Analyticity Constraints Research Articles

The stringy S-matrix bootstrap: maximal spin and superpolynomial softness

We explore the space of meromorphic amplitudes with extra constraints coming from the shape of the leading Regge trajectory. This information comes in two guises: it bounds the maximal spin of exchanged particles of a given mass; it leads to sum rules obeyed by the discontinuity of the amplitude, which express the softness of scattering at high energies. We assume that the leading Regge trajectory is linear, and we derive bounds on the low-energy Wilson coefficients using the dual and primal approaches. For the graviton-graviton scattering in four dimensions, the maximal spin constraint leads to slightly more stringent bounds than those that follow from general constraints of analyticity, crossing, and unitarity. The exponential softness at high energies is manifest in our primal approach and is not used in our implementation of the dual approach. Nevertheless, we observe the agreement between the bounds obtained from both. We conclude that high-energy superpolynomial softness does not leave an obvious imprint on the low-energy observables. We exhibit a unitary three-parameter deformation of the Veneziano amplitude for the open string case. It has a novel, exponentially soft behavior at high energies and fixed angles. We generalize the previous analysis of this regime and present a stringy version of the lower bound on high-energy, fixed-angle scattering by Cerulus and Martin.

Analysis of the ψ(3770)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\psi (3770)}$$\\end{document} resonance in line with unitarity and analyticity constraints

We study the inclusive and exclusive cross sections of e+e-→hadrons\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^+e^-\\rightarrow \ ext {hadrons}$$\\end{document} for center-of-mass energies between 3.70 and 3.83GeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3.83\\,\\,\ extrm{GeV} $$\\end{document} to infer the mass, width, and couplings of the ψ(3770)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi (3770)$$\\end{document} resonance. By using a coupled-channel K-matrix approach, we setup our analysis to respect unitarity and the analyticity properties of the underlying scattering amplitudes. We fit several models to the full dataset and identify our nominal results through a statistical model comparison. We find that, accounting for the interplay between the ψ(2S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi (2S)$$\\end{document} and the ψ(3770)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi (3770)$$\\end{document}, no further pole is required to describe the ψ(3770)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi (3770)$$\\end{document} line shape. In particular we derive from the pole location Mψ(3770)=3778.8±0.3MeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{\\psi (3770)}~=~3778.8~\\pm ~0.3~\\,\ extrm{MeV} $$\\end{document} and Γψ(3770)=25.0±0.5MeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varGamma _{\\psi (3770)}~=~25.0~\\pm ~0.5~\\,\ extrm{MeV} $$\\end{document}. Moreover, we find the decay to D+D-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^+D^-$$\\end{document} and D0D¯0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^0\\bar{D}^0$$\\end{document} to be consistent with isospin symmetry and derive an upper bound on the branching ratio B(ψ(3770)→non-DD¯)<6%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}(\\psi (3770) \\rightarrow \ ext {non-}D\\bar{D}) < 6\\%$$\\end{document} at 90%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90\\%$$\\end{document} probability.

Reaction πN→ωN in a dynamical coupled-channel approach

A refined investigation on light flavor meson-baryon scatterings is performed using a dynamical coupled-channel approach, the J\"ulich-Bonn model, that respects unitartiy and analyticity constraints. The channel space of $\pi N$, $\pi \Delta$, $\sigma N$, $\rho N$, $\eta N$, $K \Lambda$ and $K \Sigma$ is extended by adding the $\omega N$ final state. The spectra of $N^*$ and $\Delta$ resonances are extracted in terms of complex poles of the scattering amplitudes, based on the result of a global fit to a worldwide collection of data, in the energy region from the $\pi N$ threshold to center-of-mass energy $z=2.3$ GeV. A negative value of the $\omega N$ elastic spin-averaged scattering length is extracted, questioning the existence of bound states of the $\omega$ meson in the nuclear matter.

Analyticity constraints bound the decay of the spectral form factor

Quantum chaos cannot develop faster than &amp;#x03BB;&amp;#x2264;2&amp;#x03C0;/(&amp;#x210F;&amp;#x03B2;) for systems in thermal equilibrium [Maldacena, Shenker &amp; Stanford, JHEP (2016)]. This `MSS bound' on the Lyapunov exponent &amp;#x03BB; is set by the width of the strip on which the regularized out-of-time-order correlator is analytic. We show that similar constraints also bound the decay of the spectral form factor (SFF), that measures spectral correlation and is defined from the Fourier transform of the two-level correlation function. Specifically, the inflection exponent&amp;#x03B7;, that we introduce to characterize the early-time decay of the SFF, is bounded as &amp;#x03B7;&amp;#x2264;&amp;#x03C0;/(2&amp;#x210F;&amp;#x03B2;). This bound is universal and exists outside of the chaotic regime. The results are illustrated in systems with regular, chaotic, and tunable dynamics, namely the single-particle harmonic oscillator, the many-particle Calogero-Sutherland model, an ensemble from random matrix theory, and the quantum kicked top. The relation of the derived bound with other known bounds, including quantum speed limits, is discussed.

Patterns of C- and CP-violation in hadronic η and η′ three-body decays

We construct hadronic amplitudes for the three-body decays η(′) → π+π−π0 and η′ → ηπ+π− in a non-perturbative fashion, allowing for C- and CP-violating asymmetries in the π+π− distributions. These amplitudes are consistent with the constraints of analyticity and unitarity. We find that the currently most accurate Dalitz-plot distributions taken by the KLOE-2 and BESIII collaborations confine the patterns of these asymmetries to a relative per mille and per cent level, respectively. Our dispersive representation allows us to extract the individual coupling strengths of the C- and CP-violating contributions arising from effective isoscalar and isotensor operators in η(′) → π+π−π0 and an effective isovector operator in η′ → ηπ+π−, while the strongly different sensitivities to these operators can be understood from chiral power counting arguments.

Unitarity bounds for semileptonic decays in lattice QCD

In this work we discuss in detail the nonperturbative determination of the momentum dependence of the form factors entering in semileptonic decays using unitarity and analyticity constraints. The method contains several new elements with respect to previous proposals and allows to extract, using suitable two-point functions computed nonperturbatively, the form factors at low momentum transfer ${q}^{2}$ from those computed explicitly on the lattice at large ${q}^{2}$, without any assumption about their ${q}^{2}$-dependence. The approach will be very useful for exclusive semileptonic $B$-meson decays, where the direct calculation of the form factors at low ${q}^{2}$ is particularly difficult due to large statistical fluctuations and discretization effects. As a testing ground we apply our approach to the semileptonic $D\ensuremath{\rightarrow}K\ensuremath{\ell}{\ensuremath{\nu}}_{\ensuremath{\ell}}$ decay, where we can compare the results of the unitarity approach to the explicit direct lattice calculation of the form factors in the full ${q}^{2}$-range. We show that the method is very effective and that it allows to compute the form factors with rather good precision.

S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the 2 → 2 scattering matrix S2→2 such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider 3 + 1 dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves kℓ(s) that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves fℓ(s), for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.

The origin of the proton radius puzzle

The dimension of the proton, the basic building block of matter, is still object of controversy. The most precise electron-proton scattering data at low transferred momenta are re-analyzed and the extraction of the proton radius is discussed. A recent experiment from the JLAB-CLAS collaboration gives a small value for the radius (The symbol R_E^alpha stands for the root-mean-square charge radius of the proton sqrt{langle r_E^2rangle }, obtained by the experimental or theoretical Collaboration alpha .) R_E^mathrm{CLAS}= (0.831pm 0.007_mathrm{stat}pm 0.012_mathrm{syst}) fm (Xiong et al. in Nature 575:147, 2019), in contrast with previous electron scattering experiments, in particular with the MAINZ experiment (Bernauer et al. (A1 Collaboration), Phys. Rev. C 90:015206, 2014) that concluded R_E^mathrm{MAINZ}= (0.879pm 0.005_mathrm{stat}pm 0.004_mathrm{syst}pm 0.002_mathrm{model}pm 0.004_mathrm{group}) fm. The experimental results are re-analyzed in terms of different fits of the cross section and of its discrete derivative with analyticity constraints. The uncertainty on the derivative is two orders of magnitude larger than the error on the measured observable, i.e., the cross section. The systematic error associated with the radius is evaluated taking into account the uncertainties from different sources, as the extrapolation to the static point, the choice of the class of fitting functions, and the range of the data sample.

Dynamical constraints on RG flows and cosmology

Sum rules connecting low-energy observables to high-energy physics are an interesting way to probe the mechanism of inflation and its ultraviolet origin. Unfortunately, such sum rules have proven difficult to study in a cosmological setting. Motivated by this problem, we investigate a precise analogue of inflation in anti-de Sitter spacetime, where it becomes dual to a slow renormalization group flow in the boundary quantum field theory. This dual description provides a firm footing for exploring the constraints of unitarity, analyticity, and causality on the bulk effective field theory. We derive a sum rule that constrains the bulk coupling constants in this theory. In the bulk, the sum rule is related to the speed of radial propagation, while on the boundary, it governs the spreading of nonlocal operators. When the spreading speed approaches the speed of light, the sum rule is saturated, suggesting that the theory becomes free in this limit. We also discuss whether similar results apply to inflation, where an analogous sum rule exists for the propagation speed of inflationary fluctuations.

Unitarity, analyticity and duality constraints in η and π photoproduction

We report an update of the isobar model EtaMAID. A new approach is proposed to avoid double counting in the overlap region of Regge and resonances. Dispersion relation is applied on top of the isobar model, and both models describe the data equally well. Application of these ideas to pion photoproduction is discussed.

Quantum Spectral Curve of γ-Twisted 𝒩 = 4 SYM Theory and Fishnet CFT

We review the quantum spectral curve (QSC) formalism for the spectrum of anomalous dimensions of [Formula: see text] SYM, including its [Formula: see text]-deformation. Leaving aside its derivation, we concentrate on the formulation of the “final product” in its most general form: a minimal set of assumptions about the algebraic structure and the analyticity of the [Formula: see text]-system — the full system of Baxter [Formula: see text]-functions of the underlying integrable model. The algebraic structure of the [Formula: see text]-system is entirely based on (super)symmetry of the model and is efficiently described by Wronskian formulas for [Formula: see text]-functions organized into the Hasse diagram. When supplemented with analyticity conditions on [Formula: see text]-functions, it fixes completely the set of physical solutions for the spectrum of an integrable model. First, we demonstrate the spectral equations on the example of [Formula: see text] and [Formula: see text] Heisenberg (super)spin chains. Supersymmetry [Formula: see text] occurs as a simple “rotation” of the Hasse diagram for a [Formula: see text] system. Then we apply this method to the spectral problem of [Formula: see text]/CFT4-duality, describing the QSC formalism. The main difference with the spin chains consists in more complicated analyticity constraints on [Formula: see text]-functions which involve an infinitely branching Riemann surface and a set of Riemann–Hilbert conditions. As an example of application of QSC, we consider a special double scaling limit of [Formula: see text]-twisted [Formula: see text] SYM, combining weak coupling and strong imaginary twist. This leads to a new type of non-unitary CFT dominated by particular integrable, and often computable, 4D fishnet Feynman graphs. For the simplest of such models — the bi-scalar theory — the QSC degenerates into the [Formula: see text]-system for integrable non-compact Heisenberg spin chain with conformal, [Formula: see text] symmetry. We describe the QSC derivation of Baxter equation and the quantization condition for particular fishnet graphs — wheel graphs, and review numerical and analytic results for them.

Leading logarithms of the two-point function in massless O(N) and SU(N) models to any order from analyticity and unitarity

Leading (large) logarithms in non-renormalizable theories have been investigated in the recent past. Besides some general considerations, explicit results for the expansion coefficients (in terms of leading logarithms) of partial wave amplitudes and of scalar and vector form factors have been given. Analyticity and unitarity constraints haven been used to obtain the expansion coefficients of partial waves in massless theories, yielding form factors and the scalar two-point function to five-loop order in the O(4)/O(3) model. Later, the all order solutions for the partial waves in any O(N+1)/O(N) model were found. Also, results up to four-loop order exist for massive theories. Here we extend the implications of analyticity and unitarity constraints on the leading logarithms to arbitrary loop order in massless theories. We explicitly obtain the scalar and vector form factors as well as to the scalar two-point function in any O(N) and SU(N) type models. We present relations between the expansion coefficients of these quantities and those of of the relevant partial waves. Our work offers a consistency check on the published results in the O(N) models for form factors, and new results for the scalar two-point function. For the SU(N) type models, we use the known expansion coefficients for partial waves to obtain those for scalar and vector form factors as well as for the scalar two-point function. Our results for the form factor offer a check for the known and future results for massive O(N) and SU(N) type models when the massless limit is taken. Mathematica notebooks which can be used to calculate the expansion coefficients are provided as ancillary files.

Analyticity constraints for hadron amplitudes: Going high to heal low-energy issues

Analyticity constitutes a rigid constraint on hadron scattering amplitudes. This property is used to relate models in different energy regimes. Using meson photoproduction as a benchmark, we show how to test contemporary low-energy models directly against high-energy data. This method pinpoints deficiencies of the models and treads a path to further improvement. The implementation of this technique enables one to produce more stable and reliable partial waves for future use in hadron spectroscopy and new physics searches.

Vertex operators for production of baryon resonances in nucleon-gamma collisions with preserving gauge invariance and analyticity

Vertex operators for photo- and electro-production of baryon states with arbitrary spin-parity, [Formula: see text], are constructed. The operators obey gauge invariance and analyticity constraints. Analyticity is realized as a requirement of the generalized Siegert theorem for vertex form factors.

Positivity constraints for pseudolinear massive spin-2 and vector Galileons

We derive analyticity constraints on a nonlinear ghost-free effective theory of a massive spin-2 particle known as pseudo-linear massive gravity, and on a generalized theory of a massive spin-1 particle, both of which provide simple IR completions of Galileon theories. For pseudo-linear massive gravity we find that, unlike dRGT massive gravity, there is no window of parameter space which satisfies the analyticity constraints. For massive vectors which reduce to Galileons in the decoupling limit, we find that no two-derivative actions are compatible with positivity, but that higher derivative actions can be made compatible.

Predictions on the second-class current decaysτ−→π−η(′)ντ

We analyze the second-class current decays $\tau^{-}\to\pi^{-}\eta^{(\prime)}\nu_{\tau}$ in the framework of Chiral Perturbation Theory with resonances. Taking into account $\pi^{0}$-$\eta$-$\eta^{\prime}$ mixing, the $\pi^{-}\eta^{(\prime)}$ vector form factor is extracted, in a model-independent way, using existing data on the $\pi^{-}\pi^{0}$ one. For the participant scalar form factor, we have considered different parameterizations ordered according to their increasing fulfillment of analyticity and unitarity constraints. We start with a Breit-Wigner parameterization dominated by the $a_{0}(980)$ scalar resonance and after we include its excited state, the $a_{0}(1450)$. We follow by an elastic dispersion relation representation through the Omn\`{e}s integral. Then, we illustrate a method to derive a closed-form expression for the $\pi^{-}\eta$, $\pi^{-}\eta^{\prime}$ (and $K^{-}K^{0}$) scalar form factors in a coupled-channels treatment. Finally, predictions for the branching ratios and spectra are discussed emphasizing the error analysis. An interesting result of this study is that both $\tau^{-}\to\pi^{-}\eta^{(\prime)}\nu_{\tau}$ decay channels are promising for the soon discovery of second-class currents at Belle-II. We also predict the relevant observables for the partner $\eta^{(\prime)}_{\ell 3}$ decays, which are extremely suppressed in the Standard Model.

Reply to “Comment on ‘Breakdown of the expansion of finite-size corrections to the hydrogen Lamb shift in moments of charge distribution’ ”

To comply with the critique of the Comment [J. Arrington, arXiv:1602.01461], we consider another modification of the proton electric form factor, which resolves the "proton-radius puzzle". The proposed modification satisfies all the consistency criteria put forward in the Comment, and yet has a similar impact on the puzzle as that of the original paper. Contrary to the concluding statement of the Comment, it is not difficult to find an ad hoc modification of the form factor at low $Q$ that resolves the discrepancy and is consistent with analyticity constraints. We emphasize once again that we do not consider such an ad hoc modification of the proton form factor to be a solution of the puzzle until a physical mechanism for it is found.

Dalitz plot studies ofD0→KS0π+π−decays in a factorization approach

The presently available high-statistics data of the D0 --> K0S pi+ pi- processes measured by the Belle and BABAR Collaborations are analyzed within a quasi two-body QCD factorization framework. Starting from the weak effective Hamiltonian, tree and annihilation amplitudes build up the D0 --> K0S pi+ pi- decay amplitude. Two of the three final-state mesons are assumed to form a single scalar, vector or tensor state originating from a quark-antiquark pair so that the factorization hypothesis can be applied. The meson-meson final state interactions are described by K pi and pi pi scalar and vector form factors for the S and P waves and by relativistic Breit-Wigner formulae for the D waves. A combined chi^2 fit to a Belle Dalitz plot density distribution, to the total experimental branching fraction and to the tau^- --> K0S pi- nu_tau decay data is carried out to fix the 33 free parameters. These are mainly related to the strengths of the scalar form factors and to unknown meson to meson transition form factors at a large momentum transfer squared equal to the D0 mass squared. A good overall agreement to the Belle Dalitz plot density distribution is achieved. Another set of parameters fits equally well the BABAR Collaboration Dalitz plot model. The branching fractions of the dominant channels compare well with those of the isobar Belle or BABAR models.The lower-limit values of the branching fractions of the annihilation amplitudes are significant. Built upon experimental data from other processes, the unitary K pi and pi pi scalar form factors, entering our decay amplitude and satisfying analyticity and chiral symmetry constraints, are furthermore constrained by the present Dalitz plot analysis. Our decay amplitude could be a useful input for determinations of D0-D0bar mixing parameters and of the CKM angle gamma (or phi3).

Recent development in SU(3) covariant baryon chiral perturbation theory

Baryon chiral perturbation theory (BChPT), as an effective field theory of low-energy quantum chromodynamics (QCD), has played and is still playing an important role in our understanding of non-perturbative strong interaction phenomena. In the past two decades, inspired by the rapid progress in lattice QCD simulations and the new experimental campaign to study the strangeness sector of low-energy QCD, many efforts have been made to develop a fully covariant BChPT and to test its validity in all scenarios. These new endeavours have not only deepened our understanding of some long-standing problems, such as the power-counting-breaking problem and the convergence problem, but also resulted in theoretical tools that can be confidently applied to make robust predictions. Particularly, the manifestly covariant BChPT supplemented with the extended-on-mass-shell (EOMS) renormalization scheme has been shown to satisfy all analyticity and symmetry constraints and converge relatively faster compared to its non-relativistic and infrared counterparts. In this article, we provide a brief review of the fully covariant BChPT and its latest applications in the $u$, $d$, and $s$ three-flavor sector.

Ultrahigh energy predictions of proton-air cross sections from accelerator data: An update

At $\sqrt s = 57\pm 7$ TeV, the Pierre Auger Observatory (PAO) collaboration has recently measured the proton-air inelastic production cross section $\sigma_{\rm p-air}$. Assuming a helium contamination of 25%, they subtracted 30 mb from their measured value, resulting in a p-air inelastic production cross section, $\sigma_{\rm p-air}=475 \pm 22\ ({\rm stat.})\pm^{20}_{15} \ ({\rm syst.})$ mb, exclusive of helium contamination. Using this result in a Glauber calculation to obtain the $pp$ inelastic cross section, they found the inelastic $pp$ cross section $\sigma_{\rm inel}= 90\pm 7\ ({\rm stat.}) \pm^9_{11} ({\rm syst.}) \pm 1.5 {\rm \ (Glaub.})$ mb. Parameterization of the $\bar pp$ and $pp$ cross sections incorporating analyticity constraints and unitarity has allowed us to make accurate extrapolations to ultra-high energies, and using Glauber calculations, accurately predict cosmic ray results for $\spai$. In this update for 57 TeV, we predict i) a $pp$ total cross section, $\sigma_{\rm tot}=133.4\pm 1.6$ mb, using high energy predictions from a saturated Froissart bound parameterization of accelerator data on forward $\bar pp$ and $pp$ scattering amplitudes and ii) a p-air inelastic production cross section, $\sigma_{\rm p-air}=483\pm 3 $ mb, by using $\sigma_{\rm tot}$ together with Glauber theory, allowing us to determine independently that the helium contamination was 19%, in reasonable agreement with their estimate of 25%. Our predictions agree with all available cosmic ray extensive air shower measurements, both in magnitude and in energy dependence. By using our value for the $pp$ total cross section at 57 TeV, Block and Halzen \cite{blackdisk} have predicted that the $pp$ inelastic cross section is $\sigma_{\rm inel}= 92.9\pm 1.6$ mb, in agreement with the measured POA value.