Measurement of proton-proton elastic scattering into the Coulomb region at P beam = 2.5, 2.8 and 3.2 GeV/c
H. Xu, Y. Zhou, U. Bechstedt, J. Böker, A. Gillitzer, F. Goldenbaum, D. Grzonka, Q. Hu, A. Khoukaz, F. Klehr, B. Lorentz, D. Prasuhn, J. Ritman, S. Schadmand, T. Sefzick, T. Stockmanns, I.I. Strakovsky, A. Täschner, C. Wilkin, R.L. Workman, P. Wüstner
MMeasurement of proton-proton elastic scattering into the Coulomb region at P beam =2.5, 2.8 and 3.2 GeV/c
H. Xu a, ∗ , Y. Zhou a , U. Bechstedt a , J. B¨oker a , A. Gillitzer a , F. Goldenbaum a , D. Grzonka a , Q. Hu a,1 , A. Khoukaz b , F.Klehr c , B. Lorentz a,2 , D. Prasuhn a , J. Ritman a,d , S. Schadmand a , T. Sefzick a , T. Stockmanns a , I.I. Strakovsky e , A.T¨aschner b,2 , C. Wilkin f , R.L. Workman e , P. W¨ustner c a Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany b Institut f¨ur Kernphysik, Universit¨at M¨unster, 48149 M¨unster, Germany c Zentralinstitut f¨ur Engineering, Elektronik und Analytik, Forschungszentrum J¨ulich, J¨ulich, 52425, Germany d Ruhr-Universit¨at Bochum, Bochum, 44780, Germany e Data Analysis Center at the Institute for Nuclear Studies, Department of Physics, The George Washington University, Washington, D.C.20052, USA f Physics and Astronomy Department, UCL, London WC1E 6BT, UK
Abstract
The proton–proton elastic differential cross section at very small four momentum transfer squared has been measuredat three different incident proton momenta in the range of 2.5 to 3.2 GeV/c by detecting the recoil proton at polarangles close to 90 ◦ . The measurement was performed at COSY with the KOALA detector covering the Coulomb–nuclear interference region. The total cross section σ tot , which has been determined precisely, is consistent with previousmeasurements. The values of the slope parameter B and the relative real amplitude ratio ρ determined in this experimentalleviate the lack of data in the relevant energy region. This precise data on ρ might be an important check for a newdispersion analysis. Keywords:
Proton–proton elastic scattering, Coulomb-nuclear interference, total cross section
PACS:
Imf n (0) = k cm σ tot / π , where k cm is the c.m. mo-mentum of the incident particle. Forward dispersion re-lations predict the real part of the forward elastic scat-tering amplitude, Ref n (0), as well as ρ , which is definedas ρ ≡ Ref n (0) /Imf n (0). Therefore, the measurement ofproton–proton elastic scattering enables a way to deter-mine the related parameters, i.e. σ tot and ρ . This methodwas employed by many experiments, e.g. [1, 2]. However,it is difficult to extract the value of ρ accurately and theabsolute magnitude of the differential cross section can notdetermine the sign of the real part of the nuclear ampli-tude.Fortunately, the proton–proton elastic scattering is al-ways accompanied by Coulomb scattering. The interfer- ∗ Corresponding author
Email address: [email protected] (H. Xu) Present address: Institute of Modern Physics, Chinese Academyof Sciences, Lanzhou, 730000, China Present address: GSI Helmholtzzentrum f¨ur Schwerionen-forschung GmbH, Darmstadt, 64291, Germany ence between the Coulomb and the real part of the the nu-clear amplitude at small 4–momentum transfer squared, t ,allows to determine both the sign and the value of ρ . Theparameter ρ is most sensitive in the Coulomb–Nuclear In-terference (CNI) region. The determination of ρ in theCNI region provides the best chance to gain knowledgeabout it.Traditionally, the proton–proton elastic scattering isdescribed in terms of the Coulomb and nuclear amplitudes, f c and f n . As summarized in [3], at small t one obtainsd σ d t = (cid:12)(cid:12)(cid:12) f c ( t )e iαφ ( t ) + f n ( t ) (cid:12)(cid:12)(cid:12) = d σ c d t + d σ int d t + d σ n d t , (1)where d σ c d t = 4 πα G ( t )( (cid:126) c ) β t , (2)d σ int d t = − ασ tot β | t | G ( t )e − B | t | ( ρ cos( αφ ( t )) + sin( αφ ( t ))) , (3)and d σ n d t = σ (1 + ρ )e − B | t | π ( (cid:126) c ) . (4)Here, α is the fine structure constant. G ( t ) is the pro-ton dipole form factor conventionally taken as G ( t ) = Preprint submitted to Physics Letter B October 1, 2020 a r X i v : . [ nu c l - e x ] S e p − , with ∆ ≡ | t | / .
71 (GeV/c) . φ ( t ) is theCoulomb phase, that we take in the form [4, 5], φ ( t ) = − ln (cid:18) B | t | (cid:19) − γ, (5)where Euler’s constant is γ ≈ . σ tot and B and ρ involvedin the parameterization of the nuclear and the interferencecross section need to be determined by experiment. ] |t| [(GeV/c) ] / d t [ m b / ( G e V / c ) s d = 3.2 GeV/c beam P 0.4) - = r /dt ( s d 0.3) - = r /dt ( s dCoulombNuclear - = r - = r c s n s Figure 1: Calculated partial and total differential cross section ofproton–proton elastic scattering in a wide t range covering the CNIregion at 3.2 GeV/c based upon the Eqs. (1)–(4) with the values of σ tot and B taken from the world data presented below. The totaldifferential cross section for ρ = − . ρ = − . It is noted that an early measurement found the con-tribution of the spin–spin amplitude to the forward elasticscattering [6]. Further measurements show that the realparts of the spin–spin amplitude decrease rapidly with in-creasing beam energy. There is some evidence that thespin-spin amplitude might become negligible already forbeam momentum at 1.7 GeV/c [7] for the nuclear scat-tering contribution to the cross section. Therefore, thecontribution of the spin–spin amplitude to the nuclear dif-ferential cross section was neglected in this analysis.It is necessary to measure the t dependence of the elas-tic scattering differential cross section over a wide range of t in order to resolve the strong correlation between the fitparameters when they are only measured within a narrowrange of t [8]. Figure 1 indicates the calculated partialand total differential cross section of proton–proton elas-tic scattering at a beam momentum of 3.2 GeV/c with theparameterization given above. The Coulomb cross sec-tion dominates at small t . The slope parameter B maybe determined from the t dependence far above the CNIregion. The parameter ρ can be determined by analysingthe t –distribution in the CNI region, i.e. the region whered σ c / d t ≈ d σ n / d t . At a beam momentum of 3.2 GeV/c ° ValveAlphasourceTarget chamber X Recoil detectorCooling finger
Beam c m Si
BeamA)B)
Z Z H cluster beam Figure 2: A schematic view of the recoil detector system is shown inA). The sensor layout is shown in B), as seen from the interactionregion. this corresponds to | t | ≈ . . A measure-ment to even lower t , where the cross section is dominatedby Coulomb scattering, would enable us to determine theluminosity as well as the total cross section by analyz-ing the characteristic shape of the t –distribution over awide range. One goal of KOALA is to measure the | t | range of 0 . − . , which covers the CNI re-gion. This so called Coulomb normalization method wasalso pursued by high energy experiments, such as UA4 [9]and ATLAS [10].In this report, we present precision measurements ofthe proton–proton elastic differential cross section at verysmall t covering the CNI region at three beam momenta of2.5, 2.8 and 3.2 GeV/c at COSY [11]. The measurementswere made by detecting the recoil protons at polar anglesclose to 90 ◦ .Based on the detection of the recoil proton, the value of t is proportional to the laboratory kinetic energy T p of therecoil proton and is related to the recoil angle α = 90 ◦ − θ , − t = 2 m p T p = 4 m sin α (1 /β ) − sin α , (6)2here 1 /β = ( E beam + m p ) / ( E beam − m p ), and E beam and m p are the total beam energy and the proton mass,respectively. The recoil measurement technique, which hasbeen used before [12, 13], has several advantages comparedto measuring the scattered protons at small forward an-gles. It is of great importance that the recoil measurementcan achieve a larger acceptance than the forward measure-ment due to the limit of either the beam pipe or the beamemittance. Secondly, it is much easier to distinguish therecoil protons from the non-interacting protons. Thirdly,recoil protons have relatively small energies and their ki-netic energy can be precisely detected in solid state detec-tors. For instance for | t | = 0 . , T p = 54 MeV,recoil protons have a range of < <
14 mm in silicon.The measurements have been made with the recoil setupof the KOALA experiment [8], which can measure proton–proton elastic scattering in the angular range up to α =13 . ◦ for beam momenta between 2.5 to 3.2 GeV/c. TheKOALA experiment was proposed to measure antiproton–proton elastic scattering at HESR [14]. The KOALA recoildetector system was commissioned at COSY by measuringproton–proton elastic scattering since the recoil particleand the kinematics are identical for both reactions.The top part of Figure 2 schematically shows the setupused to measure the recoil protons at COSY. The circu-lating proton beam in the COSY ring intersects an in-ternal H cluster beam target (typical areal density of10 protons/cm ) to provide a luminosity of about 1 × cm − s − . The interaction region consists of the inter-section of a 5 mm diameter cylindrical proton beam withan oval shaped hydrogen gas jet with a lateral width of10 mm and a thickness of 1–2 mm along the proton beamdirection [15]. A sketch of the sensor layout is shown inthe bottom of Figure 2. The detector system has been de-scribed elsewhere [8]. As used at COSY, the KOALA recoildetector included two 76 . ×
50 mm × α = − . ◦ to 13 . ◦ . A windowless alpha source withthree isotopes could be inserted at about 38 cm abovethe detector surface to allow dedicated energy calibrationmeasurements.With the recoil technique the experimental goals con-sist of precisely measuring the energies of the recoil pro-tons and their relative yield at different recoil angles. Inthe present experiment the recoil angles typically rangedfrom α = 0 ◦ to 13.6 ◦ corresponding to | t | = 0 to | t | =0 . at P beam = 3 . T p = 0 to 54 MeV. An alpha source hasbeen used to measure the energy resolution of the siliconand germanium detectors, and was determined to be better C o un t s = 3.2 GeV/c beam P (Scaled background) (cid:176) - = a ) (|t|=0.0015 (GeV/c) (cid:176) = 1.56 a ) (|t|=0.0022 (GeV/c) (cid:176) = 1.90 a ) (|t|=0.0030 (GeV/c) (cid:176) = 2.24 a ) (|t|=0.0040 (GeV/c) (cid:176) = 2.58 a Figure 3: The energy distribution of recoil protons at small recoilangles shows the elastic scattering peak above the scaled pure back-ground measured in the unphysical region (black curve). than 20 keV and 30 keV (FWHM), respectively. The en-ergy calibration of the detectors was done when no beamwas circulating. The detector worked stably during theexperiment. The energy calibration measurement for eachsingle strip was reproducible with an uncertainty smallerthan 0.05%.Prior to the further analysis, an energy clustering al-gorithm has been implemented in order to reconstructthe proton energy deposited in more than one detectorstrip. An energy cluster consists of all relevant neigh-bouring strips, in which the deposited energy is above astrip–dependent threshold. The total energy of each eventhas been reconstructed based on the energy of the cluster.The overall precision ( δE/E ) of the reconstructed energyis better than 0.3%.Two challenges that have to be faced for gaining highprecision of the measurements are the background sub-traction in the recoil proton spectra as well as the ac-curate determination of the mean value of t . For | t | (cid:62) , the background influence is very smallwith a signal to background ratio above 100. For | t | < the recoil energy is less than 2 MeV andthe recoil peak sits on a background shown by the blackdistribution in Figure 3. The background drops quicklyand becomes negligible above around 2 MeV.By design, there are 20 strips of the first silicon sen-sor located in the unphysical region with the recoil angle α = 0 ◦ to − . ◦ , where there are no elastic events. As de-picted in Figure 3, the two main components of the back-ground consist of the secondary radiation from the sur-rounding material and minimum ionizing particles (MIP).The MIP background had a higher rate than the elasticevents and formed a peak at around 380 keV, consistentwith the energy loss of a MIP passing through 1 mm ofsilicon. It is found that the background distribution onstrips covering the unphysical recoil angle were similar toeach other and the yield in the MIP peak varied linearly3 - - ] |t| [(GeV/c) · ] - d N / d t [( G e V / c ) / ndf c tot s – – r – - Lumi 4.084e+06 – c tot s – – r – - Lumi 4.084e+06 – - - ] |t| [(GeV/c) R a t i o Data/Fit
Figure 4: The upper part shows the d N/ d t distribution at 2.8 GeV/cand the corresponding fit parameters. The ratio of the measured datato the fit is shown in the lower part. with the solid angle of the strip. This is also true forthe strips covering the physical recoil angle. Therefore,the background measured by those 20 strips were summedand subtracted from the single strip yield using a nor-malization based upon the yield in the MIP peak. Thescaling factor for those strips where the MIP peak par-tially overlapped the elastic scattering was determined bylinearly extrapolating the scaling factors along the solidangle, which was observed from the direct normalizationof the peaks without overlap. The scaled data backgroundwas then subtracted from the energy spectrum for eachstrip. The remaining peak is taken to be the real recoilprotons of elastic scattering and the yield of recoil protonsis numerically integrated. The error of the yield was de-termined based on the pure statistical error as well as thebackground error, added in quadrature.The determination of the mean value of the momen-tum transfer, | t | , is equivalent to the determination of themean value of the recoil energy T p , as indicated in Eq. (6).The t –values were calculated based on the mean energymeasured by each strip. As a consequence of the highgranularity of the detectors, the energy spectrum for asingle strip could be well described by a normal distribu-tion. The mean value of the energy was then determinedby a Gaussian fit. Due to the background subtraction, thedetermined t values for α < . ◦ have larger uncertaintiesthan others. Therefore, t was determined for those stripsfrom the recoil angle based on the geometry of the sensor,i.e. the center of the strip, instead of determining it byusing the mean energy. For the other strips, i.e. α (cid:62) . ◦ ,the t values were accurately determined by the mean en-ergy. The uncertainty of the mean energy determinationwas added in quadrature with the energy resolution for ] |t| [(GeV/c) ] / d t [ m b / ( G e V / c ) s d ) [KOALA] · · · Figure 5: The differential cross section distributions measured byKOALA together with a previous measurement of ANL [16]. each strip and taken to be the total error of the deter-mined t .In parallel to the mean energy determination the detec-tor alignment was implemented. The chamber alignmentmeasurements were done by the COSY crew based uponseveral permanent benchmark positions in the accelera-tor tunnel. Except for the finite mechanical installationprecision, the hydrogen cluster jet target has also intro-duced small off–center effects. A software based detectoralignment has been performed by comparing the measuredproton energy, i.e. the recoil angle, to the expected an-gle related to the center of each single strip. Based onthe alignment, a shift of 1–2 mm relative to the surveyhas been implemented in order for both determinations tomatch. This is consistent with the results of the positionmeasurement of the target profile and the detector cham-ber. With the implementation of the detector alignment,the solid angle for each strip was calculated based on thestrip pitch and the distance between the detector surfaceand the interaction volume. The overall uncertainty of thesolid angle determination is smaller than 0 . N /d t distribution was then reconstructed on thebasis of the determined t , the yield N of elastic events aswell as the solid angle dΩ of each strip, i.e.d N d t = πk d N dΩ . (7)Since the measurements extended into the Coulomb re-gion, i.e. | t | < , all three parameters σ tot , B and ρ as well as the luminosity, L , can be determined fromthese data. The reconstructed d N/ d t distribution was fitby the differential cross section formulas in Eqs. (1)–(4)multiplied by L , i.e. d N d t = L d σ d t . (8)Figure 4 shows the d N/ d t distribution at 2.8 GeV/cwith a fit marked in red that used Eq. (8). The lower4
10 20 30 40 50
P [GeV/c] [ m b ] t o t s / ndf c – – – - / ndf c – – – - World dataKOALA n =A+B*P tot s Figure 6: The total cross section for this measurement is shown bythe red squares in comparison with the world data in blue takenfrom [17]. The inset displays these results for a narrower range ofbeam momentum. frame of Figure 4 shows the ratio of the measured data tothe fit result, indicating that the fit very well describes thedata.The differential cross section distributions for all threemomenta, as well as a different data sample measured closeto our t and P beam range, have been plotted in Figure 5.The differential cross sections at 2.5, 2.8 and 3.2 GeV/c arepresented as solid circles with different colors. Previouslymeasured data at 3 GeV/c [16] in a similar t range are alsoshown with open squares. For display purposes factors of1.3, 1.3 and 1.3 have been applied to the data points at3.0, 2.8 and 2.5 GeV/c, respectively. In contrast to theANL data the strong rise for | t | < due tothe Coulomb interaction is clearly visible in the KOALAdata. Table 1: Forward scattering parameters determined for proton–proton elastic scattering. The first error term is the statistical errorand the second is the systematic error. P lab σ tot B ρ χ /ndfGeV/c mb (GeV/c) − ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± There are two main sources of systematic error in thedetermination of the best fit parameters. The first sourceis related to the background subtraction. As describedabove, the MIP distribution from the strips with α < ◦ was scaled and then subtracted from the distributionswith α > ◦ . The background subtraction introduces anuncertainty of the yield for each individual strip that issmaller than 0 . t , determination.The uncertainty introduced to the t determination caused P [GeV/c] ] - B [( G e V / c ) World dataKOALA
Figure 7: The slope B determined by KOALA in red, together withprevious measurements in blue [7, 18, 20], for which the same pa-rameterization was applied. The inset displays these results for anarrower range of beam momentum. by the background subtraction is smaller than 0 . ± σ , ± . σ and ± σ related to the energy peak.The combination of those two sources results in six inde-pendent fits for the parameter determination. The RMS ofthe determined parameters from the six analyses is takento be the systematic error. These values are presented asthe second error term for each parameter in Table 1.The elastic scattering parameter with the most exist-ing world data is the total cross section. These resultshave been measured by different experiments with variousmethods, e.g. transmission measurement. Very few re-sults were measured by using the Coulomb normalizationmethod discussed here. Figure 6 shows the total cross sec-tion measured by KOALA together with a compilation ofthe world data taken from [17]. The KOALA results arein very good agreement with existing data.The slope parameter B determined by KOALA is basedon Eq. (4), which assumes the nuclear differential crosssection is described by a pure exponential function. Thenuclear elastic differential cross section in the high t regionis often described asd σ n d t = d σ n d t (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( Bt + Ct ) . (9)The value of the nuclear elastic differential cross sectionextrapolated down to t = 0, i.e. d σ n / d t | t =0 , is the socalled optical point.In [18, 19] it is shown that for | t | < it ispossible to take C =0. These KOALA measurements have5 P [GeV/c] - - - r World dataKOALA - - - - - - - - Figure 8: The ρ parameter measured by KOALA in red in compari-son to previous experiments [17]. The inset displays these results fora narrower range of beam momentum. achieved a wide t -range, in which the maximum | t | is about0.1 (GeV/c) , below the cited threshold to change the pa-rameterization. Figure 7 shows the B values measuredby KOALA together with the results from other experi-ments [7, 18, 20], which used the same parameterizationfor B . The good consistency between the KOALA dataand the existing data encourages a more detailed study ofhow the slope parameter B behaves with the beam energy. Table 2: Proton–proton elastic scattering parameters when σ tot isheld fixed. The statistical error of the parameters is listed. P lab σ tot fixed B ρ χ /ndfGeV/c mb (GeV/c) − ± ± ± ± ± ± Benefitting from the excellent detector performance,the lowest values of measured | t | were 0.00083, 0.00085 and0.0009 (GeV/c) at 2.5, 2.8 and 3.2 GeV/c, respectively.Consequently both the sign and value of the parameter ρ were able to be determined. The results of ρ measuredby KOALA are plotted together with the world data inFigure 8. There are very few data sets of ρ [17] to whichthe KOALA results can be compared in the relevant energyregion.One challenge while determining the parameter ρ wasoften the correlation between σ tot and ρ in the narrow t range measured. For a test in this analysis the total crosssection was also fixed during the fit in order to check theuncertainty of the determined values of ρ . As shown in Fig-ure 6, the world data of the total cross section was fit by thecommonly–used parameterization σ tot = A + B ∗ P n [17], inwhich P is the beam momentum, to predict the σ tot at 2.5,2.8 and 3.2 GeV/c. The corresponding values are given inthe second column in Table 2 and they overlap with the ] [ m b / ( G e V / c ) t = / d t | s d IKARANKEKOALA
Figure 9: The predicted values by Grein and Kroll [22] and thelower limit given by the optical theorem are indicated with solidand dashed line, respectively. The KOALA data are shown with thequoted errors by red squares. The green squares and blue circles arethe published IKAR and ANKE values, respectively. The figure isadopted from [21].
KOALA results listed in Table 1. As a consequence the ρ values determined while σ tot was fixed are quite similarto the case where all parameters are freely varied. In thisanalysis there is no significant influence from the parame-ter correlation while determining parameter ρ .Since the largest | t | reaches 0 . , which is farabove the CNI region, it enables the extrapolation of thenuclear differential cross section d σ n / d t down to t = 0 todetermine the optical point with Eq. (4) and (9). Figure 9shows the optical point values of the IKAR [7], ANKE [21]as well as KOALA measurements. The solid and dash linesindicate the predicted values [22] and the lower limits [21]of the optical theorem, respectively. The optical pointvalues for ANKE and KOALA at 3.2 GeV/c nicely matcheach other. The data at 2.8 GeV/c show a small ( ≈ .
3% for this measurement into account. It is notedthat the ANKE points were obtained from SAID fits [21]instead of a straightforward extrapolation as carried outin this analysis.In summary, we have measured the proton–proton elas-tic scattering differential cross section over a wide rangeof t for three beam momenta using the KOALA recoil de-tector. The fit parameters σ tot , B and ρ have been de-termined precisely. The results are consistent with othermeasurements in the similar energy region. The resultscontribute to a complete understanding of the hadronicproton–proton interaction. Since there are not many datapoints on ρ available in this energy range, hopefully thenew results gained by KOALA will motivate a new dis-persion analysis to predict ρ with even higher precision,which can serve those experiments to measure the totalcross section via the optical theorem.We are grateful to the COSY crew who installed the6evice into the COSY ring and provided proton beams.We owe the ANKE collaboration for allowing us to installthe recoil detector chamber at their target station. Specialthanks for S. Mikirtytchiants, who gave excellent supportto find the optimal beam–target overlap. This work wassupported in part by the Forschungszentrum J¨ulich COSY-054 FFE under contract number 41808260 and by the U.S.Department of Energy, Office of Science, Office of NuclearPhysics, under awards DE-SC0016582 and DE-SC0016583. References [1] The TOTEM Collaboration, Europhys. Lett. 96 (2011) 21002.[2] ATLAS Collaboration, Nucl. Phys. B 889 (2014) 486.[3] M.M. Block, R.N. Cahn, Rev. Mod. Phys. 57 (1985) 2.[4] G.B. West, D.R. Yennie, Phys. Rev. 172 (1968) 1413.[5] R. Cahn, Z. Phys. C 15 (1982) 253.[6] A.V. Dobrovolsky, et al., Phys. Lett. 41B (1972) 639.[7] A.V. Dobrovolsky, et al., Nucl. Phys. B 214 (1983) 1.[8] Q. Hu, et al., Euro. Phys. J. A 50 (2014) 156.[9] D. Bernard et al., Phys. Lett. B 198 (1987) 583.[10] ATLAS Collaboration, CERN/LHCC 2008–004, ATLAS TDR018, https://cds.cern.ch/record/1095847.[11] R. Maier, Nucl. Instr. Meth. A 390 (1997) 1.[12] S. Trokenheim, et al., Nucl. Inst. and Meth. A 355 (1995) 308.[13] A.C. Melissinos, S. L. Olsen, Phys. Rep. 17 (1975) 77.[14] H. Xu, et al., DPG spring meeting 2012, http://juser.fz-juelich.de/record/130563.[15] A. Khoukaz, et al., Eur. Phys. J. D 5 (1999) 275.[16] I. Ambats, et al., Phys. Rev. D 9 (1974) 1179.[17] P.A. Zyla, et al. (Particle Data Group), Prog. Theor. Exp. Phys.2020, (2020) 083C01.[18] G.G. Beznogikh, et al., Phys. Lett. 43B (1973) 85.[19] G.G. Beznogikh, et al., Phys. Lett. 30B (1969) 274.[20] V. Bartenev, et al., Phys. Rev. Lett. 31 (1973) 1088.[21] D. Mchedlishvili, et al., Phys. Lett. B 755 (2016) 92.[22] W. Grein, P. Kroll, Nucl. Phys. A 377 (1982) 505.[23] L. Kirillova, et al., Yadern. Fiz. 1 (1965) 533.[1] The TOTEM Collaboration, Europhys. Lett. 96 (2011) 21002.[2] ATLAS Collaboration, Nucl. Phys. B 889 (2014) 486.[3] M.M. Block, R.N. Cahn, Rev. Mod. Phys. 57 (1985) 2.[4] G.B. West, D.R. Yennie, Phys. Rev. 172 (1968) 1413.[5] R. Cahn, Z. Phys. C 15 (1982) 253.[6] A.V. Dobrovolsky, et al., Phys. Lett. 41B (1972) 639.[7] A.V. Dobrovolsky, et al., Nucl. Phys. B 214 (1983) 1.[8] Q. Hu, et al., Euro. Phys. J. A 50 (2014) 156.[9] D. Bernard et al., Phys. Lett. B 198 (1987) 583.[10] ATLAS Collaboration, CERN/LHCC 2008–004, ATLAS TDR018, https://cds.cern.ch/record/1095847.[11] R. Maier, Nucl. Instr. Meth. A 390 (1997) 1.[12] S. Trokenheim, et al., Nucl. Inst. and Meth. A 355 (1995) 308.[13] A.C. Melissinos, S. L. Olsen, Phys. Rep. 17 (1975) 77.[14] H. Xu, et al., DPG spring meeting 2012, http://juser.fz-juelich.de/record/130563.[15] A. Khoukaz, et al., Eur. Phys. J. D 5 (1999) 275.[16] I. Ambats, et al., Phys. Rev. D 9 (1974) 1179.[17] P.A. Zyla, et al. (Particle Data Group), Prog. Theor. Exp. Phys.2020, (2020) 083C01.[18] G.G. Beznogikh, et al., Phys. Lett. 43B (1973) 85.[19] G.G. Beznogikh, et al., Phys. Lett. 30B (1969) 274.[20] V. Bartenev, et al., Phys. Rev. Lett. 31 (1973) 1088.[21] D. Mchedlishvili, et al., Phys. Lett. B 755 (2016) 92.[22] W. Grein, P. Kroll, Nucl. Phys. A 377 (1982) 505.[23] L. Kirillova, et al., Yadern. Fiz. 1 (1965) 533.