arXiv subjects

Martin M. Block

Publications and source records attributed to Martin M. Block.

At least 19 recordsLinked to original sources

arXiv2017

Evidence for a break in the spectrum of astrophysical neutrinos

The announcement by the IceCube Collaboration of the observation of 53 astrophysical neutrino candidates in the energy range 0.03 \alt E_ν/PeV \alt 2 has been greeted with a great deal of justified excitement. Herein we provide fits of single and a broken power-law energy-spectra to these high-energy starting events (HESEs). By comparing our statistical results from fits to (background-free) shower HESE data with the spectral shape of muon neutrinos recently reported by the IceCube Collaboration, we show that there is (3 σ) evidence for a break in the spectrum of astrophysical neutrinos. After that we use the fitted result to predict the rate of Glashow events (in the ~ 6.3 PeV region) and double-bang tau neutrino events (in the PeV region) just at the threshold of IceCube detection.

astro-ph.HE↗

arXiv2016

The slope, curvature, and higher parameters in $pp$ and $\bar{p}p$ scattering, and the extrapolation of measurements of $dσ(s,t)/dt$ to $t=0$

We study the effects of curvature in the expansion of the logarithm of the differential elastic scattering cross section near $t=0$ as $dσ(s,t)/dt=dσ(s,0)/dt\,\times\exp(Bt+Ct^2+Dt^3\cdots)$ in an eikonal model for $pp$ and $\bar{p}p$ scattering, and use the results to discuss the extrapolation of measured differential cross sections and the slope parameters $B$ to $t=-q^2=0$. We find that the curvature effects represented by the parameters $C$ and $D$, while small, lead to significant changes in the forward slope parameter relative to that determined in a purely exponential fit, and to smaller but still significant changes in the forward elastic scattering and total cross sections. Curvature effects should therefore be considered in future analyses or reanalyses of the elastic scattering data.

hep-ph↗

arXiv2016

Comment on "More on Heisenberg's model for high energy nucleon-nucleon scattering"

We comment on the treatment of asymptotic black-disk scattering in a recent paper of Nastase and Sonnenschein, Phys.\ Rev.\ D\ {\bf 92}, 015028 (2015), on scattering in an updated version of the Heisenberg model which gives $pp$ and $\bar{p}p$ cross sections which increase at very high energies as $\ln^2s$. We show that the total cross section they define does not correspond to that measured in experiments, with the result that their limit for the ratio $σ_{\rm elas}/σ_{\rm tot}$ is too small by a factor 2. The correct ratio for black-disk scattering, $σ_{\rm elas}/σ_{\rm tot} \rightarrow 1/2$ for $s\rightarrow\infty$, is strongly supported by experiment.

hep-ph↗

arXiv2015

Comprehensive fits to high energy data for $σ$, $ρ$, and $B$ and the asymptotic black-disk limit

We demonstrate that the entirety of the data on proton--proton and antiproton--proton forward scattering between 6 GeV and 57 TeV center-of-mass energy is sufficient to show that $σ_{\rm elas}/σ_{\rm tot} \rightarrow 1/2$, and that $8πB/σ_{\rm tot}\rightarrow 1$ at very high energies, where $B$ the forward slope parameter for the differential elastic scattering cross sections. The relations demonstrate convincingly that the asymptotic $pp$ and $\bar{p}p$ scattering amplitudes approach those of scattering from a black disk. This result obviously has implications for any new physics that modifies the forward scattering amplitudes.

hep-ph↗

arXiv2015

Eikonal fit to $pp$ and $\bar{p}p$ scattering and the edge in the scattering amplitude

We make a detailed eikonal fit to current data on the total and elastic scattering cross sections, the ratios $ρ$ of the real to the imaginary parts of the forward elastic scattering amplitudes, and the logarithmic slopes $B$ of the differential cross sections $dσ/dt$ at $t=0$, for proton-proton and antiproton-proton scattering at center-of-mass energies $W$ from 5 GeV to 57 TeV. The fit allows us to investigate the structure of the eikonal amplitudes in detail, including the impact-parameter structure of the energy-independent edge in the scattering amplitude shown to exist by Block {\em et al.} \cite{edge}. We show that the edge region has an essentially fixed shape with a peak at approximately the "black disk" radius $R_{\rm tot}=\sqrt{σ_{\rm tot}/2π}$ of the scattering amplitude, a constant width $t_{\rm edge}\approx 1$ fm, and migrates to larger impact parameters with increasing energy proportionally to $R_{\rm tot}$. We comment on possible physical mechanisms which could lead to the edge. We show that the eikonal results for the cross sections and $ρ$ values are described to high accuracy by analytic expressions of the forms used in earlier analyses by Block and Halzen, and extend the result to the elastic-scattering slope parameter $B$. These expressions provide simple extrapolations of the results to much higher energies. Finally, we calculate the survival probabilities for large rapidity gaps in the scattering.

hep-ph↗

arXiv2015

Evidence for a Constant `Edge' in Proton-Proton Scattering at Very High Energies

Accurate fits to $pp$ and $\bar pp$ cross section data up to Tevatron energies, incorporating the constraints imposed by analyticity and unitarity, successfully predict the results of recent LHC and cosmic ray measurements, and suggest that the cross sections approach a black disc limit asymptotically. The approach to the limit is, however, very slow. We present a simple geometric picture which explains these features in a natural way. A black disc of logarithmically growing radius is supplemented by a soft `edge' whose properties are invariant with energy. The constancy of the edge results in the prediction that the quantity $(σ^{TOT}-2σ^{El})/\surdσ^{TOT}$ approaches a constant at high energy. Using the existing fits, this prediction appears to be verified. The value of the limiting constant allows an estimate of the thickness of the edge, which turns out to be on the order of $1\,{\rm fm}$. One thus arrives at a picture where the proton-proton scattering at lower energies is dominated by what becomes the edge, while at higher energies it is dominated by the disc. The crossover between the two regimes is only at $\surd s\geq $ 10 TeV, accounting for the slow approach to asymptotic behavior. Some questions as to the nature of the edge are discussed.

hep-ph↗

arXiv2014

Connection of the virtual $γ^*p$ cross section of $ep$ deep inelastic scattering to real $γp$ scattering, and the implications for $νN$ and $ep$ total cross sections

We show that it is possible to fit all of the HERA DIS (deep inelastic scattering) data on $F_2^{γp}$ at small values of Bjorken $x$, including the data at {\em very low} $Q^2$, using a new model for $F_2^{γp}$ which both includes an asymptotic (high energy) part that satisfies a saturated Froissart bound behavior, with a vector-dominance like mass factor in the parameterization, and extends smoothly to $Q^2=0$. We require that the corresponding part of the virtual $γ^* p$ cross section match the known asymptotic part of the real $γp$ cross section at $Q^2=0$, a cross section which is determined by strong interactions and asymptotically satisfies a saturated Froissart bound of the form $α+β\ln s+γ\ln^2s$. Using this model for the asymptotic part of $F_2^{γp}$ plus a known valence contribution, we fit the asymptotic high energy part of the HERA data with $x\le 0.1$ and $W\ge 25$ GeV; the fit is excellent. We find that the mass parameter in the fit lies in the region of the light vector mesons, somewhat above the $ρ$ meson mass, and is compatible with vector dominance. We use this fit to obtain accurate results for the high energy $ep$ and isoscalar $νN$ total cross sections. Both cross sections obey an analytic expression of the type $a +b \ln E +c \ln^2 E +d \ln^3 E$ at large energies $E$ of the incident particle, reflecting the fact that the underlying strong interaction parts of the $γ^*p$, $Z^*N$ and $W^*N$ cross sections satisfy the saturated Froissart bound. Since approximately 50% of the $νN$ center of mass (cms) energy is found in $W$---the cms energy of the strongly interacting intermediate vector boson-nucleon system---a study of ultra-high-energy neutrino-nucleon cross sections would allow us, for the first time, to explore {\em strong interactions at incredibly high energies}.

hep-ph↗

arXiv2013

Implications of a Froissart bound saturation of $γ^*$-$p$ deep inelastic scattering. Part II. Ultra-high energy neutrino interactions

In Part I (in this journal) we argued that the structure function $F_2^{γp}(x,Q^2)$ in deep inelastic $ep$ scattering, regarded as a cross section for virtual $γ^*p$ scattering, has a saturated Froissart-bounded form behaving as $\ln^2 (1/x)$ at small $x$. This form provides an excellent fit to the low $x$ HERA data, including the very low $Q^2$ regions, and can be extrapolated reliably to small $x$ using the natural variable $\ln(1/x)$. We used our fit to derive quark distributions for values of $x$ down to $x=10^{-14}$. We use those distributions here to evaluate ultra-high energy (UHE) cross sections for neutrino scattering on an isoscalar nucleon, $N=(n+p)/2$, up to laboratory neutrino energies $E_ν\sim 10^{16}$-$10^{17}$ GeV where there are now limits on neutrino fluxes. We estimate that these cross sections are accurate to $\sim$2% at the highest energies considered, with the major uncertainty coming from the errors in the parameters that were needed to fit $F_2^{γp}(x,Q^2)$. We compare our results to recently published neutrino cross sections derived from NLO parton distribution functions, which become much larger at high energies because of the use of power-law extrapolations of quark distributions to small $x$. We argue that our calculation of the UHE $νN$ cross sections is the best one can make based the existing experimental deep inelastic scattering data. Further, we show that the strong interaction Froissart bound of $\ln^2 (1/x)$ on $F_2^{γp}$ translates to an exact bound of $\ln^3E_ν$ for leading-order-weak $νN$ scattering. The energy dependence of $νN$ total cross section measurements consequently has important implications for hadronic interactions at enormous cms (center-of-mass) energies not otherwise accessible.

hep-ph↗

arXiv2013

Implications of a Froissart bound saturation of $γ^*$-$p$ deep inelastic scattering. Part I. Quark distributions at ultra small $x$

We argue that the deep inelastic structure function $F_2^{γp}(x, Q^2)$, regarded as a cross section for virtual $γ^*p$ scattering, is hadronic in nature. This implies that its growth is limited by the Froissart bound at high hadronic energies, giving a $\ln^2 (1/x)$ bound on $F_2^{γp}$ as Bjorken $x\rightarrow 0$. The same bound holds for the individual quark distributions. In earlier work, we obtained a very accurate global fit to the combined HERA data on $F_2^{γp}$ using a fit function which respects the Froissart bound at small $x$, and is equivalent in its $x$ dependence to the function used successfully to describe all high energy hadronic cross sections, including $γp$ scattering. We extrapolate that fit by a factor of $\lesssim$3 beyond the HERA region in the natural variable $\ln(1/x)$ to the values of $x$ down to $x=10^{-14}$ and use the results to derive the quark distributions needed for the reliable calculation of neutrino cross sections at energies up to $E_ν=10^{17}$ GeV. These distributions do not satisfy the Feynman "wee parton" assumption, that they all converge toward a common distribution $xq(x,Q^2)$ at small $x$ and large $Q^2$. This was used in some past calculations to express the dominant neutrino structure function $F_2^{ν(\barν)}$ directly in terms of $F_2^{γp}$. We show that the correct distributions nevertheless give results for $F_2^{ν(\barν)}$ which differ only slightly from those obtained assuming that the wee parton limit holds. In two Appendices, we develop simple analytic results for the effects of QCD evolution and operator-product corrections on the distribution functions at small $x$, and show that these effects amount mainly to shifting the values of $\ln(1/x)$ in the initial distributions.

hep-ph↗

arXiv2012

Commentary on "Total Hadronic Cross Section Data and the Froissart-Martin Bound", by Fagundes, Menon and Silva

This Commentary on the paper "Total Hadronic Cross Section Data and the Froissart-Martin Bound", by Fagundes, Menon and Silva, to be published in Braz. J. of Phys., Vol. 42 (2012) (arXiv 1112.4704), was invited by the Editors of the Brazilian Journal of Physics to appear directly after the above authors' printed version, in the same journal issue. We here challenge that paper's conclusions that the Froissart bound was violated. We will show that this conclusion follows from a statistical methodology that we question, and will present compelling supplementary evidence that the latest ultra-high energy experimental $pp$ cross section data are consistent with a $\ln^2 s$ behavior that satisfies the Froissart bound.

hep-ph↗

arXiv2012

New experimental evidence that the proton develops asymptotically into a black disk

Recently, the Auger group has extracted the proton-air cross section from observations of air showers produced by cosmic ray protons (and nuclei) interacting in the atmosphere and converted it into measurements of the total and inelastic $pp$ cross sections $σ_{\rm tot}$ and $σ_{\rm inel}$ at the super-LHC energy of 57 TeV. Their results reinforce our earlier conclusions that the proton becomes a black disk at asymptotic energies, a prediction reached on the basis of sub-LHC $\pbar p$ and $pp$ measurements of $σ_{\rm tot}$ and $ρ$, the ratio of the real to the imaginary part of the forward scattering amplitude [M. M. Block and F. Halzen, Phys. Rev. Lett. {\bf 107}, 212002 (2011)]. The same black disk description of the proton anticipated the values of $σ_{\rm tot}$ and $σ_{\rm inel}$ measured by the TOTEM experiment at the LHC cms (center of mass) energy of $\sqrt s=7$ TeV, as well as those of $σ_{\rm inel}$ measured by ALICE, ATLAS and CMS, as well as the ALICE measurement at 2.76 TeV. All data are consistent with a proton that is asymptotically a black disk of gluons: (i) both $σ_{\rm tot}$ and $σ_{\rm inel}$ behave as $\ln^2s$, saturating the Froissart bound, (ii) the forward scattering amplitude becomes pure imaginary (iii) the ratio $σ_{\rm inel}/σ_{\rm tot}=0.509 \pm 0.021$, compatible with the black disk value of 1/2, and (iv) proton interactions become flavor blind.

hep-ph↗

arXiv2012

Forward hadronic scattering at 8 TeV: predictions for the LHC

The Large Hadron Collider (LHC) recently started operating at 8 TeV. In this note, we update our earlier LHC forward hadronic scattering predictions \cite{physicsreports,update7, blackdisk}, giving new predictions, including errors, for the $pp$ total and inelastic cross sections, the $ρ$-value, the nuclear slope parameter $B$, $dσ_{\rm el}/dt$, and the large gap survival probability at 8 TeV.

hep-ph↗

arXiv2012

"Soft" Hadronic Cross Sections Challenge Hidden Dimensions

High energy measurements of the inelastic proton-proton cross sections, at the LHC at $\sqrt s$=7 TeV and by Auger at 57 TeV, have validated previous evidence from data collected over a wide range of energies that the total and inelastic cross sections for $pp$ and $\bar pp$ interactions saturate the Froissart bound of $\ln^2 s$. Although the data themselves did not cover truly asymptotic energies, our recent analysis of these data obtained the asymptotic ratio $\sigin/\sigtot=0.509\pm 0.021$, consistent with the value of 1/2 required for scattering by a black disk; further, the forward scattering amplitude became purely imaginary for $s\rightarrow \infty$, confirming the black disk interpretation. In addition, the limiting black disk behavior has been independently confirmed by an analysis of Schegelsky and Ryskin including LHC data on the shrinkage of the slope of the forward elastic scattering cross section. Unless one considers these results, emerging from an analytic amplitude analysis of data over an energy range of $6\le \sqrt s\le 57000$ GeV, a complete numerical accident, we rule out any new physics thresholds that contribute higher powers of $\ln s$ or, worse, powers of $s$ to the energy dependence of cross sections {\it at any energy}. This includes theories with additional dimensions of space-time, whose existence is challenged.

hep-ph↗

arXiv2011

Applications of the leading-order Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations to the combined HERA data on deep inelastic scattering

We recently derived explicit solutions of the leading-order Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations for the $Q^2$ evolution of the singlet structure function $F_s(x,Q^2)$ and the gluon distribution $G(x,Q^2)$ using very efficient Laplace transform techniques. We apply our results here to a study of the HERA data on deep inelastic $ep$ scattering as recently combined by the H1 and ZEUS groups. We use initial distributions $F_2^{γp}(x,Q_0^2)$ and $G(x,Q_0^2)$ fixed by a global fit to the HERA data. From $F_2^{γp}(x,Q_0^2)$ we obtain the singlet quark distribution $F_s(x,Q_0^2)$---using small non-singlet quark distributions taken from either the CTEQ6L or the MSTW2008LO analyses---evolve to arbitrary $Q^2$, and then convert the results to individual quark distributions. Finally, we show directly from a study of systematic trends in a comparison of the evolved $F_2^{γp}(x,Q^2)$ with the HERA data, that the assumption of leading-order DGLAP evolution is inconsistent with those data.

hep-ph↗

arXiv2011

Comment on "Ultrahigh-Energy Neutrino-Nucleon Deep-Inelastic Scattering and the Froissart Bound": Phys. Rev. Lett. 106, 231802 (2011)

The authors of a recent paper, "Ultrahigh-Energy Neutrino-Nucleon Deep-Inelastic Scattering and the Froissart Bound", A. Illarianov, B. Kniehl and A. Kotikov, Phys. Rev. Lett. 106, 231802 (2011), derive an approximate formula for the UHE limit of $σ_{νN}(s)$ in a class of models that includes our own and assert that they are led "to the important observation that $σ_{BBT}^{νN} \propto ln^3s$, which manifestly violates the Froissart bound [2] in contrast to what is stated in Refs. [6-8]", the latter reference being to our work and the $σ_{BBT}^{νN}$ to the cross sections we reported there. We here correct their erroneous implication that $σ_{νN}(s) should satisfy the Froissart bound and their mistaken assertion that we state that $σ_{BBT}^{νN}$ satisfies it.

hep-ph↗

arXiv2011

Experimental Confirmation that the Proton is Asymptotically a Black Disk

Although experimentally accessible energies can not probe `asymptopia', recent measurements of` inelastic $pp$ cross sections at the LHC at 7000 GeV and by Auger at 57000 GeV allow us to conclude that: i) both $\sigin$ and $\sigtot$, the inelastic and total cross sections for $pp$ and $\bar p p$ interactions, saturate the Froissart bound of $\ln^2 s$, ii) when $s\rightarrow \infty$, the ratio $\sigin/\sigtot$ is experimentally determined to be $0.509\pm 0.021$, consistent with the value 0.5 required by black disk at infinite energies, and iii) when $s\rightarrow \infty$, the forward scattering amplitude becomes purely imaginary, another requirement for the proton to become a totally absorbing black disk. Experimental verification of the hypotheses of analyticity and unitarity over the center of mass energy range $6\le \sqrt s\le 57000$ GeV are discussed. In QCD, the black disk is naturally made of gluons; our results suggest that the lowest-lying glueball mass is $2.97\pm 0.03$ GeV.

hep-ph↗

arXiv2011

A new numerical method for inverse Laplace transforms used to obtain gluon distributions from the proton structure function

We recently derived a very accurate and fast new algorithm for numerically inverting the Laplace transforms needed to obtain gluon distributions from the proton structure function $F_2^{\gamma p}(x,Q^2)$. We numerically inverted the function $g(s)$, $s$ being the variable in Laplace space, to $G(v)$, where $v$ is the variable in ordinary space. We have since discovered that the algorithm does not work if $g(s)\rightarrow 0$ less rapidly than $1/s$ as $s\rightarrow\infty$, e.g., as $1/s^\beta$ for $0<\beta<1$. In this note, we derive a new numerical algorithm for such cases, which holds for all positive and non-integer negative values of $\beta$. The new algorithm is {\em exact} if the original function $G(v)$ is given by the product of a power $v^{\beta-1}$ and a polynomial in $v$. We test the algorithm numerically for very small positive $\beta$, $\beta=10^{-6}$ obtaining numerical results that imitate the Dirac delta function $\delta(v)$. We also devolve the published MSTW2008LO gluon distribution at virtuality $Q^2=5$ GeV$^2$ down to the lower virtuality $Q^2=1.69$ GeV$^2$. For devolution, $ \beta$ is negative, giving rise to inverse Laplace transforms that are distributions and not proper functions. This requires us to introduce the concept of Hadamard Finite Part integrals, which we discuss in detail.

math.NA↗

arXiv2011

Forward hadronic scattering at 7 TeV: predictions for the LHC; an update

The LHC has successfully run for a long period at half energy, 7 TeV. In this note, we update earlier full-energy Large Hadron Collider (LHC) forward hadronic scattering predictions \cite{physicsreports}, giving new predictions, including errors, for the $pp$ total and inelastic cross sections, the $ρ$-value, the nuclear slope parameter $B$, $dσ_{\rm el}/dt$, and the large gap survival probability at the current 7 TeV energy.

hep-ph↗