Experimental Confirmation that the Proton is Asymptotically a Black Disk
aa r X i v : . [ h e p - ph ] S e p Experimental Confirmation that the Proton is Asymptotically a Black Disk
Martin M. Block
Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208
Francis Halzen
Department of Physics, University of Wisconsin, Madison, WI 53706 (Dated: August 16, 2018)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 concludethat: i) both σ inel and σ tot , the inelastic and total cross sections for pp and ¯ pp interactions, saturatethe Froissart bound of ln s , ii) when s → ∞ , the ratio σ inel /σ tot is experimentally determined tobe 0 . ± . s → ∞ , the forward scattering amplitude becomes purely imaginary, another requirement forthe proton to become a totally absorbing black disk. Experimental verification of the hypotheses ofanalyticity and unitarity over the center of mass energy range 6 ≤ √ s ≤ . ± .
03 GeV.
Introduction:
We discuss the implication of three new measurements of the high energy pp inelastic cross sections, σ inel ( √ s ), where √ s is the cms (center of mass) energy. At √ s = 7000 GeV, the Atlas collaboration [1] reports σ inel = 69 . ± . . ) ± . . ) mb, with (expt.) and (extr.) the total experimental and extrapolation errors. TheCMS collaboration [2], using a completely different technique, measures σ inel = 68 . ± . . ) ± . ± (lum . ) ± (extr . ),where (syst.) is the systematic error, (lum.) the error in luminosity and (extr.) is the extrapolation error formissing single and double diffraction events. Most recently, the Pierre Auger Observatory collaboration [3] reported ameasurement of σ p − airinel , the inelastic p-air cross section at √ s = 57000 ± contamination in a cosmic ray beam consisting mostly of protons at that energy, was converted by aGlauber calculation into the pp inelastic cross section [3], σ inel = 90 ± . ) ± (syst . ) ± . . ), with (stat.)the statistical, (syst.) the systematic errors and (Glaub.) the estimated error in the Glauber calculation. With acosmic ray measurement at 57000 GeV it is likely that we are now experimentally as close to asymptopia (definedhere as the energy behavior of hadron-proton cross sections near s → ∞ ) as we will ever get.Block and Halzen (BH) [4, 5] have made an analyticity constrained amplitude fit to lower energy data (6 ≤ √ s ≤ σ tot for ¯ pp and pp asymptotically saturates the Froissart bound [7]. This note exploitsthe new higher energy measurements of σ inel in order to make accurate predictions at asymptotia based only onmeasurements of pp and ¯ pp cross sections in the energy range 6 ≤ √ s ≤ ρ -value, the ratio of real and the imaginary partsof the forward scattering amplitude, an eikonal model, dubbed the ‘Aspen’ model [11], will be used to obtain the ratioof the inelastic to total cross sections, r ( √ s ) ≡ σ inel ( √ s ) /σ tot ( √ s ). We will show that the resulting ρ -value and theratio of σ inel /σ tot at √ s = ∞ are consistent with the proton being an expanding black disk, presumably of gluons;our fits to σ inel and σ el will allow us to infer a lowest-lying glueball mass of 2 . ± .
03 GeV. Furthermore, we willshow that both the Martin-Froissart bound [6, 7] on the pp and ¯ pp total cross sections and the Martin bound [9] onthe pp and ¯ pp inelastic cross sections are saturated, from 6 ≤ √ s ≤ The Analytic Amplitude Model:
Using this approach, BH was able to claim accurate predictions of the forward pp (¯ pp ) scattering properties, σ tot ≡ πp Im f ( θ L = 0) and ρ ≡ Re f ( θ L =0)Im f ( θ L =0) , using the analyticity-constrained analyticamplitude model[5] that saturates the Froissart bound [7]; here f ( θ L ) is the pp laboratory scattering amplitude with θ L , the laboratory scattering angle and p is the laboratory momentum. By saturation of the Froissart bound, we meanthat the total cross section σ tot rises as ln s . Furthermore the use of analyticity constraints allows one to anchor fitsat 6 GeV to the very accurate low energy cross section measurements between 4 and 6 GeV in the spirit of FiniteEnergy Sum Rules (FESR)[10]. A local fit is made of the experimental values of σ ± between 4 and 6 GeV, for both¯ pp and pp , from which BH [5] derive precise 6 GeV ‘anchor-points’ for σ ± and their energy derivatives in Eq. (1). Theresults are actually consistent with those obtained with old-fashioned FESR[8]. The model parameterizes the evenand odd (under crossing) cross sections and fits [5] 4 experimental quantities, σ ¯ pp ( ν ) , σ pp ( ν ) , ρ ¯ pp ( ν ) and ρ pp ( ν ) to thehigh energy parameterizations σ ± ( ν ) = σ ( ν ) ± δ (cid:16) νm (cid:17) α − , (1) ρ ± ( ν ) = 1 σ ± ( ν ) (cid:26) π c + c π ln (cid:16) νm (cid:17) − β P ′ cot( πµ (cid:16) νm (cid:17) µ − + 4 πν f + (0) ± δ tan( πα (cid:16) νm (cid:17) α − (cid:27) , (2)where the upper sign is for pp and the lower sign is for ¯ pp , and, for high energies, ν/m ≃ s/ m . Here the evenamplitude cross section σ is given by σ ( ν ) ≡ β P ′ (cid:16) νm (cid:17) µ − + c + c ln (cid:16) νm (cid:17) + c ln (cid:16) νm (cid:17) , (3)where ν is the laboratory energy of the incoming proton (anti-proton), m the proton mass, and the ‘Regge intercept’ µ = 0 .
5. The predictions for the pp and ¯ pp total cross sections are shown in Fig. 1. The dominant ln ( s ) term in thetotal cross section (Eq. (3)) saturates the Froissart bound [7]; it controls the asymptotic behavior of the cross sections.BH made a simultaneous fit[5] to the pp and ¯ pp data for the ρ value, the ratio of the real to the imaginary forwardscattering amplitudes, shown in Fig. 2. From Eq. (2) and Eq. (3), we see that in the limit of s → ∞ , ρ → / ln s ,(albeit very slowly), a necessary condition for a black disk. Although the ρ -values are essentially the same for ¯ pp and pp for √ s >
100 GeV, at the highest accelerator energies, ρ only changes from 0.135 at 7000 GeV to 0.132 at 14000GeV. Clearly, we are no where near asymptopia, where ρ = 0.With two low energy constraints at 6 GeV and 4 parameters, precise values for c and β P ′ could be obtained[5]. Thefitted values for the coefficients of σ ( ν ) of Eq. (3) for the fit for 6 ≤ √ s ≤ σ tot = 134 . ± . pp interactions. We note that c , the coefficient of ln ( s ),is well-determined, having a statistical accuracy of ∼ σ tot for both ¯ pp and for pp in the energy interval 6 ≤ √ s ≤ FIG. 1: The fitted total cross section, σ tot , for ¯ pp (dashed curve) and pp (dot-dashed curve) from Eq. (1), in mb vs. √ s , thecms energy in GeV, taken from BH [5]. The ¯ pp data used in the fit are the (red) circles and the pp data are the (blue) squares.The fitted data were anchored by values of σ ¯ pp tot and σ pp tot , together with the energy derivatives dσ ¯ pp tot /dν and dσ pp tot /dν at 6 GeVusing FESR, as described in Ref. [5]. The lowest (red) solid curve that starts at 100 GeV is our predicted inelastic cross sectionfrom Eq. (5), σ inel , in mb, vs. √ s , in GeV. The lowest energy inelastic data, the ¯ pp (red) diamonds, were not used in the fit,nor were the 3 high energy pp inelastic measurements, the (black) circle CMS value, the (green) square Atlas measurementand the (blue) diamond Auger measurement. As clearly seen, our inelastic prediction from Eq. (5), which also asymptoticallybehaves as ln ( s ), is in excellent agreement with the new measurements of the inelastic cross section at very high energy.TABLE I: Values of the parameters for the even amplitude, σ ( ν ), using 4 FESR analyticity constraints (taken from Ref. [5]) c =37 .
32 mb, c = − . ± .
070 mb, c =0 . ± . β P ′ .=37 .
10 mb
Aspen Model:
The Aspen model [11] is an eikonal model that describes experimental ¯ pp and pp data for σ tot , ρ andthe slope parameter B ≡ d [ln dσ el /dt ] t =0 , the logarithmic derivative of the forward differential elastic scattering crosssection, where t is the square of the 4-momentum transfer. Among many other quantities, it allows one to accuratelypredict the ratio r = σ el ( ν ) /σ tot ( ν ), i.e., the ratio of the elastic to total cross section for both ¯ pp and pp , as a functionof energy, where again, the total cross sections have been anchored at 6 GeV by FESR constraints [10]. Details ofthe model are given in Ref. [4, 11]. As is the case of the total cross sections, the values for r are essentially identicalfor ¯ pp and pp for cms energies √ s ≥
100 GeV. The ratio r is plotted in Fig. 3. Again, we see that we are far fromasymptopia, where the black disk model implies a ratio r = 1 /
2, whereas at 57000 GeV, we predict r ∼ . FIG. 2: The fitted ρ -value, for ¯ pp (dashed curve) and pp (dot-dashed curve) from Eq. (1) vs. √ s , the cms energy in GeV. The¯ pp data used in the fit are the (red) circles and the pp data are the (blue) squares.FIG. 3: The r -value, the ratio of σ el /σ tot , vs. √ s , the cms energy in GeV. Inelastic cross section:
We are now ready to evaluate σ inel ( ν ) ≡ (1 − r ( ν )) σ ( ν ) numerically for √ s ≥
100 GeV,using r ( ν ) obtained above, together with the fitted even amplitude cross section σ ( ν ) of Eq. (3) determined by theparameters of Table I. Since the approach is at this point purely numerical, we decided to fit the inelastic numberswith the same analytical parameterization as was used for the total cross section σ ( ν ) in Eq. (3). The analyticexpression for the even amplitude high energy inelastic cross section σ ( ν ) given by σ ( ν ) ≡ β inel P ′ (cid:16) νm (cid:17) µ − + c inel0 + c inel1 ln (cid:16) νm (cid:17) + c inel2 ln (cid:16) νm (cid:17) (4)= 62 . (cid:16) νm (cid:17) − . + 24 .
09 + 0 . (cid:16) νm (cid:17) + 0 . (cid:16) νm (cid:17) mb (5)accurately reproduces the numerical values of σ inel ( ν ) to better than 4 parts in 10 over the energy range 100 ≤√ s ≤ σ ( ν ) implies that the Froissart bound is also saturated for the high energy inelastic cross sections in the energy interval 100 ≤ √ s ≤ pp inelastic cross sections.The LHC 7000 GeV pp inelastic cross section data points are the (black) circle from CMS [2] and the (green) squarefrom Atlas [1], slightly separated for visual purposes. The (blue) diamond is the Auger inelastic cross section [3] for a25% He contamination of their σ p − airin cross section at 57000 GeV. We emphasized that none of these experimentalinelastic cross sections were used in our fits that predicted high energy inelastic cross sections. At 7000 GeV ourprediction is σ inel = 69 . ± . σ inel = 92 . ± . ( s )fit of Eq. (5) for σ ( ν ) is in excellent agreement with all experimental data, up to the highest possible energy. Evidence for a black disk:
It is unlikely that there will ever be higher energy measurements for σ inel for either ¯ pp or pp collisions, yet our results show that present measurements are far from asymptopia. Nevertheless, the data giveus a consistent picture of asymptopia by the compelling evidence that both the elastic and inelastic cross sectionssaturate the Froissart bound. The addition of the inelastic cross section of Eq. (5) going as ln s now allows us toexplore asymptopia experimentally ; we find the limit of σ inel ( s ) /σ tot ( s ) as s → ∞ simply by taking the ratio of theln ( s ) terms in Eq. (5) and Eq. (3). We find the experimentally-determined value at infinity,lim s → ∞ σ inel ( s ) σ tot ( s ) = c inel2 c = 0 . ± . , (6)a result compatible with the ratio 1/2 predicted for a black disk. Satisfying this ratio of the inelastic to the totalcross section at infinity gives us the first experimental evidence that the proton becomes an expanding black disk atasymptopia. We have already shown that the second condition, ρ = 0, i.e., the amplitude is imaginary, is also satisfied.The model of Troshin [13] in which the elastic scattering dominates over the inelastic is thus falsified, whereas themodels [14, 15] in which the proton becomes a black disk asymptotically are now justified experimentally. Properties of a black disk : In impact parameter space b , the elastic and total cross sections are given by σ el = 4 Z d b | a ( b, s ) | , σ tot = 4 Z d b Im a ( b, s ) . (7)The amplitude a ( s, b ) of the black disk of radius R is given by a ( b, s ) = i , ≤ b ≤ R, a ( b, s ) = 0 , b > R, (8)so that (for details, see Ref. [12]) σ tot = 2 πR , σ inel = σ el = πR , σ inel σ tot = 0 . , dσ el dt = πR (cid:20) J ( qR ) qR (cid:21) , where q = − t. (9)Using analyticity and unitarity, Andre Martin has recently found a more rigorous inelastic hadron-proton bound [9],using t = (2 m π ) , i.e., σ inel < π m π ln s, so that σ tot < π m π ln s (10)where for the total cross section bound we have invoked the black disk ratio of 2 to 1. The use of m π in the two-particlemass M = 2 m π is clearly wrong experimentally, since π m π ln ( ν/m ) = 31 .
23 ln ( ν/m ) mb, whereas experimentallywe have obtained c ln ( ν/m ) = 0 . ( ν/m ) mb, a cross section two orders of magnitude smaller, implying thatthe scale is not set by the pion mass but by a mass scale one order of magnitude larger. Reinterpreting M = 2 m π inEq. (10) as the lowest-lying glueball mass which we call M glueball , we find M glueball = (2 π/c ) / = 2 . ± .
03 GeV . Obviously, the definition of this scale is still arguable.Also, if the asymptotic proton is a black disk of gluons, the high energy behavior is flavor blind and the coefficientof the ln s term is the same for all reactions, from πp to γp scattering. Support for this claim comes from both theCOMPETE group[16] and Ishida and Igi [17]. Conclusions:
We find that the ln s Froissart bound for the proton for σ tot [7] and σ inel [9] is saturated and thatat infinite s , (1) the experimental ratio σ inel /σ tot = 0 . ± . σ tot has been extrapolated up from the Tevatron, we expect no new large cross sectionphysics between 2000 and 57000 GeV.Finally, the lowest-lying glueball mass is measured to be M glueball = 2 . ± .
03 GeV. Reproducing these experi-mental results will be a task of lattice QCD.
Acknowledgments:
In part, F. H. is supported by the National Science Foundation Grant No. OPP-0236449, by theDOE grant DE-FG02-95ER40896 and by the University of Wisconsin Alumni Research Foundation. M. M. B. thanksthe Aspen Center for Physics, supported in part by NSF Grant No. 1066293, for its hospitality during this work. [1] Atlas Collaboration, arXiv:1104.0326 [hep-ex] 2 April,2011; to be published, Nature Comm. [2] CMS Collaboration, CERN Document Server, http://cdsweb.cern.ch/record/1373466?ln=en, August 27, 2011.[3] Auger Collaboration, arXiv:1107.4804 [astor-phys.HE], (2011) , part 2., ”Estimate of the proton-air cross section”, R.Ulrich; Auger Collaboration, M. Mostaf´a, XXXI Physics in Collision Conference, Vancouver, Sept 1 2011.[4] M. M. Block, Phys. Reports , 71 (2006).[5] M. M. Block and F. Halzen, Phys. Rev. D , 036006 (2005); M. M. Block and F. Halzen, Phys. Rev. D , 054022 (2006).[6] A. Martin, Nuovo Cimento , 930 (1966).[7] M. Froissart, Phys. Rev. , 1053 (1961).[8] K. Igi and M. Ishida, Phys. Rev. D , 034023 (2002).[9] A. Martin, Phys. Rev. D , 065013 (2009).[10] M. M. Block, Eur. Phys. J. C. , 697 (2006).[11] M. M. Block, E. Gregores, F. Halzen and G. Pancheri, Phys. Rev. D , 054024 (1999).[12] M. M. Block and R. H. Cahn, Rev. Mod. Phys. , 563 (1985).[13] S. M. Troshin and N. E. Tyurin, Int. J. Mod. Phys. A22 , 4437, 2007; arXiv:08103391 [hep-th], 2008.[14] T. T. Chou and C. N. Yang, Phys. Rev , 1591 (1968).[15] C. Bourrely, J. Soffer and T. T. Wu, Nucl. Phys.
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