1 New approaches to the determination of the total cross section O.V. Selyugin a∗ aBLTPh, JINR, 141980Dubna, Russia New methods have been developed for extracting the parameters of diffraction scattering amplitude, σtot and 1 ρ=ReFN/ImFN,from experimentaldata. The latter determinesthese parameters with less errors and without 0 the knowledge of the normalization coefficient. The impact of additional information from the measurement of 0 thespin-dependencecross section on thedetermination of thebasic characteristic of elastic scattering amplitude 2 isexamined. Thenewformoftheconnectionbetweensomecharacteristicsoftheanalyzingpowerandmagnitude n of thepp− total cross section is presented. a J 9 1 1. Introduction contributiontoσtot ofσobs,thedirectlymeasured v value, and of ∆σ and ∆σ , the extrapolated el inel 1 In reality, in experiment one measure dN/dt, contributionsoftheelasticandinelasticcrosssec- 7 as a result of which ”experimental” data such as tions, are shown at energies √s = 30.6,52.8 and 0 σ , slope-B, ρ are extracted from dN/dt with tot 62.7 GeV. One can see that the growth of the 1 some model asumptions. Some approaches are 0 total cross sections is due to ∆σel by 50% for pp needed to extrapolate the measured quantities 1 and nearly by 100% for pp¯scattering. 0 to t = 0. The elastic scattering data were con- The analysisofexperimentaldataalsoshowsa / strainedbyseveralconditions: theimaginarypart h possible manifestation of spin-flip amplitudes at ofthenuclearamplitude hasanexponentialform p high energies [2]. The research into spin effects - inthesmalltregion;therealandimaginaryparts willbeacrucialstonefordifferentmodelsandwill p of the nuclear amplitude have the same t depen- e help us to understand the interaction and struc- dence; spin contributions are neglected. Note h tureofparticles,especiallyatlargedistances. All : thatthevalueofρheavilycorrelatswiththenor- thisraisesthequestionaboutthemeasureofspin v malization of dσ/dt. Its magnitude weakly im- i effectsintheelastichadronscatteringatsmallan- X pacts the determination of σ only in the case tot gles at future accelerators. Especially, we would r when the normalization is known exactly. like to note the programsatRHIC where the po- a There is no any experiment on measurement larization of both the collider beams will be con- of the values separately. But in some experi- structed. ments, to reduce experimental errors, the mag- To obtainthe magnitude ofρ, we fit the differ- nitude of some quantities is taken from another entialcrosssectionseithertakingintoaccountthe experiment. Sometimes, it leads to contradic- value of σ from another experiment, as made tot tion between the basic parameters for one en- by the UA4/2 Collaboration, or taking σ as tot ergy (for example, if we calculate the imaginary a free parameter. If one does not take the nor- partofscatteringamplitude,wecanobtainanon- malization coefficient as a free parameter in the exponentialbehavior). It canleadto some errors fitting procedure, his determination requires the in the analysis based on the dispersion relations. knowledge of the behavior of the imaginary and The procedure of extrapolation of the imagi- realpartsofthescatteringamplitudeintherange narypartofthescatteringamplitudeissignificant ofsmalltransfermomentaandthemagnitudesof for determining σ . The importance of the ex- tot σ and ρ. tot trapolatedcontributionisseenfrom[1]wherethe ∗e-mail: [email protected] 2 2. The method of changing the sign and try to find the derivate of the calculated ρ on t (dρ/dt) with a small deviation of n (δn) [3]. The differential cross sections measured in the Write for that case our equation for ρ for two experiment are described by the squaredscatter- points: t1,where Fc =2ImFh,andfort2,where ing amplitude. From the equation for the differ- − Fc = ImFh; and calculate ∆ρ = ρ(t2) ρ(t1) ential cross section one can obtain the equation − − for normalization n+δn with δn << 1. So, for for ρ for every experimental point - t [3] i example, for t2 we can input in (1) 1 ρ(s,t )= ReF (s,t )+ 1dσ i ImF (s,t ) { c i (n+δn)=2ImF2(1 ρ+ρ2)(1+δn) (3) N i π dt h − 1dσ [ in (ImF (s,t )+ImF (s,t ))2]1/2 . (1) π dt − c i N i } and obtain, if we take σt−prob = σtr, that the difference between the calculated ρ will be i As the imaginary part of scattering amplitude is definedbyImF (s,t)=Hexp(B/2t)whereH = N ∆ρ(t2,t1) δn/4 (4) σ /(4π 0.389)itisevidentfrom(1)thatthereal ≃ tot ∗ part depends on n,σtot,B. and is a function of t. It is clear that if we find For the proton-proton scattering this equation the true magnitude of the total cross section and has a remarkable property. If we expand the ex- δn=0,then∆ρ(t2,t1)willbe equaltozero. The pression under the radical sign, we obtain same calculations are carried out for δσ with tr 2 2 δn = 0 and show that the sign of ∆ρ(t2,t1) de- (n 1)(ImF +ImF ) +n(ReF +ReF ) .(2) − c N c N pends on the sign of ∆σtot. Then, if we will cal- AstherealpartoftheCoulombscatteringampli- culate ∆ρ(t2,t1) with differences δn and σtot we can determine the true magnitude of σ . tude is negative and the real part of the nucleon tot scattering amplitude is positive, it is clear that this expression will have a minimum situated on 4. Connection between A and σ N tot the scale of t independent of n and σ . As we tot Letusexaminethefuturepp2ppExperimentat know the Coulomb amplitude, we estimate the √s = 500 GeV, as an example. Elastic differen- real part of the proton-proton scattering ampli- tial cross section will be regarded as having 2% tude at this point. Note that all other methods statisticalerrors,accordingtotheProposalofthe give us the real part only in a sufficiently wide PP2PP Collaboration. Now,infittingprocedure interval of the transfer momenta. wetakeintoaccountthestandardassumptionfor Thismethod worksonlyinthe caseofthe pos- highenergyelastichadronscatteringatsmallan- itive real part of the nucleon amplitude and it is gles: the simple exponential behavior with slope especially good in the case of large ρ. So, it is B of the imaginary and real parts of the scatter- interesting for the future experiment at RHIC. ing amplitude; hadron spin-flip amplitude does notexceed10%ofthehadronspin-non-flipampli- 3. Thedifferentialmethod(”tailofghost”) tude. Thedifferentialcrosssectionwascalculated Let us examine the simplest gedanken case of usingσ =63.5mb; ρ=0.15; B =15.5GeV−2 tot pp-scattering and try to determine the sign of ( in variant I with 150 points and in variant II dρ(t)/dt depending on the difference of normal- with 75 points) from t0 = 0.00075 with ∆t = ization n and n+δn and δσ =σtrue σtot−prob. 0.00025 GeV2) and then put through a special − There, σ is the magnitude of the true total random process using 2% errors. After that, the true cross section and σtot−prob is the magnitude of obtained ”experimental” data were fitted. The the total cross section which we take as our first systematic errors were taken into account as the approximation. freeparametern. TheresultispresentedinTable In first, let us suppose that in experiment we 1 for three (n = 1.- fixed) or four parameters (n find true normalization of data dN/dt (so n=1) - free). 3 Table 1. Fit of dσ/dt measurements resultsareshowninTable3. Theaddedpolariza- N σtot mb δB δρ δn tion data decrease the error in σtot only by 10% I 63.54 0.12 0.2 0.008 fix (from1.25mbto 1.1mb). Butthe determination ± ± ± I 63.6 1.25 0.3 0.02 0.04 of the magnitude of the realand imaginary parts ± ± ± ± II 63.5 0.25 0.7 0.01 fix of the hadron spin-flip amplitude become three ± ± ± II 64.05 1.4 1.0 0.03 0.05 time better. It is to be noted that the variant ± ± ± ± C3 leads to decrease of in the error of σtot from It is clearthat the most importantvalue is the 1.25 mb to 0.9 mb. coefficient normalization of the differential cros ± ± section. Its small errors lead to significant errors Table 2. Fit of A N in the σ . So, we see that the normalization of experimteonttaldataisthemostimportantproblem N σtot mb δρ δk1 δk2 A1 63.5 3.4 0.08 0.07 0.05 for the determination of σtot. A2 63.46± 3.8 ±0.15 ±0.06 ±0.1 Lacking better knowledge, we assume that the ± ± ± ± B1 63.5 3.8 0.09 0.07 0.11 hadron spin-flip amplitude is a slowly varying ± ± ± ± B2 62.7 4. 0.3 6.3 5.6 function of t apart the kinetic factor, and we ± ± ± ± C1 63.4 3.6 0.09 0.07 0.11 parametrize it as ± ± ± ± C2 63.5 -fix fix 0.015 0.011 ± ± pt C3 63.9 1.83 0.05 0.037 0.035 φh5 = | |(ρk2+ik1)Imφh1, (5) ± ± ± ± m where ρ, k1, k2 are slowly changing functions of Table 3. Fit of dσ/dt and A (300 points) s. The coefficients k1 and k2 are the ratios of the N real and imaginary parts of the spin-flip to spin- N σtot mb δρ δk1 δk2 non-flip amplitudes without the kinematic factor A 63.4 1.1 0.02 0.02 0.02 ± ± ± ± pt. As a result, the AN can be written as C 63.4 1.1 0.02 0.02 0.03 | | ± ± ± ± AN dσ = Imφh1 α (µ−1 k1) C3 63.2±0.9 ±0.015 ±0.014 ±0.02 − 8πPB dt − mpt 2 − | | 5. Connectionbetween t ofA and σ +p|t|ρ[Imφh1]2∆k, (6) max N tot m As notedabove,mostuncertaintyinthe deter- with∆k =k2 k1. We examinedalsoafew vari- mination of σtot using the measurement of AN − ants with different assumptions about the mag- came from the error in the beam polarization nitudes of the imaginary and real parts of the which plays the role of a normalization factor of hadron spin-flip amplitude at √s = 500 GeV. the differential cross section. The point of max- Variant Ai: φh5 = 0., so k1 = k2 = 0 . Variant imum of AN is independent of the magnitude of Bi: k1 = 0.1, k2 = 0.15. Variant Ci: k1 = 0.1, thebeampolarization. So,itallowsustousethis k2 = 0.15. Variant C3 was made by a random value for the extraction of the magnitude of σtot. − procedurewitherrorstwice smaller,sothe errors Here is the formula [6] (the case B1) equal 5% 10% (see, Table 2). The ”ex÷periment” with 10% 20% errors de- σ = 9.776α[√3 8 (ρI R)] (ρ αϕ)], tot ÷ t − µ 1 − − − termine the magnitudes of real and imaginary max − parts of the hadron spin flip almost with 100% where I = k1 and R = ρk2, and these coeffi- errors. The variant C3, which reflects the exper- cients are unknown. Its determination will de- iment with 5% 10% errors, twice decreases the pend on the magnitude of beam polarization. To ÷ errors of the magnitude of σ and hadron spin- reduce the impact of the hadron spin-flip ampli- tot flip. tude, it was proposed to used the new value - Let us make the fit of both data on the differ- tmax2, the place of the maximum of dσ/dtA2N [4] entialcrosssectionandtheanalyzingpower. The . The derivation of this value gives (the case A1 4 ) Table 4. σ as function of t tot max 2 σtot =8π 0.39(mb/GeV ) α/tmax2. (7) Input ρ=0.1,B =13., σtot =43.0 mb Inthecaseoftheexponentialbehaviorofthescat- k1,k2 B1 A1tm2 C1tm C2tm 0., 0. 42.0 43.2 43.3 43.3 tering amplitude, one obtains (the case A2 ) 0.1, 0.2 41.9 42.1 42.2 43.3 σtot =9.776α[1/tmax2 B/2], (8) 0.2, 0.2 42.0 43.2 43.3 43.3 − 0.0, 0.2 42.0 41.0 42.1 43.4 where B is the slope of the differential cross sec- ρ=0.075 tion. In our previous work we showed that it 0.0, 0.2 41.9 41.4 41.7 43.5 could be obtained from the measurement of A N of some ratio of the real and imaginary part of the hadron spin-flip amplitude which is indepen- Table 5. σtot as function of tmax dent of the magnitude of the beam polarization Input ρ=0.15,B =15.5, σ =63.5 mb tot ρ(k2−k1)/(1−k1). Using this ratio, we can ob- k1,k2 B1 A1tm2 C1tm C2tm tain the relation between σ and the point of tot 0., 0. 62.1 63.5 63.3 63.5 maximum A in the form (the case C2) N 0.2, 0.2 62.3 63.5 61.9 63.5 1 p3+4ρ2 0.0, 0.1 62.8 61.8 61.5 63.7 σ =9.776α( +B)[ tot t 1+ρ2 0.1, 0.2 62.0 61.3 60.7 63.3 max 0.0, 0.2 62.2 59.3 58.5 63.7 8ρ∆k (ρ αϕ)]. (9) -0.1, -0.2 61.0 64.9 64.9 63.1 (µ 1)(1+2ρ)(1 k1) − − − − ρ=0.1 The calculationof σtot by using these formulaeis 0.2, 0.2 62.0 63.5 63.6 63.6 shown in Table 4 for √s=52 GeV and in Table ρ=0.0 5 for √s = 540 GeV for different variants of the 0.2, 0.2 61.9 63.2 63.8 63.7 magnitudeofthehadronspin-flipamplitude. For comparison, the variant C1 is show without tak- Acknowledgments. I would like to express my ingintoaccountofthecontributionofthehadron sincerely thanks to the Organizing Committee spin-flip amplitude (the case C1) and especially to R. Fiore and A. Papa for the kind invitation and the financial support at such σ = tot remarkable Conference, and W. Gurin, B. Nico- 1 p3+4ρ2 9.776α( +B)[ (ρ αϕ)]. (10) lescu and E. Predazzi for fruitful discussions. t 1+ρ2 − − max REFERENCES 6. Conclusions 1. G. Carbonny, Nucl. Phys. B 254 (1985) 697. Themagnitudesofσ ,ρandslope-Bhaveto tot 2. O.V. Selyugin, Phys. Atom. Nuc. 62 (1999) be determined in one experiment and their mag- 333. nitudes depend on each other. The normaliza- 3. O.V. Selyugin, in Proc. ”Frontiers in Strong tionsofdN/dtandA aremostimportantforthe N Interactions”, ed. P. Chappetta, M. Hague- determination of these values. The new methods nauer, J. Tranh Van, Blois (1995) 87. of extracting magnitudes of these quantities are 4. 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