New analyticity constraints on the high energy behavior of hadron-hadron cross sections

We here comment on a series of recent papers by Igi and Ishida[K. Igi and M. Ishida, Phys. Lett B 622, 286 (2005)] and Block and Halzen[M. M. Block and F. Halzen, Phys. Rev D 72, 036006 (2005)] that fit high energy $pp$ and $\bar pp$ cross section and $\rho$-value data, where $\rho$ is the ratio of the real to the imaginary portion of the forward scattering amplitude. These authors used Finite Energy Sum Rules and analyticity consistency conditions, respectively, to constrain the asymptotic behavior of hadron cross sections by anchoring their high energy asymptotic amplitudes--even under crossing--to low energy experimental data. Using analyticity, we here show that i) the two apparently very different approaches are in fact equivalent, ii) that these analyticity constraints can be extended to give new constraints, and iii) that these constraints can be extended to crossing odd amplitudes. We also apply these extensions to photoproduction. A new interpretation of duality is given.Comment: 9 pages, 1 postscript figure; redone for clarity, removal of typos, changing reference; figure replace

Decoupling the coupled DGLAP evolution equations: an analytic solution to pQCD

Using Laplace transform techniques, along with newly-developed accurate numerical inverse Laplace transform algorithms, we decouple the solutions for the singlet structure function $F_s(x,Q^2)$ and $G(x,Q^2)$ of the two leading-order coupled singlet DGLAP equations, allowing us to write fully decoupled solutions: F_s(x,Q^2)={\cal F}_s(F_{s0}(x), G_0(x)), G(x,Q^2)={\cal G}(F_{s0}(x), G_0(x)). Here ${\cal F}_s$ and $\cal G$ are known functions---found using the DGLAP splitting functions---of the functions $F_{s0}(x) \equiv F_s(x,Q_0^2)$ and $G_{0}(x) \equiv G(x,Q_0^2)$, the chosen starting functions at the virtuality $Q_0^2$. As a proof of method, we compare our numerical results from the above equations with the published MSTW LO gluon and singlet $F_s$ distributions, starting from their initial values at $Q_0^2=1 GeV^2$. Our method completely decouples the two LO distributions, at the same time guaranteeing that both distributions satisfy the singlet coupled DGLAP equations. It furnishes us with a new tool for readily obtaining the effects of the starting functions (independently) on the gluon and singlet structure functions, as functions of both $Q^2$ and $Q_0^2$. In addition, it can also be used for non-singlet distributions, thus allowing one to solve analytically for individual quark and gluon distributions values at a given $x$ and $Q^2$, with typical numerical accuracies of about 1 part in $10^5$, rather than having to evolve numerically coupled integral-differential equations on a two-dimensional grid in $x, Q^2$, as is currently done.Comment: 6 pages, 2 figure

Sifting data in the real world

In the real world, experimental data are rarely, if ever, distributed as a normal (Gaussian) distribution. As an example, a large set of data--such as the cross sections for particle scattering as a function of energy contained in the archives of the Particle Data Group--is a compendium of all published data, and hence, unscreened. Inspection of similar data sets quickly shows that, for many reasons, these data sets have many outliers--points well beyond what is expected from a normal distribution--thus ruling out the use of conventional $\chi^2$ techniques. This note suggests an adaptive algorithm that allows a phenomenologist to apply to the data sample a sieve whose mesh is coarse enough to let the background fall through, but fine enough to retain the preponderance of the signal, thus sifting the data. A prescription is given for finding a robust estimate of the best-fit model parameters in the presence of a noisy background, together with a robust estimate of the model parameter errors, as well as a determination of the goodness-of-fit of the data to the theoretical hypothesis. Extensive computer simulations are carried out to test the algorithm for both its accuracy and stability under varying background conditions.Comment: 29 pages, 13 figures. Version to appear in Nucl. Instr. & Meth.

New physics, the cosmic ray spectrum knee, and pppp cross section measurements

We explore the possibility that a new physics interaction can provide an explanation for the knee just above $10^6$ GeV in the cosmic ray spectrum. We model the new physics modifications to the total proton-proton cross section with an incoherent term that allows for missing energy above the scale of new physics. We add the constraint that the new physics must also be consistent with published $pp$ cross section measurements, using cosmic ray observations, an order of magnitude and more above the knee. We find that the rise in cross section required at energies above the knee is radical. The increase in cross section suggests that it may be more appropriate to treat the scattering process in the black disc limit at such high energies. In this case there may be no clean separation between the standard model and new physics contributions to the total cross section. We model the missing energy in this limit and find a good fit to the Tibet III cosmic ray flux data. We comment on testing the new physics proposal for the cosmic ray knee at the Large Hadron Collider.Comment: 17 pages, 4 figure

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.Comment: 16 pages, 2 figures; title and abstract changed, typos correcte

A new numerical method for obtaining gluon distribution functions G(x,Q2)=xg(x,Q2)G(x,Q^2)=xg(x,Q^2), from the proton structure function F2γp(x,Q2)F_2^{\gamma p}(x,Q^2)

An exact expression for the leading-order (LO) gluon distribution function $G(x,Q^2)=xg(x,Q^2)$ from the DGLAP evolution equation for the proton structure function $F_2^{\gamma p}(x,Q^2)$ for deep inelastic $\gamma^* p$ scattering has recently been obtained [M. M. Block, L. Durand and D. W. McKay, Phys. Rev. D{\bf 79}, 014031, (2009)] for massless quarks, using Laplace transformation techniques. Here, we develop a fast and accurate numerical inverse Laplace transformation algorithm, required to invert the Laplace transforms needed to evaluate $G(x,Q^2)$, and compare it to the exact solution. We obtain accuracies of less than 1 part in 1000 over the entire $x$ and $Q^2$ spectrum. Since no analytic Laplace inversion is possible for next-to-leading order (NLO) and higher orders, this numerical algorithm will enable one to obtain accurate NLO (and NNLO) gluon distributions, using only experimental measurements of $F_2^{\gamma p}(x,Q^2)$.Comment: 9 pages, 2 figure

Impact of new collider data on fits and extrapolations of cross sections and slopes

The latest Collider data are compared with our earlier extrapolations. Fits that include the new data are made. Those for which sigma/sub tot/ grows as log/sup 2/(s/s/sub o/) indefinitely give a significantly poorer chi/sup 2/ than those for which sigma/sub tot/ eventually levels out. For the proposed SSC energy for the former fits predict sigma/sub tot/(..sqrt..s = 40 TeV) approx. =200 mb while the latter give sigma/sub tot/(..sqrt..s = 40 TeV) approx. = 100 mb. 6 refs

A new approach to calculate the gluon polarization

We derive the Leading-Order master equation to extract the polarized gluon distribution G(x;Q^2) = x \deltag(x;Q^2) from polarized proton structure function, g1p(x;Q^2). By using a Laplace-transform technique, we solve the master equation and derive the polarized gluon distribution inside the proton. The test of accuracy which are based on our calculations with two different methods confirms that we achieve to the correct solution for the polarized gluon distribution. We show that accurate experimental knowledge of g1p(x;Q^2) in a region of Bjorken x and Q^2, is all that is needed to determine the polarized gluon distribution in that region. Therefore, to determine the gluon polarization \deltag /g,we only need to have accurate experimental data on un-polarized and polarized structure functions (F2p (x;Q^2) and g1p(x;Q^2)).Comment: 12 pages, 5 figure

Total photoproduction cross-section at very high energy

In this paper we apply to photoproduction total cross-section a model we have proposed for purely hadronic processes and which is based on QCD mini-jets and soft gluon re-summation. We compare the predictions of our model with the HERA data as well as with other models. For cosmic rays, our model predicts substantially higher cross-sections at TeV energies than models based on factorization but lower than models based on mini-jets alone, without soft gluons. We discuss the origin of this difference.Comment: 13 pages, 9 figures. Accepted for publication in EPJC. Changes concern added references, clarifications of the Soft Gluon Resummation method used in the paper, and other changes requested by the Journal referee which do not change the results of the original versio

Hadronic Cross-sections in two photon Processes at a Future Linear Collider

In this note we address the issue of measurability of the hadronic cross-sections at a future photon collider as well as for the two-photon processes at a future high energy linear $e^+e^-$ collider. We extend, to higher energy, our previous estimates of the accuracy with which the \gamgam\ cross-section needs to be measured, in order to distinguish between different theoretical models of energy dependence of the total cross-sections. We show that the necessary precision to discriminate among these models is indeed possible at future linear colliders in the Photon Collider option. Further we note that even in the $e^+e^-$ option a measurement of the hadron production cross-section via \gamgam processes, with an accuracy necessary to allow discrimination between different theoretical models, should be possible. We also comment briefly on the implications of these predictions for hadronic backgrounds at the future TeV energy $e^+e^-$ collider CLIC.Comment: 20 pages, 5 figures, LaTeX. Added an acknowledgemen