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Divergent integrals in quantum field theory (QFT) can be given well defined existence as Lorentz covariant complex measures, which may be analyzed by means of a spectral calculus. The case of the photon self energy is considered and the spectral vacuum polarization function is shown to have very close agreement with the vacuum polarization function obtained using dimensional regularization / renormalization in the timelike domain. Using the spectral vacuum polarization function a potential function defined in the timelike domain is derived. The Uehling potential function, from which the Uehling contribution to the Lamb shift may be computed, is derived from an analytic continuation into the spacelike domain of this potential function. The spectral running coupling for QED is computed from this analytically continued potential function. The integral defining the spectral running coupling constant is shown to converge for all non-zero energies while that for the running coupling constant computed using dimensional regularization / renormalization is shown to diverge for all non-zero energies. It is seen that the spectral running coupling does not have a Landau pole and agrees both qualitatively and quantitatively with the results of scattering experiments at all energies.

Well-known measures of entanglement in one-dimensional many body quantum systems, such as the entanglement entropy and the logarithmic negativity, may be expressed in terms of the correlation functions of local fields known as branch point twist fields in a replica quantum field theory. In this "replica" approach the computation of measures of entanglement generally involves a mathematically non-trivial analytic continuation in the number of replicas. In this paper we consider two-point functions of twist fields and their analytic continuation in the 1+1 dimensional massive (non-compactified) free Boson theory. This is one of the few theories for which all matrix elements of twist fields are known so that we may hope to compute correlation functions very precisely. We study two particular two-point functions which are related to the logarithmic negativity of semi-infinite disjoint intervals and to the entanglement entropy of one interval. We show that our prescription for the analytic continuation yields results which are in full agreement with conformal field theory predictions in the short-distance limit. We provide numerical estimates of universal quantities and their ratios, both in the massless (twist field structure constants) and the massive (expectation values of twist fields) theory. We find that particular ratios are given by divergent form factor expansions. We propose such divergences stem from the presence of logarithmic factors in addition to the expected power-law behaviour of two-point functions at short-distances. Surprisingly, at criticality these corrections give rise to a log(logL) correction to the entanglement entropy of one interval of length L. This hitherto overlooked result is in agreement with results by Calabrese, Cardy and Tonni and has been independently derived by Blondeau-Fournier and Doyon (in preparation)., Comment: 35 pages, 6 figures (v3 is the published version which incorporates minor improvements)

We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the K\"all\'en-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the K\"all\'en-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularize this problem we implement an appropriate version of Tikhonov regularization supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two ---mathematically equivalent--- versions of the K\"all\'en-Lehmann spectral integral., Comment: 33 pages, 18 figures

We consider two integrable deformations of 2d sigma models on supercosets associated with AdS n × S n . The first, the “η-deformation” (based on the Yang–Baxter sigma model), is a one-parameter generalization of the standard superstring action on AdS n × S n , while the second, the “λ-deformation” (based on the deformed gauged WZW model), is a generalization of the non-abelian T-dual of the AdS n × S n superstring. We show that the η-deformed model may be obtained from the λ-deformed one by a special scaling limit and analytic continuation in coordinates combined with a particular identification of the parameters of the two models. The relation between the couplings and deformation parameters is consistent with the interpretation of the first model as a real quantum deformation and the second as a root of unity quantum deformation. For the AdS 2 × S 2 case we then explore the effect of this limit on the supergravity background associated with the λ-deformed model. We also suggest that the two models may form a dual Poisson–Lie pair and provide direct evidence for this in the case of the integrable deformations of the coset associated with S 2 .

In this paper, by employing the k T factorization theorem, we made the first calculation for the space-like scalar pion form factor Q 2 F ( Q 2 ) at the leading order (LO) and the next-to-leading order (NLO) level, and then found the time-like scalar pion form factor F a , I ′ ( 1 ) by analytic continuation from the space-like one. From the analytical evaluations and the numerical results, we found the following points: (a) the NLO correction to the space-like scalar pion form factor has an opposite sign with the LO one but is very small in magnitude, can produce at most 10 % decrease to the LO result in the considered Q 2 region; (b) the NLO time-like scalar pion form factor F a , I ′ ( 1 ) describes the O ( α s 2 ) contribution to the factorizable annihilation diagrams of the considered B → π π decays, i.e. the NLO annihilation correction; (c) the NLO part of the form factor F a , I ′ ( 1 ) is very small in size, and is almost independent of the variation of cutoff scale μ 0 , but this form factor has a large strong phase around − 55 ° and may play an important role in producing large CP violation for B → π π decays; and (d) for B 0 → π + π − and π 0 π 0 decays, the newly known NLO annihilation correction can produce only a very small enhancement to their branching ratios, less than 3 % in magnitude, and therefore we could not interpret the well-known ππ-puzzle by the inclusion of this NLO correction to the factorizable annihilation diagrams.

Open string disk integrals are represented as contour integrals of a product of Beta functions using Mellin transformations. This makes the mathematical problem of computing the α ′ -expansion around the field-theory limit similar to that of the ϵ -expansion in Feynman loop integrals around the four-dimensional limit. More explicitly, the formula in Mellin space obtained directly from the standard Koba–Nielsen-like representation is valid in a region of values of α ′ that does not include α ′ = 0 . Analytic continuation is therefore needed since contours are pinched by poles as α ′ → 0 . Deforming contours that get pinched by poles generates a set of ( n − 3 ) ! multi-dimensional residues left behind which contain all the field theory information. Some analogies between the field theory formulas obtained by this method and those derived recently from using the scattering equations are commented at the end.

The O ( α s 3 n f T F 2 C A , F ) terms to the massive gluonic operator matrix elements are calculated for general values of the Mellin variable N using a new summation technique. These twist-2 matrix elements occur as transition functions in the variable flavor number scheme at NNLO. The calculation uses sum-representations in generalized hypergeometric series turning into harmonic sums. The analytic continuation to complex values of N is provided.

We present formulae for the calculation of Dirac gaugino masses at leading order in the supersymmetry breaking scale using the methods of analytic continuation in superspace and demonstrate a link with kinetic mixing, even for non-abelian gauginos. We illustrate the result through examples in field and string theory. We discuss the possibility that the singlet superfield that gives the U(1) gaugino a Dirac mass may be a modulus, and some consequences of the D-term coupling to the scalar component. We give examples of possible effects in colliders and astroparticle experiments if the modulus scalar constitutes decaying dark matter., Comment: 17 pages, no figures. Published version, one reference added

An important “observable” of planar N=4 SYM theory is the scaling function f(λ) that appears in the anomalous dimension of large spin twist 2 operators and also in the cusp anomaly of light-like Wilson loop. The non-trivial relation between the anomalous dimension and the Wilson loop interpretations of f(λ) is well-understood on the perturbative gauge theory side of the AdS/CFT duality. In the first part of this paper we present the dual string-theory counterpart of this relation, i.e., the equivalence between the closed-string and the open-string origins of f(λ). We argue that the coefficient of the logS term in the energy of the closed string with large spin S in AdS5 should be equal to the coefficient in the logarithm of expectation value of the null cusp Wilson loop, to all orders in λ−1/2 expansion. The reason is that the corresponding minimal surfaces happen to be related by a conformal transformation (and an analytic continuation). As a check, we explicitly compute the leading 1-loop string sigma model correction to the cusp Wilson loop, reproducing the same subleading coefficient in f(λ) as found earlier in the spinning closed string case. The same function f(λ) appears also in the resummed form of the 4-gluon amplitude as discussed at weak coupling by Bern, Dixon and Smirnov and recently found at the leading order at strong coupling by Alday and Maldacena (AM). Here we attempt to extend the latter approach to a subleading order in λ−1/2 by computing the IR singular part of the 1-loop string correction to the corresponding T-dual Wilson loop. We discuss explicitly the 1-cusp case and comment on apparent problems with the dimensional regularization proposal of AM when directly applied order by order in strong coupling (string inverse tension) expansion.

We derive the DGLAP and BFKL evolution equations in the N =4 supersymmetric gauge theory in the next-to-leading approximation. The eigenvalue of the BFKL kernel in this model turns out to be an analytic function of the conformal spin | n |. Its analytic continuation to negative | n | in the leading logarithmic approximation allows us to obtain residues of anomalous dimensions γ of twist-2 operators in the non-physical points j =0,−1,… from the BFKL equation in an agreement with their direct calculation from the DGLAP equation. Moreover, in the multi-color limit of the N =4 model the BFKL and DGLAP dynamics in the leading logarithmic approximation is integrable for an arbitrary number of particles. In the next-to-leading approximation the holomorphic separability of the pomeron Hamiltonian is violated, but the corresponding Bethe–Salpeter kernel has the property of a Hermitian separability. The main singularities of anomalous dimensions γ at j =− r obtained from the BFKL and DGLAP equations in the next-to-leading approximation coincide but our accuracy is not enough to verify an agreement for residues of subleading poles.

The conductivity of a finite temperature 1+1 dimensional fermion gas described by the massive Thirring model is shown to be related to the retarded propagator of the dual boson sine-Gordon model. Duality provides a natural resummation which resolves infra-red problems, and the boson propagator can be related to the fermion gas at non-zero temperature and chemical potential or density. In addition, at high temperatures, we can apply a dimensional reduction technique to find resummed closed expressions for the boson self-energy and relate them to the fermion conductivity. Particular attention is paid to the discussion of analytic continuation. The resummation implicit in duality provides a powerful alternative to the standard diagrammatic evaluation of transport coefficients at finite temperature., 41 pages, 6 figures

The sine-Gordon model on the half-line with a dynamical boundary introduced by Delius and one of the authors is considered at quantum level. Classical boundary conditions associated with classical integrability are shown to be preserved at quantum level too. Non-local conserved charges are constructed explicitly in terms of the field and boundary operators. We solve the intertwining equation associated with a certain coideal subalgebra of $U_q(\hat{sl_2})$ generated by these non-local charges. The corresponding solution is shown to satisfy quantum boundary Yang-Baxter equations. Up to an exact relation between the quantization length of the boundary quantum mechanical system and the sine-Gordon coupling constant, we conjecture the soliton/antisoliton reflection matrix and boundstates reflection matrices. The structure of the boundary state is then considered, and shown to be divided in two sectors. Also, depending on the sine-Gordon coupling constant a finite set of boundary bound states are identified. Taking the analytic continuation of the coupling, the corresponding boundary sinh-Gordon model is briefly discussed. In particular, the particle reflection factor enjoys weak-strong coupling duality., Comment: 15 pages, LaTeX file with amssymb, v2: references added, Comments added, typos corrected. To appear in Nucl.Phys. B

Loop calculations involve the evaluation of divergent integrals. Usually [1] one computes them in a number of dimensions different than four where the integral is convergent and then one performs the analytical continuation and considers the Laurent expansion in powers of epsilon =n-4. In this paper we discuss a method to extract directly all coefficients of this expansion by means of concrete and well defined integrals in a five dimensional space. We by-pass the formal and symbolic procedure of analytic continuation; instead we can numerically compute the integrals to extract directly both the coefficient of the pole 1/epsilon and the finite part., 13 pages, 1 Postscript figure

The small-x_Bj limit of deep inelastic scattering is related to the high-energy limit of the forward Compton amplitude in a familiar way. We show that the analytic continuation of this amplitude in the energy variable is calculable from a matrix element in Euclidean field theory. This matrix element can be written as a Euclidean functional integral in an effective field theory. Its effective Lagrangian has a simple expression in terms of the original Lagrangian. The functional integral expression obtained can, at least in principle, be evaluated using genuinely non-perturbative methods, e.g., on the lattice. Thus, a fundamentally new approach to the long-standing problem of structure functions at very small x_Bj seems possible. We give arguments that the limit x_Bj -> 0 corresponds to a critical point of the effective field theory where the correlation length becomes infinite in one direction., 25 pages, LaTeX file (including 6 eps-figures)

We consider scattering amplitudes in non-critical string theory of $N$ external states in the limit where the energy of all external states is large compared to the string tension. We argue that the amplitudes are naturally complex analytic in the matter central charge $c$ and we propose to define the amplitudes for arbitrary value of $c$ by analytic continuation. We show that the high energy limit is dominated by a saddle point that can be mapped onto an equilibrium electro-static energy configuration of an assembly of $N$ pointlike (Minkowskian) charges, together with a density of charges arising from the Liouville field. We argue that the Liouville charges accumulate on segments of curves, and produce quadratic branch cuts on the worldsheet. The electro-statics problem is solved for string tree level in terms of hyper-elliptic integrals and is given explicitly for 3- and 4-point functions. We show that the high energy limit should behave in a string-like fashion with exponential dependence on the energy scale for generic values of $c$., Comment: 34pp, 8 figures, harvmac, epsf (TeXing problems fixed)

We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the Thermodynamic Bethe Ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one- and two- particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal space approach, and we also treat some of the ultraviolet and infrared asymptotics analytically., 25 pages, latex, 6 ps files, uses epsf.tex. Typos corrected

Thermal activation is mediated by field configurations that correspond to saddle points of the energy functional. The rate of probability flow along the unstable functional directions, i.e. the activation rate, is usually obtained from the imaginary part of a suitable analytic continuation of the equilibrium free energy. In this note we provide a real-time, non-equilibrium interpretation of this imaginary part which is analogous to the real-time interpretation of the imaginary part of the one-loop effective potential in theories with symmetry breaking. We argue that in situations in which the system is strongly out of equilibrium the rate will be time dependent, and illustrate this with an example.

We construct a topological Landau-Ginzburg formulation of the two-dimensional string at the self-dual radius. The model is an analytic continuation of the Ak+1 minimal model to k = −3. We compute the superpotential and calculate tachyon correlators in the Landau-Ginzburg framework. The results are in complete agreement with matrix model calculations. We identify the momentum one tachyon as the puncture operator, non-negative momentum tachyons as primary fields, and negative momentum ones as descendants. The model thus has an infinite number of primary fields, and the topological metric vanishes on the small phase space when restricted to these. We find a parity invariant multi-contact algebra with irreducible contact terms of arbitrarily large number of fields. The formulation of this Landau-Ginzburg description in terms of period integrals coincides with the genus zero W1+∞ identities of two-dimensional string theory. We study the underlying Toda lattice integrable hierarchy in the Lax formulation and find that the Landau-Ginzburg superpotential coincides with a derivative of the Baker-Akhiezer wave function in the dispersionless limit. This establishes a connection between the topological and integrable structures. Guided by this connection we derive relations formally analogous to the string equation.

We compute the exact spectral density of random matrices in the ground state of the quantum hamiltonian corresponding to the matrix model whose double scaling limit describes pure gravity in 2D. We show that the non-perturbative effects are very large and in certain cases dominate the semi-classical WKB contribution studied in the earlier literature. The physical observables in this model are the loop averages with respect to the spectral density. We compute their exact ground-state expectation values and show that they differ significantly from the values obtained in the WKB approximation. Unlike the alternative regularizations of the nonperturbative 2D quantum gravity, based on analytic continuation of the Painlev\'e transcendent, our solution shows no pathologies., Comment: 14 pages (LaTeX) + 4 postscript figures encoded through uufiles. PUPT-1354, TAUP-2013-92

We study the high-temperature behaviour of the n-point function to one-loop order in thermal QCD, in the analytic continuation of the imaginary-time formalism. Using ideas of Barton, we relate these functions to angular integrals of the forward-scattering amplitudes (for the thermal particles) in the high-energy limit. These amplitudes are Lorentz and gauge invariant, and the Ward identities determine all of them uniquely in terms of the two-point one. We are thus able, by gauge invariance, to write down a generating function for all n-point functions. Its relationship to a previous one (with a Lorentz and gauge non-invariant integrand) is explained.

Thermodynamic Bethe ansatz equations are suggested for RSOS scattering theories, which describe the scattering of kinks in the field theories, corresponding to the perturbations of the minimal CFT models. M p by relevant operators φ1.3 with negative coupling constant. The equations turn to be of Ap−2 type with a special choice of the mass terms. For the M 4 case (tricritical Ising) the equations are studied numerically. This gives high precision data for the Casimir energy of the finite-volume system. The analytic continuation in the coupling constant results in reasonable numerical data for positive values of coupling. The data support the interpretation of the positive-coupling field theory as the trajectory flowing from the tricritical Ising fixed-point to the critical Ising one.

We directly derive the crossing (fusion and braiding) matrices for arbitrary isospins of the WZW SU(2) model in the Feigin-Fuchs integral representation and prove explicitly that they are quantum Racah-Wigner 6j matrices under proper normalization conditions. For integer k of the Kac-Moody algebra, the matrices may be truncated and the admissible states are closed under analytic continuation. The non-admissible terms will not appear in the other channels after braiding. The consistency of crossing matrices with the locality condition is also proven. Similar results are obtained for the minimal models, which are in agreement with the results obtained by G. Felder, J. Frolich and G. Keller.

We study the high-temperature behaviour of the n -point function to one-loop order in thermal QCD. We employ an analytic continuation of the imaginary-time formalism. The leading terms are all of order T 2 . They are totally symmetric in their Lorentz indices and traceless. They are gauge independent and obey simple, QED-like, Ward identities. The n -point functions for n >3 can be expressed as linear combinations with rational coefficients of the 2- and 3-point functions - a fact which follows from the symmetry, tracelessness and Ward identities.

The conduction properties of a two-dimensional tight-binding model with on-site disorder and an applied perpendicular magnetic field with precisely one-half of a magnetic flux quantum per plaquette are studied. A continuum hamiltonian is derived which enables the construction of a field theory for the diffusive modes. The field theory is shown to be in the universality class of the O (2n, 2n) O (2n) × (2n)(n → 0) non-linear σ-model implying that all the electronic states are localized. The system is shown to be related, via an analytic continuation, to a system of self-interacting fermions in 1 + 1 dimensions, in the n → 0 limit.

The predictions of perturbative QCD are derived in the deep euclidean region, whereas the physical region for most observables is timelike. The confrontation of these predictions with experiment thus necessitates an analytic continuation. This we find introduces large higher order corrections in terms of αs(|Q2|), the usual choice ofperturbative expansion parameter. These corrections are naturally absorbed by changing to the expansion parameter a (Q 2 ) = |α s (Q 2 )|( Re α s (Q 2 )/|α s (Q 2 )|) (n−2) 3 , where αs(Q2)n is the leading term in the spacelike region. For the intermediate range of Q2 experimentally accessible at present, where a (Q 2 ) is significantly smaller than αs(|Q2|), we find the resulting phenomenology is improved. In particular, we demonstrate how the values of Λ MS obtained from analyses of quarkonium decays become consistent.

A regular covariant method is proposed for expanding the one-loop effective action in powers of curvatures. The expansions is nonlocal and corresponds to summation of the local Schwinger-DeWitt series. The effective equations are computed for both euclidean and lorentzian signatures of spacetime. The diagrammatic technique for expectation values is used to prove that the nonlocal effective equations for the mean field in the in-vacuum state can be obtained from euclidean equations by a special analytic continuation.

Renormalization group equations and the specific heat singularity of the 2D q -state Potts model with weak quenched bond randomness are obtained to two-loop order in an expansion in the parameter ( q − 2) around the Ising model, in analogy to the e-expansion for φ 4 scalar field theory around the gaussian model. The following result for the random thermal exponent is obtained: y T = 1 − 1 2 (1 − x e ) 2 + O ((1 − x e ) 3 ) where x e is the dimension of the energy operator in the pure model. The perturbation expansion involves non-gaussian correlations in the pure model which are obtained from conformal invariance. It is shown to two loops that, as a consequence of the operator product expansion, all divergences of the non-gaussian perturbation series can be absorbed into renormalization of the parameters and that analytic continuation in q may be used to obtain renormalization group equations by minimal subtraction of poles in ( q − 2). The limit q → 2 and corrections to the ln ln-form of the specific heat of the random Ising model are studied. A new universal amplitude ratio is computed for the random Ising model.

Two systems, O(n) non-linear σ-model and CP ( n −1) , are studied in the light of Elitzur's theorem, on the disappearance of infrared singularities at two dimensions. The consequences of the theorem are expressed in dimensional regularization, and issues like the proper analytic continuation to d = 2 + e , the peculiarities of momentum-space Green functions near d = 2 and their renormalization, and the exponentiation of Green functions are clarified. The analysis is applied to compute the renormalization constants, and the gauge-invariant critical exponent η associated with the wave function of CP ( n −1) at one order higher than previously done. Finally, we conjecture on a possible connection between infrared finiteness and renormalizability.

Contact interactions are shown to be required in superstring theory in order to ensure either world-sheet supersymmetry in a covariant formalism or spacetime supersymmetry in the light-cone gauge. These interactions only contribute for specific ranges of the momenta of the emitted particles. Their role is to ensure analyticity of S-matrix elements in much the same way as with contact terms in ordinary field theories with point particles. For tree diagrams the usual functional calculations, performed at sufficiently spacelike momentum transfer, give correct results. For other values of the momenta the correct answer is obtained either by analytic continuation or by adding the contribution of the contact interactions. In certain loop calculations the effect of the contact interactions must be taken into account. In the light-cone gauge formalism the contact terms ensure positivity of the hamiltonian.

A quasi-particle description of a quantum electrodynamical plasma at finite temperature and density is sought in the context of the imaginary-time formalism of quantum field theory. Functional techniques and analytic continuation are used to derive an expression for the charge density, of the same form as valid for a non-interacting system, in terms of an effective propagator similar to, but different from, the conventional many-body fermion propagator. In contrast to the latter, the effective propagator possesses a pole on the real axis. This defines a statistical quasi-particle, in the sense of Balian and de Dominicis, having a real mass, which is explicitly evaluated at the one-loop level. Remarkable features of this lowest-order mass are that it is independent of the chemical potential, and that it implies a zero-temperature mass shift.

Infrared singularities arising in some renormalized amplitudes of quantum electrodynamics are analyzed using the dimensional regularization method. We define infrared and ultraviolet convergent regions in the ν complex plane (ν is the number of dimensions of space time). It turns out that these regions do not overlap for quantum electrodynamics. Nevertheless, it is shown that there exists a unique analytic continuation from the infrared convergent region which allows us to interpret the infrared divergence in the renormalized electron self-energy amplitude as an isolated singularity at ν = 4. This statement seems to be true at all orders of perturbation theory. We also prove that the double limit μ → 0, ν → 4 (μ is the auxiliary photon mass) does not exist in quantum electrodynamics and we conjecture that this lack of uniformity provides theoretical support for the ansatz of Marciano and Sirlin.

We present a parton model interpretation of the predictions of quantum chromodynamics in the process e+e−→hadron + anything. We give thecomplete list of parameters needed for the study of the scaling violations of fragmentation functions up to the next-to-leading logarithmic approximation. This includes flavour non-singlet and flavour singlet anomalous dimensions up to order α2 and coefficient functions up to order α. We also present results for the deep inelastic scattering e−h→e− + anything. We find that in e+e− annihilation the ratio of scaling violations of second order to first order is in general bigger than the corresponding ratio for deep inelastic scattering. The Gribov-Lipatov relation is thus violated in second order. We also find that a modified Drell-Yan analytic continuation relation holds between the deep inelastic and annihilation structure functions for quarks and gluons. In x space we give detailed numerical evaluation of the QCD effects for non-singlet and singlet densities, in the space-like and time-like regions.

An expression for the one-loop self-energy contribution to excited string states on the leading trajectory, J = 2 + α′m2, of closed superstrings is derived. Since the masses of such states are above threshold for decay, a useful formula has to depend analytically on m2to allow the determination of finite real and imaginary parts by analytic continuation. This is achieved by factorization of a one-loop four-point amplitude. The procedure is tested by a calculation of δm2 for strings compactified to D dimensions in the limit of large m2. Results are obtained for D > 6, but difficulties are encountered in lower dimensions. Possible explanations of this problem are discussed.

A general non-perturbative approach to functional integrals describing systems possessing both thermal and quantum fluctuations is presented. The partition function is evaluated in the semiclassical limit by enlarging the class of paths in the functional integral to include complex-valued solutions of the classical equations of motion. Semiclassical energy eigenvalues, including tunnelling effects associated with decay of false vacua, can be obtained in this way for all states, not just the ground state. In this approach the imaginary part of an energy eigenvalue of an unstable state is directly related to physical boundary conditions at infinity; these results are compared to earlier attempts (fate of the false vacuum) to compute decay rates of metastable systems, involving an analytic continuation prescription for a negative mode of the second-order fluctuation operator. Use of complex-valued solutions obviates the need to employ the dilute-gas approximation which is generally invalid at finite temperature. The new method is checked by one-dimensional examples, and then extended to field theory, where a general expression for the partition function of a field theory is given in terms of stability angles about a complex classical solution.

We develop a technique, based explicitly on the factorization properties of mass singularities, which allows one to calculate the evolution of parton densities beyond leading order. We present the results for the evolution of hadronic structure functions as well as for parton fragmentation functions into hadrons. Within our scheme the predictions for a particular process are obtained by convoluting a universal parton density with a “short-distance” cross section specific to the process. As an application, we calculate the QCD predictions for the Q2 dependence of deep inelastic lepton-hadron scattering and of one-particle iclusive e+ e− annihilation cross sections. Our results for electroproduction agree with those obtained with the operator product expansion technique. Physical quantities in scattering are related to the corresponding ones in annihilation by analytic continuation, whereas the Gribov-Lipatov relation is strongly violated.

Functional integrals for the Green functions in the massless [gϕ4]4 theory are analytically continued from the physical domain g > 0 to the unphysical domain g = |g| e±iπ. The main contribution ∼ϵ = e−16π2/gh to the discontinuity of the functional integral on the cut along the negative g axis is shown to be made by complex instanton solutions of the classical field equations. The general structure of the Green functions in the unphysical domain is considered and it is shown that there are two expansion parameters: ϵ and gh. The method of calculating quantum corrections to the quasi-classical approximation is developed. These corrections are expressed in the form of converging integrals over the parameters of the instanton solution. We show that renormalized Green functions in the unphysical domain are ultraviolet and infrared finite. As an illustration we calculate the discontinuity of the Gell-Mann-Low function on the cut g < 0 in the gaussian approximation. The dispersion representation for the Gell-Mann-Low function gives the Borel sum of leading terms of the perturbation expansion of this function.

The structure of the non-compact gaugings of N = 8 supergravity is analyzed. The SO(p,q) gaugings are shown to be related to the SO(8) model by a form of analytic continuation. This gives a method for calculating the scalar dependence of the potential and Yukawa interactions in terms of the corresponding quantities in the de Wit-Nicolai model. The truncations to N = 4 supergravity coupled to N = 4 Yang-Mills multiplets are given.

It is shown that the light-cone operator expansion of the product of two SU(3) ⊗ SU(3) currents or divergences, the requirement of maximal analyticity of the second kind, and the mathematical method of analytic continuation can be used to shed light on the validity and meaning of the Ward-Takahashi identities, of the Bjorken-Johnson-Low limit and in general of sum rules derived from current algebra either at equal time or on the light cone. In particular we give a definite content to the PCAC hypothesis in terms of the ‘dimension’ of the axial divergence and suggest some specific values for this ‘dimension’. It is also shown that the requirement of maximal analyticity of the second kind and gauge invariance impose a non-trivial condition on the Cornwall-Corrigan-Norton sum rule.

The chiral anomaly in heterotic strings with full and partial modular invariance in D = 2n + 2 dimensions is calculated. The boundary terms which were present in previous calculations are shown to be cancelled in the modular-invariant case by contact terms, which can be obtained by an appropriate analytic continuation. The relation to the low-energy field theory is explained. In theories with partial modular invariance, an expression for the anomaly iis obtained and shown to be non-zero in general.

A study is made of theories having the unusual analyticity properties recently proposed by Lee and Wick. A prescription is given for setting up a covariant perturbation-theory expansion of the scattering amplitudes, based on Feynman graphs. It is found that the presence of a complex pole in the upper half plane of the physical sheet leads to points of non-analyticity in the physical region, such that the values of the amplitude to either side of the point are not related by analytic continuation. It is shown how this is compatible with unitarity. The nature of the non-analyticity is not fully determined by unitarity. Neither, in the case of the more complicated graphs, is it fully determined by the perturbation-theory prescription, and some extra constraint must be imposed on the theory to remove the ambiguity. It is shown that the prescription of Lee and Wick has an exactly similar ambiguity, but for their prescription different results are obtained in different Lorentz frames. An estimate is made of the extent to which the theory violates causality, and is found to be too small to measure.

A diagonal form for the continuation of a multiparticle unitarity integral to complex angular momentum is given involving the “Froissart-Gribov” continuations of multiparticle amplitudes to complex helicity and angular momentum defined in a previous paper. The form for the continuation of the three-particle unitarity integral is considered in detail and shown to converge and satisfy the Carlson condition on the asymptotic behaviour under quite general conditions.

A Lagrangian formalism is constructed for a symmetric, traceless tensor (spinor-tensor) of rank N which describes a multiplet with spin 0, 1, 2, …, N( 1 2 , 3 2 , …, N + 1 2 ) . From physical quantities that are obtainable from this formalism we derive the corresponding quantities for an infinite multiplet with integer (half-integer) spin by means of an analytic continuation which changes the original field into a basis of an infinite-dimensional, unitary representation of the Lorentz group. Feynman rules for infinite multiplets are determined in this way. Within this formalism mass spectra of infinite multiplets are always of the Majorana type, and the difficulties related to the spin-statistics theorem and spacelike solutions do not arise.

In using FESR at fixed momentum transfer an analytic continuation of the amplitude into the uhysical region near to threshold is necessary. We show how direct channel Regge poles may be used to facilatate this and give a detailed application to the π N amplitudes.

Following the Cutkosky theory of representation of data by analytic functions, a probability functional is introduced which takes account of the different statistical errors in the determination of the real and imaginary parts of a function along its boundary of analyticity. With this probability the most probable function is found and the mean value and standard deviation of any functional are determined in terms of this function. These results are applied to the problem of continuation of a function off its boundary and a modified dispersion relation derived which defines the best continuation possible given the form of the probability function.

The structure functions of deep inelastic scattering and of deep inelastic annihilation of a current in the go 3 theory are studied in the ladder approximation. To this aim an integral equation which is the scaling limit of the classical ABFST equation is used. By solving this equation in a particular case we obtain in closed form the sum of all ladder diagrams with all massless rungs except for the first and the last ones which are given arbitrary masses. The solution is a hypergeometric function in two variables and its analyticity properties in the scaling variable ω, for fixed values of the external mass, are studied. The structure function for annihilation turns out to be, in this case, the analytic continuation of the scattering structure function. Going beyond this particular solution we show in general from the integral equation that the average multiplicity of the secondary particles in the annihilation region approaches a finite limit for increasing q 2 , q μ being the four momentum of the current. Implications of this result for e + e − annihilation into hadrons are discussed.

A new inversion formula for the Stieltje's transform is derived and used for the analytic continuation of nucleon form factors. For given experimental accuracy the extent to which the spectral functions may be determined is analysed and with present data it is clear that only a very poor resolution may be obtained. An estimate of the experimental accuracy required to resolve the rho contribution is given.

The momentum space analytic function I(Z1, Z2, Z3; ϱ) corresponding to the configuration space function.

The double-pomeron coupling strength in the dual resonance model is found in both the inclusive and exclusive regions by comparison with experiments. Double-pomeron coupling occurs in inclusive experiments in the Mueller diagram for the central plateau region. Its strength can also be bounded from its non-observation in the two-particle to four-particle exclusive experiments. The dual resonance model is used to perform the analytic continuation of a six-point amplitude between these regions. The results show that the coupling strength for two forward pomerons in the exclusive region must be less than 1 300 of that in the inclusive region. This is experimental evidence for substantial forward double-pomeron decoupling in exclusive processes.

A rigorous construction for the zero-mass exponential superpropagator is given. It is based on an explicit regularization of the propagator in coordinate space and a suitable choice of the coupling constant is taken in order to damp down the regularized superpropagator to an L2 function. By applying a Sommerfeld-Watson transformation and a Fourier transformation we are able to show that the regularized superpropagator tends, in an appropriate sense, both in momentum and coordinate space, to some localizable generalized function. Analytic continuation in the coupling constant shows the appearance of a logarithmic cut. By means of the regularization of the propagator and the special choice of the coupling constant we have neither to cut divergent integrals nor to pass to Euclidean region. Extensions to superpropagators which are formally written as entire functions of the propagator are given.

Analytic continuation of the proton form factors GM and GE into the timelike region shows - independent of the special extrapolation technique - severe violations of the scaling law GE = GM/μ, and favours the relation GE = GM at the threshold for the process e+e−→pp.