Utilizing the most recent experimental data, we reanalyze high energy \pbar p
and pp data, using the asymptotic amplitude analysis, under the assumption that
we have reached `asymptopia'. This analysis gives strong evidence for a $\log
\,(s/s_0)$ dependence at {\em current} energies and {\em not} $\log^2 (s/s_0)$,
and also demonstrates that odderons are {\em not} necessary to explain the
experimental data.Comment: 7 pages in LaTeX, 4 figures and 5 files, uuencoded in file
  "sigall.uu

Block, M. M.

Margolis, B.

White, A. R.

English

arXiv

Utilizing the most recent experimental data, we reanalyze high energy pp and pp data, using the asymptotic amplitude analysis, under the assumption that we have reached {open_quotes}asymptopia{close_quotes}. This analysis gives strong evidence for a log (s/s{sub 0}) dependence at current energies and not log{sup 2} (s/s{sub 0}), and also demonstrates that odderons are not necessary to explain the experimental data

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The submitted manuscript has been authored by a contractor of the U.S. Gwsrnmant under contract No. W-31-104ENG.38. Accordingly, the U. S Govsrnment retains a nonexclurfw, royalty-frw licano to publish or reproduce the published form of this contribution, or allow others to do m, for U. S. Gourrnmant DU~POIBI. j , N.U.H.E.P. Report No. 490 AN L-HEP-CP-95-79 The High Energy Behavior of the Forward Scattering Parameters-An Amplitude Analysis Update M. M. Block * Department of Physics and Astronomy, Northwestern University, Euanston, IL 60208 B. Maxgolis Physics Department, McGill University, Montreal, Canada H3A 2T8 A. R. White High Energy Physics Diuision, Argonne National Laboratory, Argonne, IZ60439 Paper presented by Martin M. Block at  the Vlth Blois Workshop, Chateau de Blois, France, June, 1995 Abstract Utilizing the most recent experimental data, we reanalyze high energy pp and pp data, using the asymptotic amplitude analysis, under the assumption that we have reached ‘asymptopia’. This analysis gives strong evidence for a log ( S / S O )  dependence at current energies and not log2(s/so), and also demonstrates that  odderons are not necessary to  explain the experimental data. ‘Work partially supported by Department of Energy contract DA-AC02-76-Er02289 Task B. t Deceased. 4 Work supported by Department of Energy contract W-31-109-ENG-38. I Dedication This work is dedicated to the memory of Bernard Margolis, Rutherford Professor of Physics at McGill University, Montreal, Canada, who died shortly after this paper was presented in June, 1995. He was a magnificent physicist and a more magnificent friend. He will be sorely missed! I1 Asymptotic Amplitude Analysis In spite of the fact that there are excellent arguments [l] that the energy region in which present experiments are conducted-even at the Tevatron Collider-is too low to be con- sidered asymptotic, we will consider here the consequences of assuming the opposite. This allows us to test specific hypotheses using a well-defined phenomenological analysis . We caution the reader that we don’t believe we are in ‘asymptopia’ and thus don’t believe the analysis is applicable as a true asymptotic analysis. We do believe that present day energies are too low to make a truly asymptotic analysis. Nonetheless, we feel that such an analysis is valuable as a guideline to what is and is not happening at present energies. We apply a “standard” asymptotic analytic amplitude analysis procedure[l] to the now- available data on otot, the total cross section and p, the ratio of the real to the imaginary portion of the forward scattering amplitude, in the energy region 6 = 5 to 1800 GeV, including the new CDF cross sections. The data are parameterized in terms of even and odd analytic amplitudes. Consistent with all asymptotic theorems, this allows use of even amplitudes varying as fast as log2(s/so) and odd amplitudes (the ‘Odderon’ family) that do not vanish as s + 00. We show here only the large s limit of the even and odd amplitudes that are used[l]. We make five fits to the data: (i)  Fit 1: log2(s/so) energy dependence for the cross section, with no Odderon am- (ii) Fit 2: log2(s/so) energy dependence for the cross section, with an Odderon am- plitude whose cross sectional dependence is logs, the most rapid behavior allowed by asymptotic theorems, pli tude, (iii) Fit 3: log2(s/so) energy dependence for the cross section, with an Odderon am- plitude whose cross sectional dependence is constant, (iv) Fit 4: log(s/so) energy dependence for the cross section, with no Odderon am- plitude, (v) Fit 5: log(s/so) energy dependence for the cross section, with an Odderon am- plitude whose cross sectional dependence is constant, the most rapid behavior allowed by asymptotic theorems for this choice of even amplitude. In all cases, an odd amplitude which vanishes with increasing energy is also employed, as well as an even amplitude that mimics Regge behavior. - 1 -  11.1 log 2(s) Energy Behavior We introduce f+ and f-, the even and odd (under crossing) analytic amplitudes at t = 0, and define the pp and pp forward scattering amplitudes by fpp = f+ + f- and fpp = f+ - f-, giving total cross sections atot and the p-values We parameterize the ‘conventional’ even and odd amplitudes f+ and f- by : 4n DsP-l is(l-a)/2 -f- = - e P The parameter CY in Eq (3) turns out to be about 0.5, and thus this odd amplitude vanishes as s + 00. Asymptotic theorems by Eden and Kinoshita[l] prove that the digerence of cross sections can not grow faster than l0g7l2(s), when the cross section grows as log7(s). Thus, odd amplitudes which do not vanish as s + co for this case are : 2 Tf- 47r (0)  = - €(O), pf- 4% (1) = - [log ($) - i;] &), and, Ffi2) = - [log (e) - i;] d 2 ) .  The complete odd amplitude is formed by adding any one (or none) of the f!) to the conventional odd amplitude f- of Eq (3). We then fit the experimental p and atot data, for both pp and pp, for energies between 5 and 1800 GeV, to obtain the real constants A,  /?, SO, c, p, D, CY, The data used below 500 GeV are listed in [l], and the high energy points are from UA1, UA4, E710 and CDF[l]. We emphasize that what we really fit for the UA4 and CDF cross sections is the measured experimental quantity atot x (1 + p2),  which is appropriate for experiments that measure a ‘luminosity-free’ cross section, whereas for UA1 and the 1020 GeV point of E710, we fit the experimental quantity atot x Jm, which was their experimentally measured quantity (they measured a ‘luminosity-dependent’ cross section). 11.2 Fitted Results for log2(s) Behavior (i) Fit 1-This fit uses no Odderons in the odd amplitude and uses the even ampli- tude of Eq (2). The x2/d.f. (X2/degree of freedom) for the fit was 1.94, a rather large number. The fitted constants are shown in Table 1, Fit 1-the computed curves are shown in Fig. 1 (for atot) and Fig. 2 (for p).  The most obvious features of the fit are (a) the predicted value of the total cross section is much too high to fit the (b) it predicts much too high a p-value at 546 GeV. experimental values (E710 and CDF) at 1800 GeV, We conclude that a simple log2(s) fit does not fit the data. - 2 -  (iii) Fit 2- We fit the data with an additional degree of freedom, by adding Odderon 2 to f- of Eq (3), along with the even amplitude of Eq (2). The parameters are summarized as Fit 2, in Table 1. Again, we conclude that this combination doesn't fit the data, since the high energy cross section predicted at 1800 GeV is much too high. Although the p-value predicted at 540 GeV is slightly lower, the p values predicted are still too high. (iv) Fit 3-The odd amplitude added to the conventional f- of Eq (3) was Odderon 1. The parameters are given as Fit 3 in Table 1. Again, the fit suffers from the same defect as the Odderon 2 fit, giving much too high a total cross section at 1800 GeV, as well as predicting a UA4/2 p-value which was much too high. The addition of Odderon 0 can have no effect on the cross section. Since it turns out to have a negligible effect on p, we will not consider it further. We conclude that an even amplitude varying as log2(s/so) does not f;t the cross section data. We see that the experimental cross section does not rise as rapidly as log2(s/so), in the present-day energy region. The addition of an Odderon term does not change this conclusion. Figure 1: The total cross section utot, in mb, for pp and pp scattering vs. the energy, 4, in GeV, for Fit 1, described in Table I. The fit was made with a log'((s) energy variation, and no Odderon. The crosses are for the jjp experimental data and the circles indicate pp data. The dot-dashed curves are for pp, and the solid curves for pp. The pp cosmic-ray lower limit[l] is appended to the curve, but is not used in the fit. 11.3 log (s) Energy Behavior Since the experimental cross section in the energy region 5-1800 GeV did not vary as fast as log2(s/so), we now consider an asymptotic variation that goes as log(s/so). We - 3 -  Figure 2: The pvalue for pp and pp scattering us. the energy, 6, in GeV, for Fit 1, described in Table I. The fit was made with a log2(s) energy variation, and no Odderon. The crosses are for the pp experimental data and the circles indicate pp data. The dot-dashed curves are for pp, and the solid curves for pp. substitute for the even amplitude in Eq (2) a new amplitude f+ varying as log(s/so), &f+ = i ( A  + p [log (6) - i;] + c s  '+1eir(1-p)/2) . We use the conventional odd amplitude of Eq (3), along with no Odderon or Odderon 1, in Fits 4 and 5, respectively. We make the important observation that since the energy variation of the cross section is now only log(s), Odderon 2 is not allowed by the asymptotic theorems. P 11.4 Fitted Results for log(s) Behavior (i) Fit 4-The data are fitted with a log(s/so) cross section energy behavior, with no Odderon. The results are detailed in Table 1, and plotted in Fig. 3 and Fig. 4. The fit is quite satisfactory, giving a x2/d.f. of 1.22, fitting reasonably well to all cross section data over the entire range of energy. Most importantly, it now fits the UA4/2 p-value at 546 GeV, as well as the E710 p-value at 1800 GeV. (ii) Fit 5- The data are fitted with a log(s) cross section energy behavior, along with Odderon 1. The results are given in Table 2. This fit (as is Fit 4) is quite satisfactory, giving a x2/d.f. of 1.24. Indeed, it is almost indistinguishable from fit 4. We find that the experimental cross sections and p-values in the energy domain 5-1800 GeV can be reproduced using a log(s/so) energy variation. Further, the introduction of an Odderon amplitude is not needed to explain the experimental data. Also, using the new CDF cross sections does not change these conclusions. -4- 230 220 210 200 190 180 170 I60 150 - 130 120 100 90 EO 70 60 50 40 30 140 E b 110 I t I * . . . I .  1 * . , I . * * , I  1 I * I . . * . I  . . * * . I  4 10 I 00  I000 io000 100000 6 (GeV) Figure 3: The total cross section utot, in mb, for pp and pp scattering vs. the energy, 6, in GeV, for Fit 4, described in Table I. The fit was made with a log(s) energy variation, and no Odderon. The crosses are for the ijp experimental data and the circles indicate pp data. The dot-dashed curves are for pp, and the solid curves for pp. The pp cosmic-ray lower limit[l] is appended to the curve, but is not used in the fit. References For a complete bibliography, see M. M. Block et al., p. 373, Proceedings, XXIII In- ternational Symposium on Multiparticle Dynamics, Aspen, CO, 1994, World Scientific, M. M. Block and A. White, Editors. DISCLAIMER - This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, ream- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. - 5 -  .30 .25 .20 Figure 4: The pvalue for pp and pp scattering vs. the energy, 6, in GeV, for Fit 4, described in Table I. The fit was made with a log(s) energy variation, and no Odderon. The crosses are for the pp experimental data and the circles indicate pp data. The dot-dashed curves are for pp, and the solid curves for pp. I - “““I . - ’ . . . . a ,  . . . . .. Parameters I I. 4 X2/d.f. d.f. ’-------. Fit 1 40.3 f .20 -47 f .02 200 f 20 -40.9 f 1.9 .46 f .02 30.9 f 4.1 .46 1.94 82 Fit 2 40.23~ .24 .48 f .02 207 f 25 -38.8 f 2.1 .47 f .02 26.6 f 4.3 .49 -.024 f .011 1.83 81 Fit 3 41.6f .04 .57 f .01 346 f 11 -36.8 f 1.6 .49 f .02 5.9 f 2.6 .49 .035 f .040 2.58 81 Fit 4 -26.0 f 12.7 9.6 f .9 500 -43.2 f 2.1 .45 f .02 159 f 17 .86 f -01 ~ 1.22 82 Fit 5 -22.5 f 11.7 9.4 f -8 500 -43.3 f 2.0 .45 f .02 154 f 16 .86 f .01 -.016 f .043 1.22 81 Table 1: Results of fits to  total cross sections and p-values, including Odderons. Fit 1, Fit 2 and Fit 3 correspond to  an asymptotic cross section variation of log2(s/so), with no Odderon, Odderon 2 and Odderon 1, respectively, whereas Fit 4 and Fit 5 correspond to  an energy dependence of log(s/so), with no Odderon and Odderon 1, respectively. - 6 -  

The high energy behavior of the forward scattering parameters - an amplitude analysis update

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The High Energy Behavior of the Forward Scattering Parameters---An Amplitude Analysis Update

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