Pomeron pole plus grey disk model: real parts, inelastic cross sections and LHC data

I propose a two component analytic formula $F(s,t)=F^{(1)}(s,t)+F^{(2)}(s,t)$ for $(ab\rightarrow ab) +(a\bar{b}\rightarrow a\bar{b})$ scattering at energies $\ge 100 GeV$ ,where $s,t$ denote squares of c.m. energy and momentum transfer.It saturates the Froissart-Martin bound and obeys Auberson-Kinoshita-Martin (AKM) \cite{AKM1971} scaling. I choose $Im F^{(1)}(s,0)+Im F^{(2)}(s,0)$ as given by Particle Data Group (PDG) fits to total cross sections. The PDG formula is extended to non-zero momentum transfers using partial waves of $Im F^{(1)}$ and $Im F^{(2)}$ motivated by Pomeron pole and 'grey disk' amplitudes . $Re F(s,t)$ is deduced from real analyticity: I prove that $Re F(s,t)/ImF(s,0) \rightarrow (\pi/\ln{s}) d/d\tau (\tau Im F(s,t)/ImF(s,0) )$ for $s\rightarrow \infty$ with $\tau=t (ln s)^2$ fixed, and apply it to $F^{(2)}$.Using also the forward slope fit by Schegelsky-Ryskin , the model gives real parts,differential cross sections for $(-t)<.3 GeV^2$, and inelastic cross sections in good agreement with data at $546 GeV, 1.8 TeV,7 TeV$ and $ 8 TeV $. It predicts for inelastic cross sections for $pp$ or $\bar{p} p$, $\sigma_{inel}=72.7\pm 1.0 mb$ at $7TeV$ and $74.2 \pm 1.0mb$ at $8 TeV$ in agreement with $pp$ Totem experimental values $73.1\pm 1.3 mb $ and $74.7\pm 1.7 mb$ respectively, and with Atlas values $71.3\pm 0.9 mb$ and $71.7\pm 0.7mb$ respectively. The predictions at $546 GeV$ and $1800 GeV$ also agree with $\bar{p} p$ experimental results of Abe et al \cite{Abe} at $546 GeV$ and $1800 GeV$. The model yields for $\sqrt{s}>0.5 TeV$, with PDG2013 total cross sections , and Schegelsky-Ryskin slopes as input, $\sigma_{inel} (s) =22.6 + .034 ln s + .158 (ln s)^2 mb , and \sigma_{inel} / \sigma_{tot} \rightarrow 0.56, s\rightarrow \infty ,$ where $s$ is in $GeV^2$

Introduction. Precision measurements of pp cross sections at LHC [7] [8] [9][10] [11][12] [13][14] [15][16], and in cosmic rays [17] motivate me to present a model for ab → ab scattering amplitude at c.m. energies √ s > 100GeV described by an analytic formula containing very few parameters. Neglecting terms with a power decrease at high s , the Particle Data Group (PDG) fits to total cross sections [3], [4] are the sum of one constant component and another rising as (lns) 2 , corresponding to a simple pole and a triple pole at J = 1 in the angular momentum plane, σ ab tot = σ I propose that, analogously, the full amplitude F (s, t) = F (1) (s, t) + F (2) (s, t), where, F (1) is a Pomeron simple pole amplitude , ImF (2) has partial waves with a smooth cut-off at impact parameter b = R(s) corresponding to a grey disk and ReF (2) (s, t) is calculated from a theorem I prove using real analyticity and Auberson-Kinoshita-Martin (AKM) [1] [2] scaling for s → ∞ with fixed t(lns) 2 . Inelastic unitarity is tested using inputs of total cross sections, forward slopes and Pomeron parameters. * Electronic address: smroy@hbcse.tifr.res.in Only inputs leading to unitary amplitudes are accepted. Model predictions for inelastic cross sections,near forward real parts and differential cross sections agree with existing data and can be tested against future LHC experiments.
Froissart-Martin bound basics. Froissart [18],from the Mandelstam representation, and Martin [19], from axiomatic field theory, proved that the total cross-section σ tot (s) for two particles a, b to go to anything must obey the bound, where C, s 0 are unknown constants.It was proved later [20] that C = 4π/(t 0 ), where t 0 is the lowest singularity in the t−channel .This bound has been extremely useful in theoretical investigations [ [32]. Analogous bounds on the inelastic cross section have been obtained by Martin [33]and Wu et al [34]; for pion-pion case, Martin and Roy obtained bounds on energy averaged total [35] and inelastic cross sections [36] which also fix the scale factor s 0 in these bounds.
Normalization.For the ab → ab scattering amplitude F (s, t), a = b, with k = c.m. momentum, and z = 1 + t/(2k 2 ), with the inelastic unitarity constraint Ima l (s) ≥ |a l (s)| 2 . For identical particles a = b, the partial waves a l (s) → 2a l (s) in the above partial wave expansions for F (s, t) , and σ tot (s), but the odd partial waves are zero. We have the same formulae for the unitarity constraint, and the differential cross section as given above. At high energy, using a l (s) ≡ a(b, s), l = bk, where b is the impact parameter, and P l (cosθ) ∼ J 0 (2l + 1) sin(θ/2) + O(sin 2 (θ/2)), we have the impact parameter representaion, There exist very good fits to high energy data [37] [38] with a very large number of free parameters . There are also very good eikonal based models involving several free parameters [23] [39][40] uses high energy data to guess the glue-ball mass and to probe whether the proton is a black disk.
A two component partial wave model. I present a two component model with very few parameters and with analytic formulae for the total amplitude incorporating unitarity-analyticity constraints , PDG total cross sections and the AKM scaling theorem .
Imaginary parts. I use the two component PDG total cross section fit. I propose that in the impact parameter picture, the Imaginary part Ima(b, s) of the partial waves at fixed s is also a sum of two components, one part Ima (1) (b, s) a Gaussian corresponding to a Pomeron pole, and the other Ima (2) (b, s) a polynomial of degree 2n in b 2 with a smooth cut-off at b = R(s) , n being a positive integer. so that Ima (2) (b, s) is continuous and has continuous derivative at b = R(s). The second component corresponds to a "grey" disk with cross section rising as (ln s) 2 , where θ(x) = 1, f orx ≥ 0, and 0 otherwise. The unitarity constraints are, In Eq.(5) we take the simplest choice n = 1 in this paper. Using the ansatz for Ima (1) (b, s), integrating over b , and matching the result for ImF (1) (s, t) with the standard small t Pomeron amplitude , we obtain , tot is a constant, C(s) → const/(ln s), s → ∞ for α ′ = 0. Similarly, the ansatz for Ima (2) (b, s) with n = 1 yields, where J m (x) denotes the Bessel function of order m. Hence, Thus, C(s)D 2 (s) and E(s)R 2 (s) are determined from the PDF total cross section fits using Eqns. (8) and (10) respectively. A nice feature of the model is that the above unitarity constraints (6) as well as a stronger version including real parts can be readily tested, and provide acceptability criteria for extrapolations of experimental data for pp scattering.
In the c.m. energy range from 0.5T eV to 100T eV ,the model parameters are very well approximated by the following fits.

Input σ
where,x ≡ ln s. Remarkably, fits for input σ  (27) These results are close to the black disk value of 1/2 favoured by BH [39] [40].Recent detailed analysis of high energy data [51] concluded that, although consistent with experimental data, the black disk does not represent an unique solution.
Jin and Martin [52] proved that for |t| < t 0 , where t 0 is the lowest t-channel singularity,twice subtracted dispersion relations in s hold. Hence t 1 may be thought of as a phenomenological lowest t − channel singularity. Using the formulae for R 2 (s) given above, Our √ t 1 ∼ 1.4 − 1.8GeV is reminiscent of , but different from the glue-ball mass of BH [39] [40]. Given the instability of analytic continuations, its main function is to suggest that the usual Lukaszuk-Martin bound [20] is quantitatively poor as it assumes lack of t−channel singularities only upto 4m 2 π which is much smaller than t 1 . Conclusion. I presented an analytic formula for the high energy elastic amplitude F (s, t) = F (1) (s, t) + F (2) (s, t) given by Eqns. (7,18) for √ s > 100GeV , exhibiting Froissart bound saturation, AKM scaling [1] [2], inelastic unitarity , predicting differential cross sections for (−t) < 0.3GeV 2 and total inelastic cross sections, at 546GeV , 1800GeV , 7T eV and 8T eV in agreement with experimental results, as well as the real parts and inelastic cross sections upto 100T eV . An 'effective' t-channel singularity at √ t ∼ 1.4 − 1.8GeV is suggested by analytic continuation to positive t. Detailed tables and graphs of model parameters, real parts and cross sections upto 100 T eV will be published separately. The 'grey disk' component could be generalized using a smoother impact parameter cut-off, i.e. n > 1 in Eqn. (5).