to be formally of the same structure as the K\"all\'en ${\ensuremath{\Xi}}_{n}(t)$ manifolds for the $x$-space axiomatic primitive domain. The boundary of the Landau manifold is then shown to be the (DANAD)\ensuremath{'} manifold. The relationship between the (DANAD)\ensuremath{'} and the Jost (DANAD) manifold is a precise generalization of what exists between the ${{F}_{\mathrm{kl}}}^{\ensuremath{'}}$ and ${F}_{\mathrm{kl}}$ surfaces of K\"all\'en and Wightman. Since the (DANAD)\ensuremath{'} defines a natural domain of holomorphy, the axiomatic envelope of holomorphy cannot be expected to be continuable beyond (DANAD)\ensuremath{'}. This (DANAD)\ensuremath{'} result furnishes a specific conjecture to part of the envelope of holomorphy.

Second-type singularities of Feynman graphs

An analysis is made of second-type (non-Landau) singularities for arbitrary Feynman graphs. It is shown that the second-type singularities of Feynman graphs can in the general case be reduced to the conventional (Landau) singularities of the graphs reduced in the corresponding manner. A method is indicated of finding such reduced graphs from the initial Feynman graph. An elaborate study is made of the conditions of the appearance of second-type singularities for one- and two-loop graphs, and characteristic examples of such graphs are considered. Also conditions are studied for the second-type singularities in the case of the simplest three-loop graph (“triangle”).

Second-type singularities in perturbation theory

The second-type (non Landau) singularities of perturbation theory are investigated. It is shown that they can be represented by a graphical analysis similar to the dual diagrams of Karplus, Sommerfield and Wichmann and others. The discontinuity across cuts associated with second-type singularities is deduced from a generalized form of Cutkosky’s formula.