Comprehensive fits to high energy data for σ , ρ , and B and the asymptotic black-disk limit
aa r X i v : . [ h e p - ph ] N ov Comprehensive fits to high energy data for σ , ρ , and B and the asymptotic black-disklimit Martin M. Block ∗ Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208
Loyal Durand † Department of Physics, University of Wisconsin, Madison, WI 53706
Phuoc Ha ‡ Department of Physics, Astronomy and Geosciences, Towson University, Towson, MD 21252
Francis Halzen § Wisconsin IceCube Particle Astrophysics Center and Department of Physics,University of Wisconsin-Madison, Madison, Wisconsin 53706 (Dated: September 10, 2018)We demonstrate that the entirety of the data on proton–proton and antiproton–proton for-ward scattering between 6 GeV and 57 TeV center-of-mass energy is sufficient to show that σ elas /σ tot → /
2, and that 8 πB/σ tot → B the forward slope pa-rameter for the differential elastic scattering cross sections. The relations demonstrate convincinglythat the asymptotic pp and ¯ pp scattering amplitudes approach those of scattering from a black disk.This result obviously has implications for any new physics that modifies the forward scatteringamplitudes. PACS numbers: 13.85.Dz, 13.85.Lg, 13.85.-t
I. INTRODUCTION
Proton–proton ( pp ) and antiproton–proton (¯ pp ) scattering have been studied for many decades. A persistentquestion since the advent of high-energy accelerators has concerned the behavior of the cross sections at very highenergies. They are bounded theoretically to increase no more rapidly than ln s , the Froissart bound [1–3], where s = W is the square of the total energy in the center-of-mass system. Block and Halzen [4] and Igi and Ishida [5, 6]showed convincingly that the ln s behavior in fact held for the pp and ¯ pp cross sections measured up to Tevatronenergies, with this behavior leading to successful predictions for the cross sections at the Large Hadron Collider (LHC).For a review, see [7].Block and Halzen [8, 9] and Schegelsky and Ryskin [10] also presented tentative evidence that the pp and ¯ pp scattering amplitudes may asymptotically approach those for scattering from a completely absorptive or “black”disk—the “black-disk” limit—at ultra-high energies, but the results of those analyses were not definitive. This result,and the common assumption that hadronic scattering is dominated at high energies by the interactions between gluonsin the two hadrons, together imply that all hadron-hadron cross sections should approach a common black-disk limitas s → ∞ , a very interesting result.In the present paper, we present the results of a comprehensive analysis of the forward pp and ¯ pp scattering datafor center-of–mass energies from 6 GeV to 57 TeV. We discuss various constraints on the cross sections which areessential in tying down the parametrizations of the low-energy cross sections, and present a fit to the data on σ pp (¯ pp )tot , σ pp (¯ pp )elas , and σ pp (¯ pp )inel , the forward slope parameters B pp and B ¯ pp , and the ratios of the real to imaginary parts of theforward scattering amplitudes ρ pp and ρ ¯ pp , using parametrizations which reflect the established ln s behavior of thecross sections at high energies.We find that the fit to the entirety of the data gives convincing evidence that the pp and ¯ pp scattering amplitudesapproach the black-disk limit at very high energies. We use this result to obtain a final, essentially identical, fit with ∗ Electronic address: [email protected] † Electronic address: [email protected]; Mailing address: 415 Pearl Ct., Aspen, CO 81611 ‡ Electronic address: [email protected] § Electronic address: [email protected] the black-disk constraints σ elas /σ tot → /
2, and 8 πB/σ tot → ρ , and B at the higher energies which may become accessible inthe future. The present results agree well with the predictions of earlier fits II. PARAMETRIZATIONS AND CONSTRAINTSA. Parameterization of the cross sections, the real-to-imaginary ratio ρ , and the slope parameter B We will be concerned here with global fits to the high-energy total, elastic, and inelastic pp and ¯ pp scatteringcross sections, the ratios ρ = Re f ( s, / Im f ( s,
0) of the real to the imaginary parts of the forward elastic scatteringamplitudes f ( s, t ), and the forward slope parameter B = d (ln σ ( s, t )) /dt (cid:12)(cid:12) t =0 for the differential cross sections dσ/dt .We will use the parametrizations of σ tot and ρ introduced by Block and Cahn [11] and used by Block and Halzen [4, 7]in their earlier fit to the pp and ¯ pp data up to a center-of-mass energy W = √ s = 1800 GeV. That fit was excellentand gave successful predictions of the more recent, higher energy data from the Large Hadron Collider (LHC) andcosmic ray experiments [8, 9].The Block-Cahn analysis assumed a ln s bound on the growth of the cross sections at high energy as impliedby the Froissart bound [1–3] and parametrized σ pp tot and σ ¯ pp tot as quadratic expressions in the s − dependent variable ν/m = ( s − m ) / m with additional falling Regge-like terms important at lower energies. The phase of thescattering amplitude at high energies and the corresponding expression for ρ then followed from the constraintsimposed by analyticity and crossing symmetry under the transformation ν → − ν [7, 11].We will extend the parametrizations here to the elastic and inelastic cross sections and the B parameter, with σ ( ν ) = c + c ln (cid:16) νm (cid:17) + c ln (cid:16) νm (cid:17) + β (cid:16) νm (cid:17) µ − , (1) σ ± tot ( ν ) = σ ( ν ) ± δ (cid:16) νm (cid:17) α − , (2) σ ± elas ( ν ) = b + b ln (cid:16) νm (cid:17) + b ln (cid:16) νm (cid:17) + β e (cid:16) νm (cid:17) µ − ± δ e (cid:16) νm (cid:17) α − , (3) ρ ± = 1 σ ± tot ( ν ) (cid:20) π c + πc ln (cid:16) νm (cid:17) − β cot (cid:16) πµ (cid:17) (cid:16) νm (cid:17) µ − + 4 πν f + (0) ± δ tan (cid:16) πα (cid:17) (cid:16) νm (cid:17) α − (cid:21) , (4) B ± ( ν ) = a + a ln (cid:16) νm (cid:17) + a ln (cid:16) νm (cid:17) + β B (cid:16) νm (cid:17) µ − ± δ B (cid:16) νm (cid:17) α − , (5)where the upper and lower signs are for pp and ¯ pp scattering, respectively. Here ν is the laboratory energy of theincident particle, with 2 mν = s − m = W − m where W is the center-of-mass energy and m is the proton mass.The inelastic cross sections are given by the differences between the total and elastic cross sections, σ ± inel = σ ± tot − σ ± elas .They are therefore parametrized simply as the differences of the expressions in Eqs. (2) and (3); no new parametersappear.It is not obvious that the very simple parametrizations above should be adequate to describe the cross sections, ρ , and B over the entire energy range we will consider. It is also not clear that the coefficients of these terms canbe determined well enough from fits in the extant energy range to extrapolate properly into the ultra-high energyregion where the ln ( ν/m ) terms become dominant. We have studied these questions quantitatively using a detailedeikonal model which provides a very good description of the data from 4 GeV to 57 TeV [12]. In that analysis, weused the expressions above to fit “data” for the cross sections, ρ , and B derived from the eikonal model. The fits areexcellent, with errors typically smaller than the real experimental uncertainties, and those fits over the “experimental”region continue to hold to ultra-high energies. Small correction terms would certainly be present analytically in theexpressions in Eqs. (1)–(5), but these are clearly unimportant in the fitting and extrapolation.We emphasize also that the presence of the ln ( ν/m ) terms in the parametrizations is not connected directly withthe Froissart bound: these terms are consistent with the bound, but follow in the eikonal model from the power-lawgrowth of the imaginary part of the eikonal function coupled with its exponentially bounded behavior in impactparameter space. This leads to a effective radius of interaction between the nucleons that grows logarithmicallywith increasing energy, and within which the scattering is nearly completely absorptive. As a result, the scatteringapproaches the “black-disk” limit at very high energies, with consequences we discuss below. Finally, as noted in [12],the coefficients of the ln ( ν/m ) terms depend on properties of the eikonal function that are not well determined. Wetherefore argued that the best extrapolations of cross sections and other parameters to ultra-high energies are thosebased on direct fits to the data using the parametrization above. We carry out those fits here.We turn next to a discussion of the known constraints on the parameters in Eqs. (1)–(5). B. Constraints
1. Low-energy constraints
There are nominally 18 parameters ( a , a , a , b , b , b , c , c , c , β, β e , β B , δ, δ e , δ B , α , µ , and f + (0)) in the model,but these are not all independent and must satisfy certain constraints. When these are imposed, we will end up withonly 12 independent parameters in our final fit.Both the “analyticity constraints” of Block and Halzen, derived in [13] and discussed in detail in [4], and the finiteenergy sum rule (FESR2) of Igi and Ishida [5, 7], impose constraints on the parameters. The first requires that thefits reproduce the values of the total cross sections at a transition point ν far enough above the resonance regionthat the high-energy parametrizations may be expected to hold, but where the cross sections can still be evaluatedaccurately using the dense low-energy data. The second approach obtains equivalent results through a matching ofthe FESR integrals at ν . Following [4], we take ν = 7 .
59 GeV corresponding to W = √ s = 4 GeV. Their low-energyanalysis gives σ pp tot = 40 .
18 mb, σ ¯ pp tot = 56 .
99 mb.In the case of the crossing-even combination of cross sections σ = (cid:0) σ pp tot + σ ¯ pp tot (cid:1) / (cid:0) σ +tot + σ − tot (cid:1) / c + c ln( ν /m ) + c ln ( ν /m ) + β ( ν /m ) µ − = σ ( ν ) = 48 .
58 mb . (6)An essentially equivalent result numerically follows from the finite-energy sum rules of Igi and Ishida [5–7] relatingthe low- and high-energy regions [7].A second constraint holds for the crossing-odd combination of cross sections ∆ σ = (cid:0) σ +tot − σ − tot (cid:1) /
2. Matching thetheoretical and experimental results, we find that δ ( ν /m ) α − = ∆ σ ( ν ) = − .
405 mb . (7)Two further analyticity constraints hold if one matches the derivatives of the cross sections with respect to ν/m to their experimental values at ν [4]. We will not use these because they are less reliable numerically and are moresensitive than the cross sections themselves to small deviations of the high-energy expressions in Eqs. (1) and (2) fromthe actual cross sections at the rather low matching energy of 4 GeV.A rather subtle constraint holds for the coefficients β, β e , δ, δ e of the Regge-like terms. These cannot be entirelyindependent since a descending power-law term in the eikonal function in a general impact-parameter representationof the scattering amplitudes f ± ( s, t ) affects σ ± elas and σ ± inel as well as σ ± tot . We have investigated these aspects of thescattering using our detailed eikonal model for pp and ¯ pp scattering [12], which gives an accurate description of thedata over the region where the Regge-like effects are important.The cross sections are described in the eikonal model in terms of the integrals σ tot ( s ) = 4 π Im f ( s,
0) = 4 π Z ∞ db b (cid:0) − cos χ R e − χ I (cid:1) , (8) σ elas = 2 π Z ∞ db b (cid:12)(cid:12) − e iχ (cid:12)(cid:12) = 2 π Z ∞ db b (cid:0) − χ R e − χ I + e − χ I (cid:1) , (9) σ inel ( s ) = σ tot − σ elas = 2 π Z ∞ db b (cid:0) − e − χ I (cid:1) , (10)where χ = χ R + iχ I is the complex eikonal function written in terms of crossing-even and crossing-odd parts.Writing χ as χ = χ + χ Regge , we can isolate the contributions of the Regge-like terms to the crossing-even andcrossing-odd cross sections σ ( ν ) and ∆ σ ( ν ) by subtracting the expression for the cross section for χ Regge = 0 fromthe full result. The effect of the factor cos χ R in Eq. (2) is small enough that we can neglect it for this purpose. If wedo so, the contribution of the crossing-even Regge term to the total cross section σ ( ν ) is given by the expression4 π Z ∞ db b cosh (cid:16) χ Regge , odd I (cid:17) e − χ , even I (cid:16) − e − χ Regge , even I (cid:17) . (11)We note that the contribution of χ Regge , odd I through the cosh function is second order in that quantity and can bedropped without significant loss of accuracy. Similar expressions hold for the other cross sections.Despite the somewhat different effects of the energy-dependent eikonalization in the different cross sections, we findthat the input power in a Regge-like term ( m/ν ) γ in the eikonal function χ Regge is reproduced to a percent or betterin output power-law fits to the various integrals over the energy interval 6–1000 GeV, where those outputs are to beidentified with the Regge-like terms in Eqs. (1)–(5). The powers are therefore stable across the expressions in Eqs.(1)–(5), as assumed.Importantly, we find that the ratios of the crossing-even and crossing-odd Regge-like contributions to σ inel to thecorresponding contributions to σ tot vary only slowly over the most important important energy range, 6 to 100 GeV(and beyond), with the even ratio in the range 0.684–0.657 and the odd ratio in the range 0.802–0.787. Averagesweighted by the even- and odd cross sections give ratios 0.678 and 0.797.Converting these results on the Regge-like terms to the elastic and total cross sections Eq. (2) and Eq. (3), we findthat β e = 0 . β, δ e = 0 . δ (12)as averaged over the interval 6–100 GeV, with only very small variations from these values. These relations give ournew, and not-very-obvious, constraints on the β and δ parameters in Eqs. (1) and (3). The smallness of the elastic-to-total ratios is easily understood: the Regge-like terms enter the elastic cross section in Eq. (3) only in second orderin χ Regge , but appear to first order in σ tot and σ inel .With the imposition of the 4 low-energy constraints in Eqs. (6), (7), and (12), 14 parameters are left to fit all datausing the parametrizations introduced above. These constraints are quite important: the results anchor the total crosssections accurately at the starting energies and in the Regge region, removing extra parameters which can otherwisemix with and affect the values of the high-energy parameters of primary interest. We note that only 9 of the remainingparameters appear in the expressions for the total, elastic, and inelastic cross sections and ρ ; the remaining 5 are inthe expression for B .
2. High-energy constraints
As noted above, we expect the pp and ¯ pp scattering amplitudes to approach the black-disk limit at ultra-highenergies, with the scattering amplitudes approaching those for scattering from a completely absorbing disk with aradius R which increases logarithmically with energy. In that limit, χ R → e − χ I vanishes for impact parameters0 ≤ b ≤ R and is equal to 1 for b > R . As a result, from Eq. (8), σ tot → πR up to edge effects of order R [12], whilefrom Eq. (9), σ elas → πR , also up to edge effects, and σ elas /σ tot → / f ( s,
0) is associated at high energies with peripheral scatteringoutside the region of strong absorption and, as an edge effect, is proportional to R for finite-range forces. It thereforedecreases as 1 /R relative to the imaginary part which is proportional to σ tot ∝ R , and ρ ∝ /R ∝ / ln W → f ( s, ≪ Im f ( s, B can be written as [12] B = 12 Z ∞ db b (cid:0) − e − χ I (cid:1) (cid:30) Z ∞ db b (cid:0) − e − χ I (cid:1) . (13)With the conditions above, the integrals can be evaluated simply in the black-disk limit, and we find that B → R / σ tot / π. (14)The same result for B can be derived less rigorously if it is assumed that the differential scattering cross section ispurely exponential in t , with dσ elas /dt = π | f ( s, | e Bt . Integrating over t from −∞ to 0, then using the the relation | f ( s, | = 16 π (cid:0) ρ (cid:1) σ and rearranging, we find that [11] B = σ (cid:0) ρ (cid:1) (cid:14) π σ elas , or, with ρ → σ elas /σ tot → / B → σ tot / π .It is an important question as to whether there is evidence of an approach to the black-disk limit in present data.If so, it is reasonable to impose the black disk constraints σ inel /σ tot → / B → σ tot / π in a final fit to the data.This leads in the parametrization above to the constraints b = c / , a = c / . × π (15)where the numerical factor arises from the conversion of c in mb to units of GeV − . This leaves 12 free parameters.The approach to the black-disk limit was investigated for σ tot and σ inel in [9] using a hybrid approach in which theparametrization for σ inel was determined from that for σ tot by multiplying the latter by the ratio σ inel /σ tot found inan earlier eikonal model [14] and fitting the result to an expression of the form in Eq. (2). The result agreed verywell with the measured high-energy inelastic cross sections. The ratio of the coefficients of the ln ( ν/m ) terms gavea value 0.509 ± σ inel /σ tot has the asymptotic value 1/2 automaticallyin the eikonal model used to get the parametrization for σ inel from that for σ tot . However, the excellent agreementof the predicted and measured inelastic scattering cross sections suggests that the same ratio should be found in afree fit to the data using the parametrization which follows from Eqs. (2) and (3). We will examine this in the nextsection.The asymptotic behavior of B was studied by Schegelsky and Ryskin [10] who used a simple a + b ln ( s/s ) formwith s = 1 GeV to fit the high-energy data. The coefficient in their result, equivalent to a = 0 . ± . − in Eq. (5), and the relation in Eq. (15) predicted the value c = 0 . ± .
005 mb for the leading coefficientin σ tot , closely matching the value c = 0 . ± . χ per degree for freedom of 1.5, not a remarkably good fit. We will reexamine the fit to B in the following section. III. FITS TO HIGH ENERGY PROTON - PROTON AND ANTIPROTON-PROTON DATAA. Data and method of fitting
The data we will use in our analysis consists of results on σ tot for W ≥ σ inel for W ≥
540 GeV, σ elas for W ≥
30 GeV, and ρ and B for W ≥
10 GeV. The energy ranges for σ tot , σ inel , and ρ are the same as used in theBlock-Halzen fits [4, 7, 8], but we include the newer data at very high energies from the LHC [15–17] and the Auger[18] and HiRes [19] collaborations. As noted, we include the extensive data on σ elas and B in our fits; these quantitieshave not been used before in fits of this type.The data on σ elas can be extended to 10 GeV or below without changingthe final results significantly, but the data are somewhat less accurate in that region, and we prefer to emphasize thehigher energies given our focus on the behavior of the cross sections and B at ultra-high energies.We used the sieve algorithm [4, 20] to identify outlying points and remove them from the data set used in the finalfits. There are two underlying assumptions in this procedure. We assume, first, that the parametrization used in thefit, with the parameter set α = { a , a , . . . , f + (0) } , can give a good description of theory, a point checked theoreticallyin [12] for the present case. Second, we assume that the complete data set consists mostly of datum points whichhave a normal Gaussian distribution with respect to the actual theoretical distribution, plus some outlying pointswhich have a much broader distribution than reflected in their quoted (Gaussian) uncertainties, the result of unknownexperimental problems. These outlying points can unduly influence a χ fit based on Gaussian statistics, but havemuch-reduced impact in a fit based on a broader statistical distribution.The sieve procedure is based on a Lorentzian probability distribution adjusted to give results that agree very wellwith those from a Gaussian distribution in the absence of outliers, but which still eliminates the latter efficiently whenthey are present. The details of the analysis are given in the appendix to [20].We first make a fit to the complete data set by minimizing Λ , the Lorentzian squared with respect to the parameterset α in the fit function over the datum points y i at the set W of center-of-mass energies W i at which the observationsare made, Λ ( α , W ) = N X i =1 ln (cid:2) . χ i ( W i ; α ) (cid:3) . (16)Here ∆ χ i ( W i ; α ) = [ y i − y i ( W i , α )] /σ i ( W ) where y i is the value of the quantity of interest measured at energy W i , y i ( W i , α ) is the theoretical value of that quantity for the parameters α , and σ i is the experimental error. Because ofthe intrinsically long tails of the Lorentzian distribution, this fit should be robust in the sense that points that lie farfrom the fitted distribution are accorded relatively little weight in the fitting, and do not influence the fit unduly.We next eliminate datum points for which ∆ χ i ( W i ; α ) is “too large,” with a value larger than a chosen ∆ max , takenhere as ∆ max = 6 [20]. These points lie well away from the theoretical fit and are presumed to be outliers relative tothe “good” Gaussian-distributed data. We then make a conventional Gaussian χ fit to the remaining points. If ourassumptions about the nature of the distribution are correct, the parameters α should not change significantly in thissecond fit, and the points identified as outliers should not change relative to the fit except possibly for those on theboundary with ∆ χ i ( W i ; α ) ≈ ∆ max .We note that 98.6% (99.7%) of the points in a normal Gaussian distribution would survive cuts with ∆ max = 6 (9).However, the normal points eliminated would contribute significantly to the Gaussian χ , and we must renormalizethe result χ found for the fit by a factor R = 1 .
110 (1.027) for ∆ max = 6 (9) to get the expected Gaussian result χ = R × χ . This renormalized χ has the usual statistical interpretation.Our original data set contained 167 datum points. In the analyses discussed in the next sections, we found thesame 8 outlying points in fits performed with and without the high-energy constraints in Eq. (15). Only 2.3 pointswith ∆ χ i ( W i ; α ) > χ was essentially the same in the two cases. These outliers, if included, would increase the final χ of the fits by about 57% relative to that of the points retained. For example, for the final 12 parameter fit usingthe high-energy constraints, χ = 161 . χ per point of 1.01. The extra contribution of the outlyingpoints in the original Lorentzian fit was 91.5, an average χ per point of 11.4 with actual values ranging from 6.6,slightly above the cutoff, to 28. We note finally that the outlying points are not concentrated in a way likely to affectour conclusions about high-energy scattering, with one point each in ρ for pp and ¯ pp scattering and three points in B pp distributed over the range 6 . ≤ W ≤ . σ ¯ pp tot at 8.76 GeV, one in σ ¯ pp elas at 900 GeV, and one in σ pp inel at 1800 GeV. B. Fit without high-energy constraints
We first consider the results of a global fit to the data on σ tot , σ elas , σ inel , ρ , and B which is not constrained by theblack-disk conditions in Eq. (15) at very high energies. We did use the low-energy constraints on the cross sections inEqs. (6) and (7), and the new ratio constraints on the coefficients of the Regge-like terms in Eq. (12); these constraintsare essential in tying down the cross sections at low energies. The sieve algorithm was used to filter the data resultingin the elimination of 8 outliers among 167 datum points as noted above. Combined plots of the cross sections fromthe fit are shown in Fig. 1. We do not show the fits to ρ and B ; the curves are nearly indistinguishable from those inFig. 4 shown later. áá æ ææææææææææææææ æ ææ æ æ æææ ææ æ æà ààààààààà à àà à à àà àà àààææææææææ æææææææææææææææææ æææ ææææ ææ æàà àà àà à àà ààà áá ì ì íí ò òòòò ò
10 100 1000 10 W, GeV Σ , m b FIG. 1: Fits, top to bottom, to the total, inelastic, and elastic scattering cross sections using the low-energy analyticityconstraints in Eq. (6) and Eq. (7) and the ratio constraints on the Regge-like contributions to the low-energy cross sectionsin Eq. (12): σ ¯ pp tot and σ ¯ pp elas (red) squares and dashed (red) line; σ pp tot and σ pp elas (blue) dots and solid (blue) line; σ ¯ pp inel (black)diamonds and line; σ pp inel (purple) triangles. The fit used only data on σ tot for W ≥ σ elas for W ≥
30 GeV, and σ inel for W ≥
540 GeV. The curve for σ elas includes data down to 10 GeV to show how the cross section is tied down at lower energies.Outlying points not used in the fit are shown with large open symbols surrounding the central points; the size of those symbolsdoes not reflect the quoted errors of the measurement. Table I shows the results of this 14-parameter χ fit. As seen from the table, the raw χ per degree of freedomis 1.11, while the renormalized χ per degree of freedom is 1.23. This is a very good fit, especially considering theamount of data used.It is very interesting to use the results from this fit, constrained only at low energies, to examine the very-high-energy Parameters ∆ χ i max = 6 c (mb) 23 . ± . c (mb) 0 . ± . c (mb) 0 . ± . b (mb) 7 . ± . b (mb) − . ± . b (mb) 0 . ± . a (GeV − ) 10 . ± . a (GeV − ) 0 . ± . a (GeV − ) 0 . ± . β (mb) 45 . ± . β e (mb) 14 . ± . β B (GeV − ) 0 . ± . f (0) (mb GeV) 2 . ± . δ (GeV − ) − . ± . δ e (GeV − ) − . ± . δ B (GeV − ) − . ± . α . ± . µ . ± . χ R × χ R × χ /d.o.f. 1.231TABLE I: The results for our 14-parameter χ fit to the ¯ pp and pp total, elastic, and inelastic cross sections, ρ values and slopeparameters B using expressions in Eqs. (1)–(5), the low-energy constraints in Eqs. (6), (7), and (12), and the cut ∆ χ i max = 6in the sieve analysis of the data. The renormalized χ /d.o.f., taking into account the effects of the ∆ χ i max cut, is given inthe row labeled R × χ /d.o.f., with R (6) = 1 . behavior projected for the cross sections and B . We find from Table I that σ elas σ tot → b c = 0 . . . ± . , as s → ∞ . (17)The deviation of this value of the ratio from the expected value 1/2 for for black-disk scattering at infinity energy iswell within the uncertainty of the fit.We find that the ratio of the fitted value of the ratio of B to its black-disk value σ tot / π also agrees very well withits expected value of 1 at high energies,(0 . π a c = 0 . ± . , as s → ∞ . (18)We conclude that these results, obtained from a fit which used only the low-energy constraints in Eqs. (6), (7), and(12), give strong evidence both that pp and ¯ pp scattering can be described asymptotically as black-disk scattering,and that the limiting ln s behavior is already evident at present energies. The use of the constraints ties down thelow-energy part of the fit, fixing the values of the total cross sections at 4 GeV and the ratios of the coefficients of theRegge-like terms in the cross sections. The low energy fit is excellent, and gives slopes of the total cross sections withrespect to ν/m at 4 GeV which agree reasonably well with those estimated from lower energy data [7] even thoughthe data used in the fit was confined to energies above 6 GeV. C. Fit using the black disk constraints
We have used the general parametrizations in Eqs. (2)–(5), with the low-energy constraints in Eqs. (6), (7) and(12), and the high-energy black-disk constraints Eq. (15) all imposed, to fit the combined pp and ¯ pp data over thesame energy ranges as above. The sieve algorithm was again used to eliminate the same 8 outliers among 167 datumpoints. There are now only 12 parameters.The result of the fit is excellent as seen in the last lines in Table II, with a χ of 161 for 147 degrees of freedomfor a raw χ per d.o.f. of 1.10, and a renormalized χ / d . o . f . of 1.22. As would be expected, the parameters of the fithave smaller uncertainties than in the previous fit using only the low-energy constraints, and with the exception of a , change only within the previous uncertainties.We give combined plots of the total, inelastic, and elastic cross sections at high energies in Fig. 2 and show thelower-energy behavior of σ tot in Fig. 3. The fitted curves for ρ and B are compared with those data in Fig. 4. All thedata are shown, including the two cross section points, the two values of ρ , and the three values of B dropped in thesieve analysis. We also show the statistical error bands for the fit; these show that the fit is very tightly constrainedover the region of the data. The consistency with the fit without the high-energy constraints and the rather small11% uncertainty in c = 0 . ± . dσ tot /d ( ν/m ) = ( m /W ) dσ tot /dW at ν = 7 .
59 GeV or W = 4 GeV were not used in the fitting by imposing the second set of analyticity constraints in [4, 13], the calculatedslopes, respectively -1.38 (-0.169) mb for ¯ pp ( pp ), match well with the slopes -1.45 (-0.231) determined from the densedata around 4 GeV [7].The present fits agree well with those of earlier work based on more limited data. The results of Block and Halzen[4, 9] used only the total cross sections and ρ values up to 1.8 TeV, without including the elastic or inelastic crosssections or measured values of B . Their results gave c = 0 . ± . . ± . . ± . . ± . W = 7, 8, and 57 TeV, in substantial agreement with the values98 . ± .
2, 101 ± . ±
13) (stat) + 17( −
20) (sys) ±
16 (Glauber)) mb found by TOTEM [16, 21] andAUGER [18].Our results for the completely constrained fit using the total, elastic, and inelastic cross sections, ρ , and B give c = 0 . ± . σ tot = 97 . ± .
86 mb and 99 . ± .
97 mb at 7 and 8 TeV, and 136 . ± . W large. áá æ ææææææææææææææ æ ææ æ æ æææ ææ æ æà ààààààààà à àà à à àà àà àààææææææææ æææææææææææææææææ æææ ææææ ææ æàà àà àà à àà ààà áá ì ì íí ò òòòò ò
10 100 1000 10 W, GeV Σ , m b FIG. 2: Fits, top to bottom, to the total, inelastic, and elastic scattering cross sections using high-energy black-disk constraintsin Eq. (15) as well as the the low-energy analyticity constraints in Eq. (6) and Eq. (7) and the ratio constraints on the Regge-likecontributions to the low-energy cross sections in Eq. (12): σ ¯ pp tot and σ ¯ pp elas (red) squares and dashed (red) line; σ pp tot and σ pp elas (blue) dots and solid (blue) line; σ ¯ pp inel (black) diamonds and line; σ pp inel (purple) triangles. The fit used only data on σ tot for W ≥ σ elas for W ≥
30 GeV, and σ inel for W ≥
540 GeV. The curve for σ elas includes data down to 10 GeV to showhow the cross section is tied down at lower energies. Outlying points identified in the sieve analysis and not used in the fit areshown with large open symbols surrounding the central points; the size of those symbols is not connected to the quoted errors.The statistical error bands determined by the error analysis are shown. áá æ æææææææ æ æ æææææ æ æ æ æ æ æ æ æà à à àààà ààà à à à à à W, GeV Σ t o t , m b FIG. 3: Curves showing the fits to σ pp tot , (blue) dots and solid (blue) line, and σ ¯ pp tot , (red) squares and dashed (red) line, atlow energies, extending the curves for the total cross sections in Fig. 2. The fits used the low-energy analyticity constraintsin Eqs. (6) and (7), the ratio constraints on the Regge-like contributions to the low-energy cross sections in Eq. (12), and theblack-disk high-energy constraints in Eq. (15). The ¯ pp outlier eliminated in the sieve analysis is shown with a large open symbolsurrounding the central point; the size of the symbol does not reflect the quoted accuracy of the measured value. The fixedvalues of the cross sections at 4 GeV from the low-energy data are also shown. The crossing-even high energy inelastic cross section σ ( ν ), valid in the energy domain √ s ≥
100 GeV where theodd Regge-like terms are very small and σ pp tot and σ ¯ pp tot are essentially equal, is given by σ ( ν ) = (19 . ± .
03) + (0 . ± . (cid:16) νm (cid:17) +(0 . ± . (cid:16) νm (cid:17) + (29 . ± . (cid:16) νm (cid:17) − . mb , (19)the difference of the expressions for σ tot and σ elas with the coefficients in Table II.For the convenience of the reader, we give the numerical predictions from the fit for the high energy pp (or ¯ pp )total, inelastic, and elastic cross sections, ρ , and B in Table III.We remark finally that, although the pp and ¯ pp scattering amplitudes approach the black-disk limit at very highenergies in the sense that σ elas /σ tot → / B → σ tot / π , there is not a sharp cutoff in those distributions inimpact parameter space as in the classic black-disk model with unit amplitudes for b < R and zero amplitudes for b > R , R = p σ tot / π . Rather, as observed in [22] and studied in detail in [12], the scattering amplitudes have asmooth edge region of approximately constant width t edge ≈ t edge ≈ (2 σ inel − σ tot ) / p πσ tot / . (20)We show this in Fig. 5 using the parameters in Table II for the fit with the black-disk constraints imposed. Given theaccuracy of the fit, we conclude that there is no evidence in the present data that the edge width shrinks significantlyat very high energies, with t edge → .
018 fm for s → ∞ . IV. CONCLUSIONS
We have shown that we can obtain a very good fit to all the high-energy data on the total, elastic, and inelastic pp and ¯ pp scattering cross sections, the ratios ρ of the real to the imaginary parts of the forward scattering amplitudes,and the logarithmic slopes B of the elastic scattering cross sections, using expressions quadratic in ln s with addedfalling Regge-like terms at low energies. The use of these expressions, introduced in [11] on the basis of the Froissart0 áá æææææææ æ ææææææææææææ æ ææ ææà àààà à à à
10 50 100 500 1000 5000 1 ´ - - W, GeV Ρ ææ ææææææææææææææææææææææææææææææ æ æææ æ æàà à à à à àà à àà ççç
10 50 100 500 1000 5000 1 ´ W, GeV B , H G e V (cid:144) c L - FIG. 4: Top panel: fits to the ratios ρ of the real to the imaginary parts of the forward scattering amplitudes for pp (blue dotsand solid blue line), and ¯ pp scattering (red squares and dashed red line). Lower panel: fits to the logarithmic slope parametersfor the elastic differential scattering cross sections dσ/dt for pp (blue dots and solid line) and ¯ pp (red squares and dashed line)scattering. The fits to ρ and B used only data above 6 GeV, and imposed the low-energy constraints on the parameters inEqs. (6), (7), and (12), and the high-energy asymptotic black-disk constraints in Eq. (15). In both cases, the datum pointseliminated in the sieve analysis are shown with large open symbols surrounding the central point; the size of the open symbolsdoes not reflect the quoted accuracy of the measurement. The error bands estimated from the uncertainties in the parametersare too narrow to show in the figure. bound, was justified in [12] for detailed eikonal descriptions of the scattering in which the eikonal function grows asa power of s . The Froissart bound is satisfied but is not an input in that analysis, nor is it directly a motivation forthe forms chosen here for the cross sections, ρ , and B in Eqs. (1)–(5).The initial fit we presented here used constraints on the values of the cross sections at W = 4 GeV, and new relationsfor the ratios of coefficients of the the Regge-like terms in the cross sections, to fix the fit at low energies. The resultsshow that the cross sections and values of B obtained using the present data satisfy the conditions σ elas /σ tot → / B → σ tot / π expected for black-disk scattering within the uncertainties in the fit. We regard these results as,first, a demonstration that data at the energies currently accessible already reflect the asymptotic ln s behavior ofthe cross sections, and second, as convincing evidence for black-disk behavior of the pp and ¯ pp scattering amplitudesat very high energies.We then presented a second fit in which we imposed the black-disk behavior as a constraint at high energies. This1 Parameters ∆ χ i max = 6 c (mb) 26 . ± . c (mb) − . ± . c (mb) 0 . ± . b (mb) 7 . ± . b (mb) − . ± . b (mb) 0 . ± . a (GeV − ) 10 . ± . a (GeV − ) 1 . ± . a GeV − ) 0 . ± . β (mb) 43 . ± . β e (mb) 14 . ± . β B (GeV − ) − . ± . f (0) (mb GeV) 2 . ± . δ (mb) − . ± . δ e (mb) − . ± . δ B (GeV − ) − . ± . α . ± . µ . ± . χ R × χ R × χ /d.o.f. 1.216TABLE II: The results for our 12-parameter χ fit to the ¯ pp and pp total, elastic, and inelastic cross sections, ρ values and slopeparameters B using expressions in Eqs. (1)–(5), the low-energy constraints in Eqs. (6), (7), and (12), the black-disk constraintsin Eq. (15), and the cut ∆ χ i max = 6 in the sieve filtering of the data which eliminated 8 outlying points. The renormalized χ /d.o.f., taking into account the effects of the ∆ χ i max cut, is given in the row labeled R × χ /d.o.f., with R (6) = 1 . √ s (GeV) σ tot , pp (mb) σ inel , pp (mb) σ elas , pp (mb) ρ pp B pp (GeV/c) −
540 61 . ± .
10 48 . ± .
10 12 . ± .
03 0 . ± .
000 15 . ± . . ± .
15 52 . ± .
15 14 . ± .
05 0 . ± .
000 16 . ± . . ± .
26 58 . ± .
24 18 . ± .
09 0 . ± .
000 17 . ± . . ± .
86 71 . ± .
52 25 . ± .
32 0 . ± .
000 19 . ± . . ± .
97 72 . ± .
56 26 . ± .
36 0 . ± .
000 19 . ± . . ± . . ± .
72 29 . ± .
56 0 . ± .
000 20 . ± . . ± . . ± .
75 30 . ± .
60 0 . ± .
000 20 . ± . . ± . . ± .
63 40 . ± .
87 0 . ± .
000 23 . ± . . ± . . ± .
21 45 . ± .
77 0 . ± .
000 25 . ± . pp total, inelastic, and elastic cross sections, ρ -values and B , using the parameters ofTable II in the expressions in Eqs. (1)–(5). gives nearly identical results, provides predictions for the cross sections at energies higher than those accessible now,and sharpens the analysis of results on the soft edge region in the scattering amplitudes discussed earlier [12, 22].It is known from the proton structure functions of deep inelastic scattering, and theoretically, that the protoninteractions at high energies are determined mainly by the gluonic and associated flavor-independent sea quarkstructure of the proton. We expect the same asymptotic structure for other hadrons, with a universal color confinementvolume, implying that all hadronic cross sections, e.g. , the π ± p and K ± p cross sections, should approach the sameblack-disk limit as found for the pp and ¯ pp cross sections. This picture is supported by the analysis of Ishida andBarger [23] who fit the π ± p and K ± p cross sections and ρ values using a parametrization equivalent to that used here,2
10 1000 10 W, GeV t e dg e , f m FIG. 5: Solid curve: plot of the width t edge of the soft edge in the crossing-even part of the pp and ¯ pp scattering amplitudes asa function of energy from 10 to 10 GeV. The horizontal dashed line is a t edge = 1 fm. and with the fitted cross sections similarly constrained to agree with the low-energy data through continuous momentsum rules. Their results and those here are consistent with the existence of a universal black-disk limit. For extensivereferences on the possible theoretical origin of the universality, beginning with L.L. Jenkovszky, B.V. Struminsky andA.N. Vall [24], see [23, 25].These results could be modified with the advent of new physics at higher energies which significantly changes thenature of the hadronic interactions. There is no evidence of such changes in the present scattering data. Acknowledgments
M.M.B., L.D., and F.H. would like to thank the Aspen Center for Physics for its hospitality and for its partialsupport of this work under NSF Grant No. 1066293. F.H.’s research was supported in part by the U.S. NationalScience Foundation under Grants No. OPP-0236449 and PHY-0969061 and by the University of Wisconsin ResearchCommittee with funds granted by the Wisconsin Alumni Research Foundation. P.H. would like to thank TowsonUniversity Fisher College of Science and Mathematics for support. [1] M. Froissart, Phys. Rev. , 1053 (1961).[2] A. Martin, Phys. Rev. , 1432 (1963).[3] A. Martin, Phys. Rev. D , 065013 (2009).[4] M. M. Block and F. Halzen, Phys. Rev. D , 036006 (2005).[5] K. Igi and M.Ishida, Phys. Lett. B , 286 (2005).[6] K. Igi and M.Ishida, Prog. Theor. Phys. , 601 (2006).[7] M. M. Block, Phys. Rep. , 71 (2006).[8] M. M. Block and F. Halzen, Phys. Rev. Lett. , 212002 (2011).[9] M. M. Block and F. Halzen, Phys. Rev. D , 051504 (2012).[10] V. A. Schegelsky and M. G. Ryskin, Phys. Rev. D , 094024 (2012), arXiv:1112.3243 [hep-ph].[11] M. M. Block and R. N. Cahn, Rev. Mod. Phys. , 563 (1985).[12] M. M. Block, L. Durand, P. Ha, and F. Halzen, Phys. Rev. D , 014030 (2015), arXiv:1505.04842 [hep-ph].[13] M. M. Block, Eur. J. Phys. C , 697 (2006).[14] M. M. Block, E. M. Gregores, F. Halzen, and G. Pancheri, Phys. Rev. D , 054024 (1999). [15] G. Antchev et al. (TOTEM Collaboration), Euro. Phys. Lett. , 41001 (2011).[16] G. Antchev et al. (TOTEM Collaboration), Euro. Phys. Lett. , 21002 (2013).[17] G. Antchev et al. (TOTEM Collaboration), Euro.Phys. Lett. , 21004 (2013).[18] P. Abreu et al. (Pierre Auger Collaboration), Phys. Rev. Lett. , 062002 (2012), arXiv:1208.1520 [hep-ex].[19] R. Abbasi et al. (HiRes Collaboration), Ap. J. , 790 (2008).[20] M. M. Block, Nucl. Inst. and Meth. A. , 308 (2006).[21] G. Antchev et al. (TOTEM Collaboration) (2015), arXiv:1503.08111v2 [hep-exp].[22] M. M. Block, L. Durand, F. Halzen, L. Stodolsky, and T. Weiler, Phys. Rev. D , 011501(R) (2015), arXiv:1409.3196[hep-ph].[23] M. Ishida and V. Barger, Phys. Rev. D , 014027 (2011).[24] L. Jenkovszky, B. Struminsky, and A. Vall, Yad. Fiz. , 1519 (1987), (English translation Sov. J. Nucl. Phys. , 1519(1987)).[25] M. Giordano and E. Maggiolaro, Phys. Lett. B744