Elastic Amplitudes and Observables in pp Scattering
aa r X i v : . [ h e p - ph ] N ov Elastic Amplitudes and Observables in pp Scattering
A. Kendi Kohara, Erasmo Ferreira and Takeshi Kodama
Instituto de Física, Universidade Federal do Rio de Janeiro, C.P. 68528, Rio de Janeiro 21945-970, RJ, Brazil
Abstract.
Using a unified analytic representation for the elastic scattering amplitudes of pp scattering valid for all high energy region,the behavior of observables in the LHC collisions in the range √ s = 2.76 - 14 TeV is discussed. Similarly to the case of 7TeV data, the proposed amplitudes give excellent description of the preliminary 8 TeV data. We discuss the expected energydependence of the observable quantities, and present predictions for the experiments at 2.76, 13 and 14 TeV. Keywords:
PACS:
GENERAL INFORMATION AND DATA ANALYSIS
We establish explicitly disentangled real and imaginary amplitudes for pp elastic scattering based on a QCD motivatedmodel. With impact parameter representation ( s ,~ b ) and its Fourier transform in ( s ,~ q ) space both represented by simpleanalytical forms, we are able to control unitarity and dispersion relation constraints, and provide geometric interpreta-tion of the interaction range. The regularity obtained in the description of the data and the physical interpretation givereliability to the proposed amplitudes.The amplitudes of pp elastic scattering originally constructed through profile functions in b -space are written e T K ( s ,~ b ) = a K b K e − b / b K + l K ( s ) e y K ( g K ( s ) , b ) , (1)with the usual Gaussian forms plus the characteristic shape functions e y K ( s , b ) = e g K − √ g K + b / a a q g K + b / a h − e g K − √ g K + b / a i . (2)The label K = R , I indicates either the real or the imaginary part of the complex amplitude.For large b , corresponding to peripheral collisions, the amplitudes fall down with a Yukawa-like tail, ∼ ( / b ) exp ( − b / b ) , that reflects effects of virtual partons (modified gluon field) at large distances in the StochasticVacuum Model [1].The comparison with d s / dt data and determination of parameters are made with the amplitudes in t -space. Thequantities Y K ( g K ( s ) , t = − ~ q T ) obtained by Fourier transform of Eq. (1) are written T NK ( s , t ) = a K ( s ) e − b K ( s ) | t | + l K ( s ) Y K ( g K ( s ) , t ) , (3)and the shape functions converted to t − space take the form Y K ( g K ( s ) , t ) = g K (cid:20) e − g K √ + a | t | p + a | t | − e g K e − g K √ + a | t | p + a | t | (cid:21) . (4)The complete analysis of elastic scattering [2, 3, 4] requires also the contributions from the Coulomb interactionat small | t | and from perturbative 3-gluon exchange at large | t | . The fixed parameter a = .
39 GeV − is given bythe correlation length of the gluon condensate. All parameters have been determined as smooth functions of s and theproperties of the amplitudes (magnitudes, signs, zeros) have been described in detail [5].In our normalization the elastic differential and the total cross sections are written d s ( s , t ) dt = ( ¯ hc ) [ T I ( s , t ) + T R ( s , t )] = d s I ( s , t ) dt + d s R ( s , t ) dt , s ( s ) = ( ¯ hc ) √ p T NI ( s , t = ) . (5) ifferential Cross Sections and Amplitudes in the 1.8 to 14 TeV Range In Fig. 1 we show the predictions of d s / dt for the LHC energies 2.76 , 8 , 13 and 14 TeV. In the RHS we use theenergy √ s = T NI ( s , t ) , T NR ( s , t ) as functions of | t | as predicted by Eq.(3,4). The interplay of the imaginary and real amplitudes at mid values of | t | is responsible forthe dip-bump structure of the differential cross section, that was shown before [3] for √ s = | t | ≥ . the real part becomes dominant, with positive sign.Our description [3] of the elastic scattering data at 7 TeV from the TOTEM Collaboration [6] reproduces N=165points in d s / dt with an impressive squared average relative deviation < c > = .
31. Characteristic quantities atthis energy are s = .
65 mb, s el = .
39 mb , B = .
90 GeV − , that compare extremely well with the values s = . ± . s el = . ± . B = . ± . − published by TOTEM. Preliminary data for d s / dt at 8 TeV The preliminary TOTEM data of d s / dt at 8 TeV seem to be regular enough for our analysis. The data discussedbelow are taken (reading by eye) from slides of talks by members of the TOTEM Collaboration [7].As far as we can read from the presentation slides, we identify 212 data points in three sets : 1) N=97 points in theforward interval 6 × − ≤ | t | ≤ .
02 GeV ; 2) N=45 points in an intermediate interval 0 . ≤ | t | ≤ . ; 3)N=70 points in a mid | t | range 0 . ≤ | t | ≤ .
95 GeV . This information is transferred to the plot in Fig. 2, togetherwith our calculation. The quality of the representation seems to be equivalent to that of our treatment of the 7 TeVdata.Our values for B I and B R lead to the d s / dt effective slope B = [ B I + r B R ] / [ + r ] equal to B = .
405 GeV − .Hopefully the analysis of the final TOTEM measurements will be more precise and will confirm the validity of ourdescription for 8 TeV and predictions for 13 and 14 TeV . Inelastic and Total Cross Sections
For the inelastic cross section we assume the difference s inel = s − s el and then we have 73.26 mb at 7 TeV..Published values of the TOTEM Coll. using different methods are 73 . ± .
26 [6], 73 . ± . . ± . s inel = . ± . s inel = . ± . ± . √ s = .
76 TeV value of ALICE Coll., that gives s inel = . ± . s inel by CDF and E811 in Fermilab [13] suggeststhe value ( + r ) s inel = ( . ± . r value gives s inel = ( . ± . s inel = ± . √ s (in the pp system) up to 100 TeV.For 8 TeV we have predictions s = .
00 mb , s el = .
18 mb , s inel = .
82 mb , s el / s = .
26. The measure-ments by TOTEM [16] give for the same quantities s = . ± . s el = . ± . s inel = . ± . s el / s = . ± . d s / dt at 8 TeV and for the energy dependence of s and s inel in ppscattering work very well. Remarks and Comments
The proposed amplitudes have simple forms, evaluated with few operations with elementary functions. The shape ofthe dip-bump behavior results from a delicate interplay of the imaginary and real amplitudes. All intervening quantitiesnd derived properties present smooth energy dependences. The zeros of the real and imaginary parts have very regulardisplacements, converging to finite limits as the energy increases, showing remarkable connection with positions andheights of dips, bumps and inflections in d s / dt .The slopes B I and B R at the origin, with their characteristic difference in values, together with the ratio r , areessential quantities in the definition, through the unique analytical forms of the amplitudes, of the properties of theobserved d s / dt in the whole t range. Their values are thus fixed with high accuracy. It is important that the slopesshow quadratic dependence in log s , instead of the linear dependence suggested by Regge phenomenology.The integrated elastic cross sections are evaluated in their separate parts, obtained from the real and imaginaryamplitudes, and are also represented by simple parabolic forms in log s .The properties of ratios (with respect to the total cross section) of slopes and of integrated elastic cross sections, thattend to finite asymptotic limits, show that the hypothesis of a black disk limit in the behaviour of the pp interactionseems to be excluded by phenomenology.Taking into account previous publications at 1.8 and 7 TeV, we obtain cross sections and amplitudes at 2.76, 8 , 13and 14 TeV, with no free numbers. Future data will test.We also discuss the geometrical interpretation of our amplitudes, showing that the effective interaction radius in b -space increases with the energy. Our amplitudes obey a geometric scaling in asymptotic energies, and indicate thatthe profile function d s inel / d ~ b tends to a universal (energy independent) function with respect to a scaling variable, x ∼ b / √ s . This universal function exhibits a considerable diffused surface, indicating a scenario different from thecommonly accepted black disk. At LHC energies, the saturation seems to start (the central value of d s inel / d ~ b isalmost unity), but the asymptotic profile is still far and only can be reached for √ s > TeV. The connection betweenthe diffused surface of long range and inelastic diffractive processes will be an interesting line of investigation.We believe that our analytic representation of the scattering amplitudes will serve as important guidance for thefuture measurements in LHC, and also for a theoretical understanding of the intermediate region of partonic saturationphenomena.At this Diffraction 2014 conference related work on the black-disk behaviour [17] and on the determination of ppamplitudes at LHC energies [18, 19] were presented, showing that this is an important field of research at the present.
ACKNOWLEDGMENTS
The authors wish to thank the Brazilian agencies CNPq, PRONEX , CAPES and FAPERJ for financial support.
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14 TeV13 TeV8 TeV2.76 TeV |t| (GeV ) d s / d t ( m b / G e V ) √ s = 2.76 , 8 , 13 and 14 TeVpp elastic scatteringforward range -0.3-0.2-0.100.10.20.30.4 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 |t| (GeV ) T I ( s , t) a nd T R ( s , t) ( G e V ) √ s = 8 TeVT I T R |t| (GeV ) n o r m a li ze d a t | t | = T I /T I (0)T R /T R (0) -4 -3 -2 -1 FIGURE 1. (LHS) - Values of d s / dt obtained for energies of LHC experiments. The positions of dips and bump peaks, markedwith dots and squares, displace to the left as the energy increases, and can be connected with straight lines. The inset shows the low | t | range, with Coulomb interaction effects included. (RHS) - Real and imaginary parts of elastic pp scattering amplitude at 8 TeV,as functions of | t | . The general behaviour is the same for all energies, with one and two zeros respectively for the imaginary andreal parts. The behaviour for small | t | is shown in the inset, indicating the difference of slopes B R and B I , and the deviations of theexponential forms that occur as | t | increases, each amplitude going towards its zero. A second zero of the imaginary part occurs atmuch higher | t | . -2 -1 |t| (GeV ) d s / d t ( m b / G e V ) pp elastic scattering √ s = 8 TeVLHC - TOTEMPRELIMINARY DATA -1 |t| (GeV ) d s / d t ( m b / G e V ) pp elastic scattering √ s = 8 TeVLHC - TOTEMPRELIMINARY DATAsolid : full t amplitudesdashed : pure exp amplitudes FIGURE 2.
Preliminary data at 8 TeV extracted by eye from presentation slides of the TOTEM collaboration plotted togetherwith our predicted representation for d s / dtdt