Unitarity Relation Research Articles

On asymptotic power corrections to differential fluxes and generalization of optical theorem for potential scattering

In a wide class of potentials, the exact asymptotic dependence on finite distance R from scattering center is established for outgoing differential flux. It is shown how this dependence is eliminated by integration over solid angle for total flux, unitarity relation, and optical theorem. Thus, their applicability domain extends naturally to the finite R.

Dispersion relation for hadronic light-by-light scattering: two-pion contributions

In this third paper of a series dedicated to a dispersive treatment of the hadronic light-by-light (HLbL) tensor, we derive a partial-wave formulation for two-pion intermediate states in the HLbL contribution to the anomalous magnetic moment of the muon (g − 2)μ, including a detailed discussion of the unitarity relation for arbitrary partial waves. We show that obtaining a final expression free from unphysical helicity partial waves is a subtle issue, which we thoroughly clarify. As a by-product, we obtain a set of sum rules that could be used to constrain future calculations of γ∗γ∗ → ππ. We validate the formalism extensively using the pion-box contribution, defined by two-pion intermediate states with a pion-pole left-hand cut, and demonstrate how the full known result is reproduced when resumming the partial waves. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box, aμπ − box = − 15.9(2) × 10− 11. As an application of the partial-wave formalism, we present a first calculation of ππ-rescattering effects in HLbL scattering, with γ∗γ∗ → ππ helicity partial waves constructed dispersively using ππ phase shifts derived from the inverse-amplitude method. In this way, the isospin-0 part of our calculation can be interpreted as the contribution of the f0(500) to HLbL scattering in (g − 2)μ. We argue that the contribution due to charged-pion rescattering implements corrections related to the corresponding pion polarizability and show that these are moderate. Our final result for the sum of pion-box contribution and its S-wave rescattering corrections reads aμπ ‐ box + aμ,J = 0ππ,π ‐ pole LHC = − 24(1) × 10− 11.

Erratum to: Radiative Processes in Graphene and similar Nanostructures in Strong Electric Fields

Low-energy single-electron dynamics in graphene monolayers and similar nanostructures is described by the Dirac model, being a 2+1 dimensional version of massless QED with the speed of light replaced by the Fermi velocity v_{F}=c/300. Methods of strong-field QFT are relevant for the Dirac model, since any low-frequency electric field requires a nonperturbative treatment of massless carriers in case it remains unchanged for a sufficiently long time interval. In this case, the effects of creation and annihilation of electron-hole pairs produced from vacuum by a slowly varying and small-gradient electric field are relevant, thereby substantially affecting the radiation pattern. For this reason, the standard QED text-book theory of photon emission cannot be of help. We construct the Fock-space representation of the Dirac model, which takes exact accounts of the effects of vacuum instability caused by external electric fields, and in which the interaction between electrons and photons is taken into account perturbatively, following the general theory (the generalized Furry representation). We consider the effective theory of photon emission in the first-order approximation and construct the corresponding total probabilities, taking into account the unitarity relation.

Radiative Processes in Graphene and Similar Nanostructures in Strong Electric Fields

Low-energy single-electron dynamics in graphene monolayers and similar nanostructures is described by the Dirac model, being a 2+1 dimensional version of massless QED with the speed of light replaced by the Fermi velocity vF ≃ c/300. Methods of strong-field QFT are relevant for the Dirac model, since any low-frequency electric field requires a nonperturbative treatment of massless carriers in the case it remains unchanged for a sufficiently long time interval. In this case, the effects of creation and annihilation of electron-hole pairs produced from vacuum by a slowly varying and small-gradient electric field are relevant, thereby substantially affecting the radiation pattern. For this reason, the standard QED text-book theory of photon emission cannot be of help. We construct the Fock-space representation of the Dirac model, which takes exact accounts of the effects of vacuum instability caused by external electric fields, and in which the interaction between electrons and photons is taken into account perturbatively, following the general theory (the generalized Furry representation). We consider the effective theory of photon emission in the first-order approximation and construct the corresponding total probabilities, taking into account the unitarity relation.

One loop mass renormalization of unstable particles in superstring theory

Most of the massive states in superstring theory are expected to undergo mass renormalization at one loop order. Typically these corrections should contain imaginary parts, indicating that the states are unstable against decay into lighter particles. However in such cases, direct computation of the renormalized mass using superstring perturbation theory yields divergent result. Previous approaches to this problem involve various analytic continuation techniques, or deforming the integral over the moduli space of the torus with two punctures into the complexified moduli space near the boundary. In this paper we use insights from string field theory to describe a different approach that gives manifestly finite result for the mass shift satisfying unitarity relations. The procedure is applicable to all states of (compactified) type II and heterotic string theories. We illustrate this by computing the one loop correction to the mass of the first massive state on the leading Regge trajectory in SO(32) heterotic string theory.

Diffraction and unitarity

I begin with a tribute to V.N. Gribov and then come to a particular problem which would be of interest for him. His first paper on reggeology was devoted to elastic scatterings of hadrons. Here, using the unitarity relation in combination with experimental data about the elastic scattering in the diffraction cone, I show how the shape and the darkness of the interaction region of colliding protons change with the increase of their energies. In particular, the collisions become fully absorptive at small impact parameters at LHC energies that results in some special features of inelastic processes as well. The possible evolution with increasing energy of the shape from the dark core at the LHC to the fully transparent one at higher energies is discussed. It implies that the terminology of the black disk would be replaced by the black torus.

Constraints on a fine-grained AdS/CFT correspondence

For a boundary conformal field theory to give a good approximation to the bulk flat-space $S$-matrix, a number of conditions need to be satisfied: some of those are investigated here. In particular, one would like to identify an appropriate set of approximate asymptotic scattering states, constructed purely via boundary data. We overview, elaborate, and simplify obstacles encountered with existing proposals for these. Those corresponding to normalizable wave functions undergo multiple interactions; we contrast this situation with that needed for a flat-space Lehmann-Symanzik-Zimmermann treatment. Non-normalizable wave functions can have spurious interactions, due either to power-law tails of wave packets or to their non-normalizable behavior, which obscure $S$-matrix amplitudes we wish to extract, although in the latter case we show that such gravitational interactions can be finite, as a result of gravitational redshift. We outline an illustrative construction of arbitrary normalizable wave packets from boundary data that also yields such spurious interactions. Another set of nontrivial questions regard the form of unitarity relations for the bulk $S$-matrix, and in particular its normalization and multiparticle cuts. These combined constraints, together with those found earlier on a boundary singularity structure needed for bulk momentum conservation and other physical/analytic properties, are a nontrivial collection of obstacles to surmount if a fine-grained $S$-matrix, as opposed to a coarse-grained construction, is to be defined purely from boundary data.

Testing the consistency of theωπtransition form factor with unitarity and analyticity

We perform a dispersive analysis of the $\ensuremath{\omega}\ensuremath{\pi}$ electromagnetic transition form factor, using as input the discontinuity provided by unitarity below the $\ensuremath{\omega}\ensuremath{\pi}$ threshold and including for the first time experimental data on the modulus measured from ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\pi}}^{0}$ at higher energies. The input leads to stringent parametrization-free constraints on the modulus of the form factor below the $\ensuremath{\omega}\ensuremath{\pi}$ threshold, which are in disagreement with some experimental values measured from $\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\gamma}}^{*}$ decay. We discuss the dependence on the input parameters in the unitarity relation, using for illustration an $N/D$ formalism for the P partial wave of the scattering process $\ensuremath{\omega}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$, improved by a simple prescription which simulates the rescattering in the crossed channels. Our results confirm the existence of a conflict between experimental data and theoretical calculations of the $\ensuremath{\omega}\ensuremath{\pi}$ form factor in the region around 0.6 GeV and bring further arguments in support of renewed experimental efforts to measure more precisely the $\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\gamma}}^{*}$ decay.

Possible New Resonance from W_{L}W_{L}-hh Interchannel Coupling.

We propose and theoretically study a possible new resonance caused by strong coupling between the Higgs-Higgs and the W_{L}W_{L} (Z_{L}Z_{L}) scattering channels, without regard to the intensity of the elastic interaction in either channel at low energy (that could be weak as in the standard model). We expose this channel-coupling resonance from unitarity and dispersion relations encoded in the inverse amplitude method, applied to the electroweak chiral Lagrangian with a scalar Higgs boson.

Gravitational spin Hamiltonians from theSmatrix

We utilize generalized unitarity and recursion relations combined with effective field theory(EFT) techniques to compute spin dependent interaction terms for inspiralling binary systems in the post newtonian(PN) approximation. Using these methods offers great computational advantage over traditional techniques involving feynman diagrams, especially at higher orders in the PN expansion. As a specific example, we reproduce the spin-orbit interaction up to 2.5 PN order as also the leading order $S^2$(3PN) hamiltonian for an arbitrary massive object. We also obtain the unknown $S^3$(3.5PN) spin hamiltonian for an arbitrary massive object in terms of its low frequency linear response to gravitational perturbations, which was till now known only for a black hole. Furthermore, we derive the missing $S^4$ Hamiltonian at leading order(4PN) for an arbitrary massive object and establish that a minimal coupling of a massive elementary particle to gravity leads to a black hole structure. Finally, the Kerr metric is obtained as a series in $G_N$ by comparing the action of a test particle in the vicinity of a spinning black hole to the derived potential.

The Profile of Inelastic Collisions from Elastic Scattering Data

Using the unitarity relation in combination with experimental data about the elastic scattering in the diffraction cone, it is shown how the shape and the darkness of the inelastic interaction region of colliding protons change with increase of their energies. In particular, the collisions become fully absorptive at small impact parameters at LHC energies that results in some special features of inelastic processes. Possible evolution of this shape with the dark core at the LHC to the fully transparent one at higher energies is discussed that implies that the terminology of the black disk would be replaced by the black toroid. The approach to asymptotics is disputed. The ratio of the real to imaginary parts of the nonforward elastic scattering amplitude is briefly discussed. All the conclusions are only obtained in the framework of the indubitable unitarity condition using experimental data about the elastic scattering of protons in the diffraction cone without any reference to quantum chromodynamics (QCD) or phenomenological approaches.

Pp interactions in extended air showers

Applying the recently constructed analytic representation for the pp scattering amplitudes, we present a study of p-air cross sections, with comparison to the data from Extensive Air Shower (EAS) measurements. The amplitudes describe with precision all available accelerator data at ISR, SPS and LHC energies, and its theoretical basis, together with the very smooth energy dependence of parameters controlled by unitarity and dispersion relations, permit reliable extrapolation to higher energies and to asymptotic ranges. The comparison with cosmic ray data is very satisfactory in the whole pp energy interval from 1 to 100 TeV. High energy asymptotic behaviour of cross sections is investigated in view of the geometric scaling property of the amplitudes. The amplitudes predict that the proton does not behave as a black disk even at asymptotically high enegies, and we discuss possible non-trivial consequences of this fact for pA collision cross sections at higher energies.

Generalized unitarity and reciprocity relations for -symmetric scattering potentials

We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, and T(k), of general -symmetric scattering potentials. We use these identities to give a general proof of the relations, and , conjectured in Ahmed (2012 J. Phys. A: Math. Theor. 45 032004), and the generalized unitarity relation: , and show that it is a common property of both real and complex -symmetric potentials. The same holds for and .

Dispersive evaluation of the D-term form factor in deeply virtual Compton scattering

We present a dispersive representation of the D-term form factor for hard exclusive reactions, using unsubtracted t-channel dispersion relations. The t-channel unitarity relation is saturated with the contribution of two-pion intermediate states, using the two-pion distributions amplitude for the γ⁎γ→ππ subprocess and reconstructing the ππ→NN¯ subprocess from available information on pion-nucleon partial-wave helicity amplitudes. Results for the D-term form factor as function of t as well as at t=0 are discussed in comparison with available model predictions and phenomenological parametrizations.

Pp interaction at very high energies in cosmic ray experiments

An analysis of p-air cross section data from extensive air shower measurements is presented, based on an analytical representation of the pp scattering amplitudes that describes with high precision all available accelerator data at ISR, SPS and LHC energies. The theoretical basis of the representation, together with the very smooth energy dependence of parameters controlled by unitarity and dispersion relations, permits reliable extrapolation to high energy cosmic ray (CR) and asymptotic energy ranges. Calculations of based on Glauber formalism are made using the input values of the quantities , , BI and BR at high energies, with attention given to the independence of the slope parameters, with . The influence of contributions of diffractive intermediate states, according to Good–Walker formalism, is examined. The comparison with CR data is very satisfactory in the whole pp energy interval from 1 to 100 TeV. High energy asymptotic behaviour of cross sections is investigated in view of the geometric scaling property of the amplitudes. The observed energy dependence of the ratio between p-air and pp cross sections in the data is shown to be related to the nature of the pp cross section at high energies, that does not agree with the black disk image.

Spectral singularity in confined ${\cal {PT}}$PT symmetric optical potential

We present an analytical study for the scattering amplitudes (Reflection |R| and Transmission |T|), of the periodic ${\cal {PT}}$PT symmetric optical potential $V(x) = \displaystyle W_0 \left( \cos ^2 x + i V_0 \sin 2x \right)$V(x)=W0cos2x+iV0sin2x confined within the region 0 ⩽ x ⩽ L, embedded in a homogeneous medium having uniform potential W0. The confining length L is considered to be some integral multiple of the period π. We give some new and interesting results. Scattering is observed to be normal (|T|2 ⩽ 1, |R|2 ⩽ 1) for V0 ⩽ 0.5, when the above potential can be mapped to a Hermitian potential by a similarity transformation. Beyond this point (V0 > 0.5) scattering is found to be anomalous (|T|2, |R|2 not necessarily ⩽1). Additionally, in this parameter regime of V0, one observes infinite number of spectral singularities ESS at different values of V0. Furthermore, for L = 2nπ, the transition point V0 = 0.5 shows unidirectional invisibility with zero reflection when the beam is incident from the absorptive side (Im[V(x)] < 0) but with finite reflection when the beam is incident from the emissive side (Im[V(x)] > 0), transmission being identically unity in both cases. Finally, the scattering coefficients |R|2 and |T|2 always obey the generalized unitarity relation : $||T|^2 - 1| = \sqrt{|R_R|^2 |R_L|^2}$‖T|2−1|=|RR|2|RL|2, where subscripts R and L stand for right and left incidence, respectively.

An Impact Parameter Formalism

A relativistic impact-parameter formalism is introduced without any approximation. It is applicable for all physical values of energy and scattering angle. An impact parameter amplitude is defined from an invariant scattering amplitude by the integral with the Bessel function, in place of the Legendre function in the definition of a partial-wave amplitude. We find that this impact parameter amplitude is an example of a new integral expansion. It is proved that one of the necessary and sufficient conditions that this new expansion is permissible, is that the amplitude should satisfy the Kapteyn equation, which is required in the Webb-Kapteyn theory of the Neumann series expansion of an odd function. The physical interpretation of the amplitude is discussed and cross sections are expressed by simple integrals over the impact parameter without any approximation. The unitarity relation for the amplitude is expressed in the simpler form. Comparison is made with three other relativistic formalisms proposed by Blankenbecler and Goldberger, Ida, and Cottingham and Peiels. In contrast with our formalism, the applications of these three formalisms are limited either to the extremely high energy region or to only the forward angle scattering.

Generalized Continuity Equation for Quasi-Hermitian Hamiltonians

The continuity relation is generalized to quasi-Hermitian one-dimensional Hamiltonians. As an application we show that the reflection and transmission coefficients computed with the generalized current obey the conventional unitarity relation for the continuous double delta function potential.

Bethe-Schwinger Effective Range Theory and Lehmann and Weinberg Chiral Perturbation Theories

This paper is a brief review of low energy soft hadronic physics,starting from the invention of the low energy effective rangetheory in the late 40's due to Bethe and Schwinger fornucleon-nucleon scattering, and its generalization to the static Chew-Low model for pion nucleonscattering, to the present development of the Lehmann and WeinbergChiral Perturbation Theories. It is pointed out that a consistentlow energy calculation can be achieved with the incorporation ofthe unitarity relation in the Chiral Perturbation Theory.

Lower limits on the strengths of gamma ray lines from WIMP dark matter annihilation

We study the spectra of gamma ray signals that arise from dark matter annihilation in the universe. We focus on the large class of theories where the photon spectrum includes both continuum spectrum of gamma rays that arise from annihilation into Standard Model states at tree level, as well as monochromatic gamma rays arising from annihilation directly into two photons at the one loop level. In this class of theories we obtain lower bounds on the ratio of the strength of the gamma ray line relative to the gamma ray continuum as a function of the dark matter mass and spin. These limits are obtained from the unitarity relation between the tree level amplitude of the primary annihilation channel and the imaginary part of the loop level amplitude for annihilation directly into photons, with the primary decay products running in the loop. These results are exact in the limit that dark matter annihilation is exclusively to a single Standard Model species, occurs through the lowest partial wave and respects CP. Away from this limit the bounds are approximate. Our conclusions agree with the known results in the literature in the case of the Minimal Supersymmetric Standard Model (MSSM). We use the Fermi-LAT observations to translate these limits into upper bounds on the dark matter annihilation cross section into any specific Standard Model state.