Dumbrajs, O V

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CERN LIBRARIES, GENEVA CM-P00100681 JOINT INSTITUTE FOR NUCLEAR RESEARCH, DUBNA Report P2 - 5448 MODEL-INDEPENDENT DETERMINATION OF THE REAL PARTS OF THE pp FORWARD SCATTERING AMPLITUDES NOT ALLOWINC FOR SPIN EFFECT O.V. Dumbrajs Dubna 1970 Translated at CERN by R. Luther (Original: Russian) Not revised by the Translation Service (CERN Trans. 71-28) Geneva April 1971 1. I n t r o d u c t i o n When a real part of the pp and p forward scattering amplitude is discussed, it is normally assumed that there is no dependence whatsoever on spin (viz., for example, / 1 /). The scat­tering amplitude (in the laboratory system) may then be represented in the form of an ordinary complex number, let us say, f ± ( ω ) = D ± ( ω ) + iA ( ω ) . (1) Normalization was selected so that the optical theorem took the form σ±(ω) = 4π A ± ( ω ) / k , where k and ω are the momentum and energy (hp) of the nucleon respectively. The relationship D ± / A ± ≡ a ± for this amplitude may be determined from its interference with the Coulomb amplitude, by studying scattering at very small angles (viz., for example,/2/). From previous papers, we managed to find 35 values "a" determined by this method for pp scattering (table 1, fig. 1 ) * . In general, the experimental situation is not at all satisfactory, as can be clearly seen from the wide range of values in fig. 1. Attempts have been made to determine theoretically the real parts of these amplitudes on the basis of dispersion relations / 1 3 - 2 0 / . However, all these calculations call for certain assump­tions concerning the unphysical and asymptotic regions/15-18/ or just the asymptotic region/19-20/. This is undesirable, especially since the appearance of the Serpukhov data on the total cross-sections *For p scattering, we know of only one experimental value a, at 11,9 GeV/c / 7 /. - 2 -of interactions between various particles, as these have raised certain doubts as to the validity of the Pomeranchuk theorem. It is therefore advisable to take a fresh look at the calculation of the real parts of these amplitudes, taking into account the new experimental data on the total cross-sections/21/ of p scattering and on the real parts of pp forward scattering/9/*. This paper uses a new model-independent method for determining the real parts of amplitudes. We have already used this method in a modified form/22/ for Κ ± p scattering. 2. Description of the method The analytic structure of amplitude (1) and of the complex energy p l a n e ω i s shown in fig. 2, where -mp and mp denote the beginning of the physical thresholds for pp and p scattering respectively, m 2 π is the beginning of the unphysical region and π is the pole position. Little is known experimentally about p scattering in the low-energy physical region between mp and ω 0 . We now define the function: Δ(ω) = D-(ω) - D+(mp) + ω + mp [Ι(ω) + J ( ω ) ] . (2) π Where Ι(ω) = 1 P w [ σ + (ω') -σ- (ω') ]k'dω' , (3) ∫ 4π (ω' - mp)(ω' + ω) (ω' + mp)(ω' - ω) mp *Viz paper/2/ for a good survey of the present state of experimental and theoretical data in terms of a for pp scattering. - 3 -J ( ω ) = -Im ∫ -(ω') dω' • (4) (ω' + mp) (ω'- ω) S(W) W is the highest energy (≈ 50 GeV) at which the experimental value for the total cross-section σ - ( ω ) of p scattering is known / 2 1 / , S(w) signifies that integration is done in terms of a semicircle of radius |ω|=W in the upper half-plane. Let us consider the closed contour in the ω-plane, con­sisting of the straight line joining the points -W + iε and W + iε(ε→  0 + ) , and the semicircle S(w). By applying Cauchy's theorem to the function F ( ω ' : ω) = f-(ω') (5) (ω'+mp)(ω' - ω ) around this contour and using the well-known analyticity and cross­ing properties of f ± ( ω ) , it is easy to show that Δ(ω) = ω + mp [ π Χ (π) + mp Λ-(ω') d ω' ]. (6) ∫ π (ωπ + mp)(ωπ-ω) (ω' + mρ) (ω' - ω ) ω2π where ωπ m 2 π - 2m2p . 2mp Χ.(π) = g2π [(mπ — mp)2 — m2p] 4 m2p and g2π is the pπ coupling constant. If we assume the Pomeranchuk theorem to be valid, then in the limit W → ∞ the relation obtained by equating expressions (2) and (6) reduces to a conventional dispersion relation for f - ( ω ) with one subtraction at the pp threshold. However, the validity of the relation obtained by this method is entirely independent of the asymptotic behaviour of the scattering amplitude at finite energy W. - 4 -The total cross-sections σ + ( ω ) f o r pp scattering are known experimentally in the range 0,06 GeV/c ≤ k ≤ 24 GeV/c. Below 0,06 GeV/c they can be adequately calculated for our pur­poses on the basis of an effective radius approximation with a = - 17±0,2 fm, r = 2,83±0,03 fm / 2 5 /. σ - ( ω ) are known experimentally /21,23,24,26/ in the interval ω0 ≤ ω ≤ ω0W, where ω0 = 0,996 GeV. If we assume that σ + ( ω ) does not vary substantially in the interval from 24 GeV/c to 50 GeV/c and that the value D+(mp) is known, and if we ignore the, fact that the σ - ( ω ) values are not known in the range m p ≤ ω ≤ ω 0 and that J ( ω ) is also unknown, then the experimental values of function may be calculated from formula (2) at all points where the D± values are known. On the other hand, equation (6) expresses Δ ( ω ) a s a sum of the contributions from the unphysical region. It is clear from equation (6) that the function Δ(ω) is relatively smooth in the energy region where it can be calculated from the experimental data. This is due to the fact that Δ(ω) at moderate or high energies |ω| is expressed as a sum of the contributions from relatively distant energies. Therefore, we may expect that Δ(ω) can be expressed by means of a small number of parameters. The essence of our method is the fact that, by virtue of equation (2), a fit to Δ(ω) also signifies a fit to D±(ω). A rapidly convergent expansion may be obtained for function Δ(ω) by means of one of the variants of the conformai mapping technique. It can be seen from fig. 2 that the only singularities o f Δ ( ω ) in the ω - plane are the π-pole and the unphysical out joining points ω3π and mp. We now introduce the new variable: ξ = √ω - ω0 - √ω - ω2π (7) √ω - ω0 + √ω - ω2π - 5 -With an appropriate choice of square roots, equation (7) represents a conformal mapping of ξ ( ω ) , which transforms the entire ω-plane into the unit circle |ξ| = 1 . ξ is the plane shown in fig. 3. We wish to point out that by mapping out the interval mp ≤ ω ≤ ω0 on to the circumference, we include it in the unphysical region and thus avoid the need to know σ- (ω) in this region. The function Δ ( ξ ) i s analytic within the unit circle, except for the pole at ξ = ξπ. If this pole is removed, by multiplying Δ(ξ) by ξ - ξ π , the new function||(ξ) may be expanded in a power series ||(ξ) = (ξ - ξπ) Δ(ξ) = Σ ~ n = 0 a n ξ n , (8) which is convergent for |ξ| < 1. Moreover, by introducing the variable η = √W + ω - √ W - ω (9) √W + ω + √W - ω the ω-plane may be mapped in a unit c i r c l e | η | = 1 so that the asymptotic region, i . e . | ω | > | W | , falls on the circum­ference (fig. 4 ) . Therefore, the asymptotic contribution ω + mp/π J ( ω ) t o function Δ ( ω ) m a y be expressed as a power series ω + mp J(ω) = ∞ bn η n . (10) Σ π n=0 which is convergent fo |η| < 1. Thus, it can be said that our method involves a fit within the circumference of the experimentally measured function G (ω) = ∆(ω) - ω + mp J ( ω ) (11) π - 6 -in the following form: G(ω) = 1 ∞ an ξn -∞ bn ηn. (12) Σ Σ ξ - ξπ n=0 n=0 It should be noted that not one of the aeries in (12) can express G ( ω ) by itself. It can be seen from (4), (6) and (12) that G ( ω ) has both a low-energy singularity ( π is the pole and unphysical cut) and a high-energy cut (with branch points at ω = ±W). In the energy range with which we are concerned and where G ( ω ) is known from experiment, the first series in (12) cannot alone express G ( ω ) due to the high-energy cut, whilst the second series cannot alone express G ( ω ) due to the low-energy singularities· General considerations and also the experiment des¬ cribed in paper/22/ indicate clearly that the series in (12) converge more rapidly close to the circle centres. Only the real parts of pp scattering hare been calculated in this paper, and therefore the situation is unfavourable (viz. figs. 3 and 4) as the region within which G ( ω ) is known experimentally is situated in both circles to the right (to the left) of the centre and use is not made of the whole region within which there is good convergence adjacent to the centre. Therefore, additional conformal mapping techniques were used ζ = ξ - r • (13) 1 - rξ ν = η - q . (14) 1 - qη where r and q are constants. These techniques enable the region where G ( ω ) is known from experiment to be shifted along the real axes at fixed end-points ξ ( ω 0 ) , ξ ( ω 2 π ) , η(-W) and η(+W). - 7 -Thus, an optimal position can be found on the real axes for regions where G(ω) is known from experiment. Moreover, the term D+(mp) in (2) was omitted from the practical calculations as the analytic properties of G(ω) do not vary, and the error is reduced. 3. C a l c u l a t i o n During the calculations, and in accordance with conven­tional statistical criteria, we ascertained that the optimal number of parameters is 6 - a 0 - a2 and b 0 - b2. The total value of X2 equals 145,4 for 37 points. Of these points 6 - at p = 1,29; 1,39; 1,54; 10,11; 19,33 and 26,42 GeV/c - have abnormally large X2 values: 24,8; 42,7; 16,5; 10,1; 7,3 and 9,1 respectively. There­fore, these points were rejected and the calculations repeated. The value of X2 proved to be 13,9 for the same number of parameters (the optimal number as before). The X2 values for the separate points do not normally exceed one. 4. Results and conclusions The results of the calculations are shown in table 2 and as a curve in fig. 1. As regards the points which were rejected, the values of a at p = 10,11 GeV/c;19,33 GeV/c and 26,42 GeV/c/10/ obviously have reduced errors and the a values at ρ = 1,29 GeV/c; 1,39 GeV/c and 1,54 GeV/c/4/ are either just wrong or, if the validity of dis­persion relations is not rejected, then pp scattering in this energy range involves an abnormal dependence on spin. Therefore, there is a need for fresh measurements of the a value in the range p = 1-2 GeV/c. In conclusion, it must be noted that it would be possible in principle to continue analytically this pp scattering amplitude - 8 -into the p scattering region and also to the π-pole, thus obtaining information on D -(ω) and g π 2 . However, the practical calculation showed that, as with K±p scattering/22/, this is impossible due to the great instability of such a procedure. I wish to express my deep gratitude to N.M. Queen for his many comments and observations, and also to V.L. Lyuboshits, V.A. Nikitin, M.I. Podgoretskij and L.N. Strunov for their useful remarks. - 9 -B i b l i o g r a p h y 1. A. Vezerell. UFN, 101, 471 (1970) 2. V.A. Nikitin. Problemy fiziki elementarnykh chastits i atomnogo yadra, (Problems of the physics of elementary particles and the atomic nucleus) Bk. 1, Ρ 9, Dubna, 1970. 3. J.D. Dowell et al,m Proc. Sienna Conf., 683, 1963; Phys. Lett., 12., 252 (1964). 4. L.M.C. Dutton, H.B. Van der Raay. Phys. Lett., 26B, 11 (1968); 25B, 245 (1967). 5. L.F. Kirillova et al. Zh. Ε.Τ.F. 50, 77 (1966). 6. P. Grossmann et al. Czech. J. Phys., B17, 830 (1967). 7. K.J. Foley et al. Phys. Rev. Lett., 19, 857 (1967); 14, 74 (1965). 8. A.E. Taylor et al. Phys. Lett., 14, 54 (1965). 9. V.A· Nikitin. Report to the XV International Conference on High-Energy Physics, Kiev, 1970. 10. G. Belletini et al. Phys. Lett., 19, 705 (1966); 14 164 (1965). 11. E. Lohrmann et al. Phys. Lett., 13, 78 (1964). 12. G. Baroni et al. Nuovo Cimento, 38, 95 (1965). 13. V.S. Barashenkov. Fortschr. der Physik, 10, 205 (1962). 14. P. Söding. Phys. Lett., 8, 285 (1964). 15. I.I. Levintov. Phys. Lett., 7, 41 (1963). 16. I.I. Levintov, G.M. Adelson-Velsky. Phys. Lett., 13, 185 (1964). 17. D.V. Bubb et al. Phys. Rev., 146, 980 (1966). 18. V.S. Barashenkov, V.D. Toneev. JINR Preprint, P2-3850, Dubna 1968. 19. G. Bialkowski, S. Pokorski. Nuovo Cimento, 61, 225 (1969). 20. P. Pascual and F.J. Yndurain. Nuovo Cimento, 61, 225 (1969). 21. J.V. Allaby et al. Phys. Lett., 30B, 500 (1969). 22. O.V. Dumbrajs, T. Yu. Dumbrajs and N.M. Queen. Preprint JINR, E2-5216, Dubna, 1970. - 10 -23. V.S. Barashenkov. Cross-sections of elementary particle inter­actions, Moscow, 1966. 24. Y. Sumi and T. Yoshida. Suppl. Pr. Theor. Phys., 41; 42, 53 (1967). 25. E.M. Henley. Isospin in Nuclear Physics, ed. Wilkinson (North-Holland. Amsterdam, 1969), p.15. 26. U. Amaldi jr., et al. Nuovo Cimento, 46A, 171 (1966). - 11 -Table 1 Values for dpp ≡ (Re ft/Im ft) obtained by the interference method P l a b (GeV/c) dPP ± ∆dPP Reference P l a b (GeV/c) dPP ± ∆dPP Reference 0,44 1,85 ± 0,20 3 x ) 9,94 -0,302 ± 0,071 7 0,54 1,55 ± 0,12 3 x ) 10,00 -0,330 ± 0,035 10 0,56 1,36 ± 0,15 3 x ) 10,11 -0,430 ± 0,043 10 0,85 0,71 ± 0,16 3 x ) 10,89 -0,26 ± 0,05 5 0,93 0,62 ± 0,15 3 x ) 11,94 -0,290 ± 0,033 7 1,29 - 0,76 ± 0,13 4 12,14 -0,258 ± 0,067 7 1,39 - 0,58 ± 0,06 4 14,03 -0,272 ± 0,033 7 1,54 - 0,32 ± 0,07 4 17,82 -0,307 ± 0,065 7 1,69 0,10 ± 0,16 4 19,22 -0,210 ± 0,024 9 1,70 - 0,007 ± 0,070 3 19,33 -0,330 ± 0,033 10 2,78 - 0,12 ± 0,07 5 20,24 -0,205 ± 0,033 7 3,83 - 0,24 ± 0,08 6 24,00 -0,19 ± 0,09 11 4,85 - 0,38 ± 0,10 5 24,12 -0,157 ± 0,038 7 6,87 - 0.30 ± 0,07 5 26,12 -0,154 ± 0,045 7 7,81 - 0,331 ± 0,034 7 26,42 -0,320- ± 0,033 10 7,85 - 0,25 ± 0,03 8 27,50 -0,23 ± 0,13 12 7,92 - 0,247 ± 0,082 7 39,93 =0,187 ± 0,024 9 8,89 - 0,33 ± 0,08 5 50,93 -0,137 ± 0,030 9 9,39 - 0,319 ± 0,040 9 56,93 -0,135 ± 0,040 9 9,86 - 0,345 ± 0,038 7 70,93 -0,119 ± 0,017 9 x) value dPP obtained from phase-shift analysis. - 12 -Table 2 Calculated values for dpp P l a b (GeV/c) d P P P l a b (GeV/c) d P P 1,01 4,0 -0, 0,6 1,41 5,0 -0,24 0,7 1,13 6,0 -0,26 0.8 0,85 7,0 -0,28 0,9 0,67 8,0 -0,30 1,0 0,55 9,0 -0,31 1,1 0,42 10,0 -0,32 1,2 0,30 1 1 , 0 -0,31 1,3 0,18 12,0 -0,31 1,4 0,10 13,0 -0,29 1,5 0,08 1 4 , 0 -0,28 1,5 0.08 15,0 -0,25 1,7 -0,05 16,0 -0,25 1,8 -0,06 17,0 -0,24 1,9 -0,07 18,0 -0,23 2,0 -0,09 19,0 —0,21 2,1 -0,10 20,0 -0,21 2,2 -0,11 21,0 -0,20 2,3 -0,13 22,0 -0,20 2,4 -0,14 23,0 -0,10 2,5 -0,15 24,0 -0,19 2,6 -0,17 25,0 -0,19 2,7 -0,18 30,0 -0,19 2,8 -0,19 35,0 -0,19 2,9 -0,19 39,0 -0,18 3,0 -0,20 Fig. 1 Experimental values of app. The curve represents the theoretical values for app obtained in this report. Fig. 2 Analytic structure of amplitude in a complex energy ω-plane.The black dot and strip signify the pole and unphysical cut respectively. The hatched areas corres­pond to the physical regions. Fig. 3 Structure of the Δ function in the ξ-plane. Fig. 4 Structure of the Δ function in the ξ-plane. 

Model-independent determination of the real parts of the pp forward scattering amplitudes not allowing for spin effect

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