Odderon effects in the differential cross-sections at Tevatron and LHC energies
EEPJ C manuscript No. (will be inserted by the editor)
Odderon effects in the differential cross-sections at Tevatronand LHC energies
Evgenij Martynov · Basarab Nicolescu
Received: date / Accepted: date
Abstract
In the present paper, we extend the Froissa-ron-Maximal Odderon (FMO) approach at t differentfrom 0. Our extended FMO approach gives an excel-lent description of the 3266 experimental points consid-ered in a wide range of energies and momentum trans-ferred. We show that the very interesting TOTEM re-sults for proton-proton differential cross-section in therange 2.76-13 TeV, together with the Tevatron data forantiproton-proton at 1.8 and 1.96 TeV give further ex-perimental evidence for the existence of the Odderon.One spectacular theoretical result is the fact that thedifference in the dip-bump region between ¯ pp and pp differential cross-sections is diminishing with increas-ing energies and for very high energies (say 100 TeV),the difference between ¯ pp and pp in the dip-bump re-gion is changing its sign: pp becomes bigger than ¯ pp at | t | about 1 GeV . This is a typical Odderon effect.Another important - phenomenological - result of ourapproach is that the slope in pp scattering has a differ-ent behavior in t than the slope in ¯ pp scattering. Thisis also a clear Odderon effect. The Odderon is certainly one the most important prob-lems in strong interaction physics. It was introduced[1] in 1973 on the basis of asymptotic theorems [2],
Evgenij MartynovBogolyubov Institute for Theoretical Physics, Metrologichna14b, Kiev, 03680 UkraineE-mail: [email protected] NicolescuFaculty of European Studies, Babes-Bolyai University, Em-manuel de Martonne Street 1, 400090 Cluj-Napoca, RomaniaE-mail: [email protected] [3] and was rediscovered later in QCD [4,5,6,7,8]. Inspite of the fact that its theoretical status is very solid,its experimental evidence from half a century is stillscarce. This situation is not astonishing, The clear evi-dence for Odderon has to come by comparing the dataat the same energy in hadron-hadron and antihadron-hadron scatterings. But we have not such accelerators!We therefore have to limit our search for evidence forthe Odderon only in an indirect way. The search forthe Odderon is crucial in order to confirm the validityof QCD. It is very fortunate that the TOTEM datum ρ pp = 0 . ± .
01 at 13 TeV [9] is the first experimen-tal discovery of the Odderon at t = 0, namely in itsmaximal form [10]. Moreover, we checked recently thatjust the Maximal Odderon in FMO approach is pre-ferred by the experimental data. We generalized theFMO approach by relaxing the ln s constraints bothin the even- and odd-under-crossing amplitude and weshow that, in spite of a considerable freedom of a largeclass of amplitudes, the best fits bring us back to themaximality of strong interaction [11].In the present paper, we extend the FMO approachat t different from 0. We show that the very interest-ing TOTEM results for proton-proton differential cross-section in the range 2.76-13 TeV, together with the D0data for antiproton-proton at 1.96 TeV give further ex-perimental evidence for the existence of the Odderon. t differentfrom zero - General definitions In general amplitude of pp forward scattering is F pp ( s, t ) = F + ( s, t ) + F − ( s, t ) (1) a r X i v : . [ h e p - ph ] M a y Evgenij Martynov, Basarab Nicolescu and the amplitude of antiproton-proton scattering is F ¯ pp ( s, t ) = F + ( s, t ) − F − ( s, t ) . (2)In this model we used the following normalization ofthe physical amplitudes. σ t ( s ) = 1 (cid:112) s ( s − m ) Im F ( s, ,dσ el dt = 164 πks ( s − m ) | F ( s, t ) | (3)where k = 0 . · GeV . With this normaliza-tion the amplitudes have dimension mb · GeV .Strictly speaking crossing-even (CE), F + ( s, t ), andcrossing-odd (CO), F − ( s, t ), parts of amplitudes are de-fined as functions of z t = ( t +2 s − m ) / (4 m − t ), where m is proton mass, with the property F ± ( − z t , t ) = ± F ± ( z t , t ) . (4)In the FMO model CE and CO terms of amplitudesare defined as sums of the asymptotic contributions F H ( s, t ), F MO ( s, t ) and Regge pole contributions whichare important at the intermediate and relatively low en-ergies F + ( z t , t ) = F H ( z t , t ) + F R + ( z t , t ) ,F − ( z t , t ) = F MO ( z t , t ) + F R − ( z t , t ) (5)where F H ( z t , t ) denotes the Froissaron contribution and F MO ( z t , t ) denotes the Maximal Odderon contribution.Their specified form will be defined below. In the FMO model in the terms F R ± ( s, t ) we con-sider not only single Regge pole contributions but alsotheir double rescatterings or double cuts. Their contri-butions, F Rpp ( z t , t ) , F R ¯ pp ( z t , t ), are the following F Rpp ( z t , t ) = F + ( z t , t ) + F − ( z t , t ) ,F R ¯ pp ( z t , t ) = F + ( z t , t ) − F − ( z t , t ) (6)where z t = − s/ (4 m − t ) ≈ s/ (4 m − t ). Forconvenience in further work with parameterizations inFMO model at t = 0 and t (cid:54) = 0 contrary to standarddefinition of z t we put opposite sign for it. F + ( z t , t ) = F P ( z t , t ) + F R + ( z t , t ) + F P P ( z t , t )+ F OO ( z t , t ) ,F − ( z t , t ) = F O ( z t , t ) + F R − ( z t , t ) + F P O ( z t , t ) . (7)Here F P ( z t , t ) , F O ( z t , t ) are simple j -pole Pomeron andOdderon contributions and F R + ( z t , t ) , F R − ( z t , t ) areeffective f and ω simple j -pole contributions, where j is an angular momenta of these reggeons. F P P ( z t , t ), F OO ( z t , t ) , F P O ( z t , t ) , are double P P, OO, P O cuts,correspondingly. We consider the model at t (cid:54) = 0 andat energy √ s >
19 GeV, so we neglect the rescatteringsof secondary reggeons with P and O . In the consideredkinematical region they are small. Besides, because f and ω are effective, they can take into account smalleffects from the cuts. The standard Regge pole contri-butions have the form F R ± ( z t , t ) = − (cid:18) i (cid:19) m C R ± ( t )( − iz t ) α ± ( t ) (8)where R ± = P, O, R + , R − and α P (0) = α O (0) = 1.The factor 2 m is inserted in amplitudes F R ± ( z t , t ) inorder to have the normalization for amplitudes and di-mension of coupling constants (in mb) coinciding withthose in [10]. The same is made for all other ampli-tudes, including Froissaron and Maximal Odderon (seebelow).For the coupling function C R ± ( t ) we have consid-ered two possibilities. The first one is a simple exponen-tial form. It is used for the secondary reggeons, becausewe did not consider low energies where terms R ± ( s, t )are more important. C R ± ( t ) = C R ± e b R ± t , C R ± (0) = C R ± . (9)The second case is a linear combination of exponentsfor Standard Pomeron and Odderon terms which al-low to take into account some possible effects of non-exponential behavior of coupling function. C P,O ( t ) = C P,O (cid:2) Ψ P,O ( t ) (cid:3) ,Ψ P,O ( t ) = d p,o e b P,O t + (1 − d p,o ) e b P,O t . (10)We have added as well the double pomeron and odd-eron cuts, P P, OO, P O in their exact form without anynew parameters. Namely, F P P ( z t , t ) = − i ( z t C P ) πs (cid:112) − m /s (cid:40) d p B p exp( tB p / d p (1 − d p ) B p + B p exp (cid:18) t B p B p B p + B p (cid:19) + (1 − d p ) B p exp( tB p / (cid:27) (11) F OO ( z t , t ) = − i ( z t C O ) πs (cid:112) − m /s (cid:26) d o B o exp( tB o / d o (1 − d o ) B o + B o exp (cid:18) t B o B o B o + B o (cid:19) +(1 − d o ) B o exp( tB o / (cid:27) (12) dderon effects in the differential cross-sections at Tevatron and LHC energies 3 where B p,ok = b P.Ok + α (cid:48) P, ln( − iz t ) , k = 1 , , b P,Ok are the constants from single pomeron and odderon con-tributions. F P O ( z t , t ) = z t C P C O πs (cid:112) − m /s × (cid:26) d p d o B p + B o exp (cid:18) t B p B o B p + B o (cid:19) + d p (1 − d o ) B p + B o exp (cid:18) t B p B o B p + B o (cid:19) + (1 − d p ) d o B p + B o exp (cid:18) t B p B o B p + B o (cid:19) +(1 − d p )(1 − d o ) B p + B o exp (cid:18) t B p B o B p + B o (cid:19)(cid:27) (13)We have found that for a better description of the datait is reasonable to add to the amplitudes the contribu-tions which mimic some properties of ”hard“ pomeron( P H ) and odderon ( O H ). We take them in the simplestform P H ( t ) = i C P H z t (1 − t/t P ) µ P , µ P ≤ . (14) P O ( t ) = C OH z t (1 − t/t O ) µ O P , µ O ≤ . (15) t (cid:54) = 0 s, t )-representation at high s . The amplitude can be ex-panded in the series of partial amplitudes φ ( ω, t ). Inaccordance with the standard definition of partial am-plitude F ( z t , t ) = 16 π ∞ (cid:88) j =0 (2 j + 1) P j ( − z t ) φ ( j, t ) . (16)With such definition partial amplitude satisfies the uni-tarity equation in the formIm φ ( j, t ) = ρ ( t ) | φ ( j, t ) | + inelastic contribution ,ρ ( t ) = (cid:112) − m /t (17)We use of the Sommerfeld-Watson transform am-plitude (here and in what follows ω = j − j iscomplex angular momentum) which can be written as follows F ζ ( z t , t ) = 16 π (cid:80) ξ = − , (cid:82) C dω πi (2 ω + 3) 1 − ξe − iπω − sin( πω ) × φ ξ ( ω, t ) P ω ( z t )= 16 π (cid:80) ξ = − , (cid:82) C dω πi (2 ω + 3) × e − iπω/ e iπω/ − ξe − iπω/ − sin( πω ) φ ξ ( ω, t ) P ω ( z t )= z t (cid:80) ξ = − , (cid:82) C dω πi e ωζ ϕ ξ ( ω, t ) . (18)where ω = j − ξ is the signature of the term, contour C is a straight line parallel to imaginary axis and at theright of all singularities of φ ξ ( ω, t ), ζ = ln( z t ) − iπ/ ≡ ln( − iz t ) and ϕ ξ ( ω, t ) = 16 π (2 ω + 3) e iπω/ − ξe − iπω/ − sin( πω ) π − / ω +1 × Γ ( ω + 3 / Γ ( ω + 2) φ ξ ( ω, t ) (19)Thus for crossing even amplitude ( ξ =+1) we have ϕ + ( ω, t ) = i √ π (2 ω + 3) Γ ( ω + 3 / Γ ( ω + 2) 2 ω φ + ( ω, t )cos( πω/ ξ =-1) ϕ − ( ω, t ) = − √ π (2 ω + 3) Γ ( ω + 3 / Γ ( ω + 2) 2 ω φ − ( ω, t )sin( πω/ . (21)Inverse transformation is ϕ ± ( ω, t ) = ∞ (cid:90) dζe − ωζ F ± ( z t , t ) , z t = e ζ . (22)One can show that in order to have maximal growth oftotal cross section σ tot ( s ) ∝ ξ at s → ∞ , to have agrowing elastic cross section bounded by σ el ( s ) /σ tot ( s ) → const at s → ∞ and to provide the correct analytical properties of am-plitude at t ≈ φ ( ω, t ) in the following form (more details are given inthe Appendics, Section A) ϕ ± ( ω, t ) = (cid:18) i − (cid:19) β ± ( ω, t )[ ω + r ± q ⊥ ] / . (23)where r ± are some constants, q ⊥ = − t and β ( ω, t ) hasnot singularity at ω + R q ⊥ = 0. In fact a choice of Evgenij Martynov, Basarab Nicolescu the sign in φ − ( ω, t ) does nor matter because the cross-ing odd terms contribute to pp and ¯ pp amplitude withthe opposite signs. In order to have agreement withparametrization and parameters which we used in thepapers devoted to analysis of the data at t = 0, weshould replace -1 for for +1 in front of φ − ( ω, t ).At ω = 0, function ϕ − ( ω, t ) has singularity in t if β − (0 , t ) (cid:54) = 0, namely, φ − (0 , t ) ∝ ( − t ) / . One of argu-ments against the Maximal Odderon is that this singu-larity in partial amplitude means the existing of mass-less particle in the model. However as we seen above ϕ − ( ω, t ) is not the real physical partial amplitude whichis φ − ( ω, t ) = (cid:20) √ π (2 ω + 3) Γ ( ω + 3 / Γ ( ω + 2) 2 ω (cid:21) − × sin( πω/ ϕ − ( ω, t ) (24)and it equals to 0 at ω = 0 because of sin( πω/
2) comingfrom signature factor.Now let us suppose that in accordance with thestructure of the singularity of ϕ ± ( ω, t ) at ω + ω ± = 0( ω ± = R ± q ⊥ ) the functions β ± ( ω, t ), depending on ω through the variable κ ± = ( ω + ω ± ) / , can be ex-panded in powers of κ ± ϕ ± ( ω, t ) = (cid:18) i (cid:19) β ± ( t ) + κ ± β ± ( t ) + κ ± β ± ( t ) κ ± (25)Then making use of the table integrals (see the Sec-tion A) we obtain the expressions for F ± ( z t , t ) whichare written in the next Section.4.2 Froissaron and Maximal Odderon in( s, t )-representationAt t = 0 Froissaron and Maximal Odderon have theuniversal form independently of any extension to t (cid:54) = 0: F H ( z t , t = 0) = iz [ H ln ( − iz t ) + H ln( − iz t ) + H ] , (26) F MO ( z t , t = 0) = z [ O ln ( − iz t ) + O ln( − iz t ) + O ](27)where z = 2 m z t . At t = 0 we have z t = ( s − m ) / (2 m ).The Froissaron and the Maximal Odderon definedat t = 0 by above Eqs. (26, 27) allow various extensionsto analytical t -dependences. Probably it is impossible apriory to choose the best of them. In the present work we consider an extension of Eqs. (26, 27) in accordancewith Eq. (25). − iz F H ( z t , t ) = H ζ J ( r + τ ζ ) r + τ ζ Φ H, ( t )+ H ζ sin( r + τ ζ ) r + τ ζ Φ H, ( t ) + H J ( r + τ ζ ) Φ H, ( t ) ,Φ H,i ( t ) = exp( b Hi q + ) , i = 1 . , q + = 2 m π − (cid:112) m π − t. (28)1 z F MO ( z t , t ) = O ζ J ( r − τ ζ ) r − τ ζ Φ O, ( t )+ O ζ sin( r − τ ζ ) r − τ ζ Φ O, ( t ) + O J ( r − τ ζ ) Φ O, ( t ) ,Φ O,i ( t ) = exp( b Oi q − ) , i = 1 , , ,q − = 3 m π − (cid:112) m π − t. (29)where z = 2 m z t , ζ = ln( − iz t ) , τ = (cid:112) − t/t , t =1GeV .Due to the factor z (instead of z t ) the amplitudes F H ( z t , t ) and F MO ( z t , t ) have the required normaliza-tion with additional factor 2 m . We give here the results of the fit to the data in thefollowing region of s and | t | .for σ tot ( s ) , ρ ( s ) at 5 GeV ≤ √ s ≤
13 TeV , for dσ ( s, t ) /dt at 9 GeV ≤ √ s ≤
13 TeVand at 10 − GeV ≤ | t | ≤ . We add the recent data at t = 0 of TOTEM Collabora-tion [9,12,13,14] to data set published by Particle DataGroup [22].We have performed two alternative fits of the FMOmodel and experimental data from the above mentionedkinematic region. In the Fit I we take into account all the data att=0, i.e. σ tot and ρ are calculated from the FMO model,free parameters are determined from the fit to all t , cho-sen in such a way that we can ignore in given region thecontribution of the Coulomb part of amplitudes whichis less than 1% of the nuclear amplitude. Thus, t -region0 . < | t | < .
05 GeV is excluded in the Fit I, and theCoulomb part of amplitudes put to zero. In the Fit II all experimental data on σ tot and ρ are excluded and fit is made at energies √ s >
19 GeVand 0 < | t | < . Taking into account that in thiskinematic region parameters of CE and CO secondaryreggeons are badly determined, we put all the parame-ters of these contributions as fixed from the results ofFit I. dderon effects in the differential cross-sections at Tevatron and LHC energies 5 For 13 TeV TOTEM data we used the data at t = 0for σ tot [12] and ρ [9], as well as the data on differen-tial cross sections [16,17,18,19]. We add also recentlypublished data on dσ/dt at √ s = 2 .
76 TeV obtained byTOTEM [20].5.1 Coulomb amplitude, one of the simplestparameterizationsCoulomb terms in the pp and ¯ pp amplitudes are writtenin the well known form F CN ( s, t ) = ± πs αt F ( t ) exp( iαφ ( s, t )) (30)where α = 7 . × − = 1 / .
035 is the fine-structure constant and F ( t ) = 4 m p − µ p t m p − t − t/ . ,µ p = 2 . µ p is the magnetic momemtum of proton. Forthe phase φ ( s, t ) we nave φ ( s, t ) = ± (cid:20) ln (cid:18) B ( s )2 | t | (cid:19) + γ (cid:21) (32)where γ = 0 . B ( s ) is calculated through a fit makinguse of the equation < B ( s ) > = (1 /∆ t ) t min (cid:82) t max dt ddt (ln( dσ ( t ) /dt ))= (1 /∆ t ) (cid:20) ln (cid:18) dσ ( t min ) /dtdσ ( t max ) /dt (cid:19)(cid:21) (33)where ∆ t = t max − t min . We put (in accordance withthe TOTEM estimations [17]), t max = − .
07 GeV , t min = − .
005 GeV .5.2 pp and ¯ pp differential cross sections dσ/dt Here we present results for both methods of the datadescription.
Fit I : FMO model without Coulumb termfitted to the whole set of data excluding lowest | t | < .
05 GeV . Fit II : FMO model with Coulomb termfitted to the whole set of the data at t (cid:54) = 0. In thelegends of Fig. 1-12 these fits are labeled as ”FMO”and ”FMO+C”, correspondingly. The curves shown atthe Figs. 13 - 16 were calculated in the FMO modelwithout Coulomb terms ( Fit I ).Number of experimental points in pp and ¯ pp totalcross sections σ ppt , σ ¯ ppt , ratios ρ pp , ρ ¯ pp and differential cross sections used in the Fit I and quality of fit areshown in the Table 1.Numbers of the data points and obtained values of χ in the Fit II are given in the Table 2.
FMO without Coulomb termsProcess Observable N, number χ /N of data pp σ tot
110 0.857213E+00¯ pp σ tot
59 0.992282E+00 pp ρ
67 0.169032E+01¯ pp ρ
12 0.836012E+00 pp dσ/dt pp dσ/dt
389 0.121600E+01 χ tot = 3718 . χ / NDF = 1 . Table 1
Number of experimental points and the quality oftheir description when the usual minimization in FMO modelis applied FMO with Coulomb termsProcess Observable N, number χ /N of data pp dσ/dt pp dσ/dt
536 0.121288E+01 χ tot = 4790.652 χ / NDF=1.584
Table 2
Number of experimental points and the quality oftheir description when the fit with FMO+Coulomb terms ismade. The data on σ tot ( s ) and ρ ( s ) has been excluded fromthis fit The values of parameters and their errors obtainedin these two fits within the FMO model are given inthe Table 3 (parameters of the Froissaron and MaximalOdderon terms, of the standard Pomeron and Odderon,of the ”hard“Pomeron and Odderon, and of the sec-ondary reggeons).To avoid a possible negative cross sections in thelarge partial waves, j , (at the edge of the disk) we putin the fit the restriction r − ≤ r + . However, we observedthat in the various considered modifications of the FMOmodel these parameters are almost equal each to other.Based on this fact we put r − = r + in the model pre-sented here. Also we have fixed the parameters b ± at 0because in all considered fits b + has the error compara-ble with the value of parameter and b − has value closeto 0.Fig. 1 demonstrates a behavior of the pp and ¯ pp totalcross sections and ratios real part to imaginary part ofthe amplitudes at t = 0 obtained in the both Fit I and
Fit II . We would like to notice the interesting odderoneffect: the change of sign in the differences between totalcross section and ρ ’s between √ s ≈
50 and √ s ≈ Evgenij Martynov, Basarab Nicolescu
40 60 80 100 120 140 160 10 σ t o t ( s ) ( m b ) s (GeV)pppp-ATLASpp-cosmicpapFMO-ppFMO-papFMO+C ppFMO+C pap -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 10 ρ ( s ) = R e F ( s , ) / I m F ( s , o ) s (GeV)pppapFMO-ppFMO-papFMO+C ppFMO+C pap Fig. 1
Total cross sections and ratios ρ in FMO model with the P P, P O, OO terms added theorems. A detailed dynamic model for this effect wasnot yet invented.In Figs. 2 and 3 we show the differential cross-sectionsat energies bigger than 19 GeV. In Fig. 4 we show thedifferential cross-sections at the LHC energies 7, 8 and13 TeV and in Figs. 5, 6, 7 we show differential pp and¯ pp at lowest | t | . In Fig. 8, we show in a magnified waythe differential cross-sections at 53 GeV.As one can see from these figures our description ofthe data in a wide range of energies is very good. InFig. 9 we show the evolution of the dip-bump structurein pp and ¯ pp differential cross sections with increasingenergy. In Fig. 10 we show in a magnified way the dip-bump region at different energies and in Fig. 11 we showthe evolution of the ratio R σ = ( dσ (¯ pp ) /dt ) / ( dσ ( pp ) /dt )with increasing energy. A remarkable prediction can beseen from these last three figures: the difference in thedip-bump region between ¯ pp and pp differential crosssections is diminishing with increasing energies and, forvery high energies (say 100 TeV, see Fig. 10), the ratioin the dip-bump region goes to 1. At ISR energies until ∼
60 GeV the ratio R σ > t m . Aftermaximum the value of R σ is decreasing and equals to 1at some t which is going to lower t with increasing en-ergy. At higher t however R σ is oscillating around of 1when t increases. This is a spectacular Odderon effect.One can see also the clear Odderon effects and theirevolution with energy in Fig. 12. -7 -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 d σ / d t ( m b / G e V ) |t| (GeV )pp 19.4GeV(X10 )pp 19.5-19.9GeV(X10 )pp 20.8-23.5GeV(X10 )pp 27.43GeV(X10 )pp 30.7-31GeV(X10 )pp 44.7GeV(X10 )pp 52.8-53.1GeV(X10 )pp 62.0-62.5GeVFMO-modelFMO-model-C Fig. 2 pp differential cross sections at √ s >
19 GeV B ( s, t ) The slope B ( s, t ) is a very interesting quantity in thesearch for Odderon effects. It is defined by B ( s, t ) = ddt ln( dσ/dt ) . (34)If we consider the dependence of slope on energy andcompare this dependence with available experimental dderon effects in the differential cross-sections at Tevatron and LHC energies 7 -8 -7 -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 d σ / d t ( m b / G e V ) |t| (GeV )pap 19.42 GeV (X 10 )pap 31 GeV (X 10 )pap 52.6-53 GeV (X10 )pap 540-546 GeV (X10 )pap 630 GeV (X10 )pap 1.8-1.96 TeVFMO-modelFMO+C-model Fig. 3 ¯ pp differential cross sections at √ s from 19 GeV upto 1.96 TeV data we have to take into account that slopes in anyrealistic model depend on t . Dependence of slope on t at various energies in the FMO model is illustratedin Fig. 13 (left panel). Therefore we must to calculatethe slope < B ( s ) > averaged in some interval of t . Wedid that in the interval | t | ∈ (0 . , . for GeVenergies which approximately is in agreement to theintervals from which the experimental data on B aredetermined. < B ( s ) > = (1 /∆ t ) t min (cid:82) t max dt ddt (ln( dσ ( t ) /dt ))= (1 /∆ t ) (cid:20) ln (cid:18) dσ ( t min ) /dtdσ ( t max ) /dt (cid:19)(cid:21) (35)where ∆ t = t max − t min .We show in Table 4 our predictions for the averagedslopes in the TeV region of energy as compared withexperiments at Tevatron and LHC.In Fig. 13 (right panel) we show the increasing of theaveraged slopes at t=0 with increasing energy. One cansee that the slopes are approaching the ln s increase athigh energies.In Fig. 14 we plot the slopes as function of t in pp and ¯ pp scatterings. We discover from the t -dependenceof the slopes an extremely interesting phenomenon. Theslope in pp scattering has a different behaviour in t thanthe slope in ¯ pp scattering. In the left panel of Fig. 14we see that in pp scattering the slopes are first nearlyconstant and after that they fall sharply, they cut a firsttime the B ( t ) = 0 line, reach a deep minimum negative value, after that they increase and cut a second time the B ( t ) = 0 line and finally they reach an approximatelyconstant value for higher t . The two crossing points ofthe B ( t ) = 0 line move towards smaller t when energyincreases. In the right panel of Fig. 14 we see a verydifferent behaviour in ¯ pp scattering. In this case, at en-ergies higher than ISR ones, B(t) marginally crosseszero, but no so deeply and sharply as in pp scattering. -7 -6 -5 -4 -3 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4 d σ / d t ( m b / G e V ) |t| (GeV )pp 2.76 TeV low-t (X10 )pp 2.76 TeV high-t (X10 )pp 7 TeV (X10 )pp 8 TeV (X10 )pp 13 TeV (X10 )pp 13 TeV(X10 )FMO pp 2,76 TeVFMO-modelFMO+C-modelFMO+C pp 2,76 TeV Fig. 4 pp differential cross sections at √ s = 7 , ,
13 TeV
40 50 60 70 80 90 100 110 120 130 140 150 160 170 0.01 0.02 0.03 0.04 0.05 d σ / d t ( m b / G e V ) |t| (GeV )pp 19.4 GeVFMO+C model 40 50 60 70 80 90 100 110 120 130 140 150 160 170 0.01 0.02 0.03 0.04 0.05 d σ / d t ( m b / G e V ) |t| (GeV )pp 23.5 GeVFMO+C model 40 50 60 70 80 90 100 110 120 130 140 150 160 170 0.01 0.02 0.03 0.04 0.05 d σ / d t ( m b / G e V ) |t| (GeV )pp 31 GeVFMO+C model 40 50 60 70 80 90 100 110 120 130 140 150 160 170 0.01 0.02 0.03 0.04 0.05 d σ / d t ( m b / G e V ) |t| (GeV )pp 44.7 GeVFMO+C-model 40 50 60 70 80 90 100 110 120 130 140 150 160 170 0.01 0.02 0.03 0.04 0.05 d σ / d t ( m b / G e V ) |t| (GeV )pp 52.8 GeVFMO+C model 40 50 60 70 80 90 100 110 120 130 140 150 160 170 0.01 0.02 0.03 0.04 0.05 d σ / d t ( m b / G e V ) |t| (GeV )pp 62.5 GeVpp 62.5 GeVFMO+C model Fig. 5
Differential pp cross sections at the lowest | t | and atISR energies Evgenij Martynov, Basarab NicolescuFMO modelMinimization without Minimization withCoulomb terms Coulomb termsName (dimension) Value Error Value Error α (cid:48) P (GeV : 2) 0.18845E+00 0.15606E-03 0.16274E+00 0.12999E-03 C P (mb) 0.67305E+02 0.50925E-01 0.67098E+02 0.40739E-01 b P (GeV ) 0.57234E+01 0.54856E-02 0.60451E+01 0.48533E-02 d p b P (GeV ) 0.23392E+01 0.31028E-02 0.24359E+01 0.26420E-02 C HP (mb) -0.61825E+02 0.35377E-01 0.69984E+02 0.30999E-01 t HP (GeV ) 0.41803E+00 0.14780E-03 0.41377E+00 0.12323E-03 α (cid:48) O (GeV − ) 0.15673E-01 0.11401E-03 0.12298E-01 0.10018E-03 C O (mb) 0.29156E+02 0.25668E-01 0.31654E+02 0.24212E-01 b O (GeV − ) 0.50899E+01 0.46679E-02 0.52749E+01 0.39244E-02 d o b O (GeV − ) 0.21098E+01 0.22047E-02 0.20931E+01 0.21404E-02 C HO (mb) 0.37930E+02 0.37256E-01 0.42175E+02 0.36723E-01 t HO (GeV ) 0.58624E+00 0.36266E-03 0.55774E+00 0.30146E-03 α + (0) 0.47754E+00 0.51446E-02 0.47754E+00 fixed α (cid:48) + (GeV − ) 0.80000E+00 0.31788E-02 0.80000E+00 fixed C + (mb) 0.47341E+02 0.11590E+01 0.47341E+02 fixed b + (GeV − ) 0.00000E+00 0.00000E+00 0.00000E+00 fixed α − (0) 0.32715E+00 0.13892E-01 0.32715E+00 fixed α (cid:48) − (GeV − ) 0.11000E+01 0.33881E-01 0.11000E+01 fixed C − (mb) 0.33528E+02 0.13387E+01 0.33528E+02 fixed b − (GeV − ) 0.00000E+00 0.00000E+00 0.00000E+00 fixed H (mb) 0.31370E+00 0.16934E-03 0.33974E+00 0.14696E-03 H (mb) -0.21950E+01 0.12102E-01 0.27105E+01 0.50719E-02 H (mb) 0.39935E+02 0.98913E-01 0.50953E+02 0.62230E-01 b H (GeV − ) 0.25927E+01 0.97184E-03 0.26824E+01 0.82689E-03 b H (GeV − ) 0.72045E+01 0.27693E-01 0.61736E+01 0.13102E-01 b H (GeV − ) 0.48405E+01 0.10107E-01 0.44076E+01 0.52826E-02 r + (GeV − ) 0.26818E+00 0.57931E-04 0.26436E+00 0.50348E-04 O (mb) -0.44278E-01 0.20397E-03 0.42841E-01 0.17151E-03 O (mb) 0.93254E+00 0.14218E-01 0.83063E+00 0.14265E-01 O (mb) -0.17655E+02 0.80820E-01 0.17510E+02 0.76993E-01 b O (GeV − ) 0.15832E+01 0.41271E-02 0.15684E+01 0.38186E-02 b O (GeV − ) 0.28034E+01 0.20216E-01 0.26724E+01 0.19453E-01 b O (GeV − ) 0.28929E+01 0.59137E-02 0.28842E+01 0.56380E-02 r − (GeV − ) 0.26818E+00 0.57931E-04 0.26436E+00 0.50348E-04 Table 3
Parameters of standard Pomeron and Odderon, of their double rescatterings, of secondary Reggeons and their errorsin FMO model determined from the fits to the data on dσ/dt . Total cross sections σ tot and ratios ρ were included in the fitwithout the Coulomb termEnergy (TeV) Experiment < B pp ( s ) > (GeV − ) < B ¯ pp ( s ) > (GeV − )Experimental data FMO model Experimental data FMO model1.8 E710 - 16.70 16.3 ± ± ± ± ± ± ± ± ± .
01 20.50 - 20.25
Table 4
Experimetal values of slopes of pp and ¯ pp differential cross sections at TeV energies and the averaged slopes calculatedin FMO modeldderon effects in the differential cross-sections at Tevatron and LHC energies 9 For completeness, we show in Fig. 15 the slope parame-ter for pp scattering at 7 and 13 TeV as compared withthe slope parameter in ¯ pp scattering at 1.96 TeV, wherewe can see the same phenomenon.This phenomenon is a clear Odderon effect. Theodd-under crossing amplitude makes the difference be-tween pp and ¯ pp scatterings and this amplitude is dom-inated at high energy by the Maximal Odderon. d σ / d t ( m b / G e V ) |t| (GeV ) pp 7 TeV (X10)pp 8 TeV (X3)pp 13 TeVpp 13 TeVFMO+C-model Fig. 6
Differential pp cross sections at the lowest | t | and atLHC energies d σ / d t ( m b / G e V ) |t| (GeV )pap 1.8-Amos TeVpap 540-546 GeV (X10 )pap 52.6-53 GeV (X10 )pap 31 GeV (X 10 )pap 19.42 GeV (X 10 )FMO+C-model Fig. 7
Differential ¯ pp cross sections at the lowest | t | To our knowledge, the present model is the only modelwhich fits forward and non forward data in a wide rangeof energies (including TeV region), without theoreticaldefects (like the violation of the unitarity).However, it is important to note that our resultsconcerning the slopes are in complete agreement with -7 -6 -5 -4 -3 -2 -1 pp vs pap 53 GeV d σ / d t ( m b / G e V ) |t| (GeV )pap 53 GeV-Breakstone-1pap 53 GeV-Breakstone-2pap 53 GeV-Erhanpp-52.8-53.1pp 52.8 GeV-Nagypap-fmo 53GeVpp-fmo 53GeVpap-fmo+C 53GeVpp-fmo+C 53GeV Fig. 8 pp and ¯ pp differential cross sections at √ s = 53 GeV -8 -6 -4 -2
0 1 2 3 4 5
Dip-bump evolution with energy d σ / d t ( m b / G e V ) |t| (GeV ) pap-19(X10 )pap-53GeV(X10 )pap-62GeV(X10 )pap-546GeV(X10 )pap-630GeV(X10 )pap-1.96TeV(X10 )pap-2.76TeV(X10 )pap-7.0TeV(X10 )pap-13.0TeV(X10 )pap-100TeVpp-19(X10 )pp-53GeV(X10 )pp-62GeV(X10 )pp-546GeV(X10 )pp-630GeV(X10 )pp-1.96TeV(X10 )pp-2.76TeV(X10 )pp-7.0TeV(X10 )pp-13.0TeV(X10 )pp-100TeV Fig. 9
Evolution of pp and ¯ pp differential cross sections withincreasing energy0 Evgenij Martynov, Basarab Nicolescu those obtained recently by Cs¨org¨o et al. [21], who per-formed a very useful mirroring between the discontinu-ous experimental data (points) and continuous analyticfunctions (scattering amplitudes) by using an expansionin terms of L´evy polynomials. In such a way they get avery clear Odderon effect concerning the slopes. Theiranalysis have no dynamical content: it is a parametriza- -5 -4 -3
19 GeV d σ / d t ( m b / G e V ) FMO papFMO ppFMO+C papFMO+C pp -5 -4 -3
53 GeV d σ / d t ( m b / G e V ) FMO papFMO ppFMO+C papFMO+C pp -5 -4 -3
62 GeV d σ / d t ( m b / G e V ) FMO papFMO ppFMO+C papFMO+C pp -3 -2
546 GeV d σ / d t ( m b / G e V ) FMO papFMO ppFMO+C papFMO+C pp -3 -2
630 GeV d σ / d t ( m b / G e V ) FMO papFMO ppFMO+C papFMO+C pp -3 -2 -1 d σ / d t ( m b / G e V ) FMO papFMO ppFMO+C papFMO+C pp -2 -1 d σ / d t ( m b / G e V ) |t| (GeV ) FMO papFMO ppFMO+C papFMO+C pp -2 -1
13 TeV d σ / d t ( m b / G e V ) |t| (GeV ) FMO papFMO ppFMO+C papFMO+C pp -2 -1
100 TeV d σ / d t ( m b / G e V ) |t| (GeV ) FMO papFMO ppFMO+C papFMO+C pp
Fig. 10 pp and ¯ pp differential cross sections in and aroundthe dip region d σ p a p / d t / d σ pp / d t FMO 19 GeVFMO+C 19 GeV
FMO 53 GeVFMO+C 53 GeV
FMO 62 GeVFMO+C 62 GeV d σ p a p / d t / d σ pp / d t FMO 546 GeVFMO+C 546 GeV
FMO 630 GeVFMO+C 630 GeV
FMO 1.96TeVFMO+C 1.96 TeV d σ p a p / d t / d σ pp / d t |t| (GeV ) FMO 2.76 TeVFMO+C 2.76 TeV ) FMO+C 7 TeVFMO 7 TeV ) FMO+C 13 TeVFMO 13 TeV
Fig. 11
Evolution of the ratio of differential cross sections R σ = ( dσ (¯ pp ) /dt ) / ( dσ ( pp ) /dt ) with energy -6 -5 -4 -3 -2 -1
0 1 2 3 4 5
FMO at 20 GeV | I m F ( R e F )( s , t ) / s | |t| (GeV ) Im F pp Re F pp Re F pap (X10 -2 )Im F pap (X10 -2 ) 10 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5
FMO at 50 GeV | I m F ( R e F )( s , t ) / s | |t| (GeV ) Im F pp Re F pp Re F pap (X10 -2 )Im F pap (X10 -2 ) 10 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5
FMO at 500 GeV | I m F ( R e F )( s , t ) / s | |t| (GeV ) Im F pp Re F pp Re F pap (X10 -2 )Im F pap (X10 -2 )10 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5
FMO at 2 TeV | I m F ( R e F )( s , t ) / s | |t| (GeV ) Im F pp Re F pp Re F pap (X10 -2 )Im F pap (X10 -2 ) 10 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5
FMO at 7 TeV | I m F ( R e F )( s , t ) / s | |t| (GeV ) Im F pp Re F pp Re F pap (X10 -2 )Im F pap (X10 -2 ) 10 -6 -5 -4 -3 -2 -1
0 1 2 3 4 5
FMO at 13TeV | I m F ( R e F )( s , t ) / s | |t| (GeV ) Im F pp Re F pp Re F pap (X10 -2 )Im F pap (X10 -2 ) Fig. 12
Partial contributions of the real and imaginary partsof even and odd terms to pp and ¯ pp scattering amplitudes atvarious energies tion of experimental data in terms of big number ofparameters.This agreement is very important from two points ofview. On one side, the Odderon existence is reinforcedby two quite different analysis, one model-independentand the other one having a dynamical content.On another side, the fact that the Maximal Odderonis in agreement with a model-independent analysis re-inforce the status of the Maximal Odderon. In our paper we present an extension of the Froissaron-Maximal Odderon (FMO) approach for t different fromzero, which satisfies rigorous theoretical constraints. Ourextended FMO approach gives an excellent descriptionof the 3266 experimental points considered in a widerange of energies and momentum transferred. One spec-tacular theoretical result is the fact that the differencein the dip-bump region between ¯ pp and pp differentialcross sections is diminishing with increasing energiesand for very high energies (say 100 TeV), the differencein the dip-bump region between ¯ pp and pp is changingits sign: pp becomes bigger than ¯ pp at | t | about 1 GeV .This is a typical Odderon effect.Another important - phenomenological - result ofour approach is that the slope in pp scattering has adifferent behaviour in t than the slope in ¯ pp scattering.This is a clear Odderon effect.Let us emphasize that the FMO model is in a goodagreement with the data in a wide interval of energy.However, there is a some discrepancy of the data andmodel in a region around √ s =2 TeV (it is illustrated inthe Fig. 16). At the same time agreement with the dataat lower and at higher energies is really very good. Thisproblem requires a special investigation which we will Experimental data at t = 0 were taken from [22], with therecent TOTEM and ATLAS points being added. Set of dataat t (cid:54) = 0 will be send after personal request to E. Martynov.dderon effects in the differential cross-sections at Tevatron and LHC energies 11 -20-15-10-5 0 5 10 15 20 25 30 35 40 0.2 0.4 0.6 0.8 1 FMO model without Coulomb term B pp ( t ) , B p a p ( t ) ( G e V - ) |t| (GeV )pp-23.5 GeVpp-53.0 GeVpp-540 GeVpp-2.76 TeVpp-7 TeVpp-13 TeVpap-23.5 GeVpap-53.0 GeVpap-540 GeVpap-2.76 TeVpap-7 TeVpap-13 TeV 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 10 B pp , B p a p ( G e V - ) s (GeV) FMO FMO FMO+C FMO+C B pp dataB pp ATLAS dataB pap data
Fig. 13
Slopes B pp ( t ) and B ¯ pp ( t ) at increasing energy(left panel) and the s -dependence of the averaged slopes < B pp ( s ) > , < B ¯ pp ( s ) > together with experimental data (right panel) -35-30-25-20-15-10-5 0 5 10 15 20 25 30 35 40 45 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 B pp ( t ) ( G e V - ) |t| (GeV )pp-23.5GeVpp-31GeVpp-44.7GeVpp-52.8GeVpp-62.5GeVpp-2.76TeVpp-7TeVpp-13TeV -35-30-25-20-15-10-5 0 5 10 15 20 25 30 35 40 45 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 B p a p ( t ) ( G e V - |t| (GeV )pap-52.8GeVpap-546GeVpap-630GeVpap-1.96TeV Fig. 14
Slope B ( s, t ) for pp (left panel) and ¯ pp ) (right panel) at selected energies perform after the publication of the common TOTEM/D0paper [23].New ways of detecting Odderon effects, e. g. in anElectron-Ion Collider, were recently explored on the ba-sis of a general QCD light front formalism [24]. Acknowledgment.
The authors thank Prof. Si-mone Giani for a careful reading of the manuscript. Oneof us (E.M.) thanks the Department of Nuclear Physicsand Power Engineering of the National Academy ofSciences of Ukraine for support (continuation of theproject No 0118U005343).
A Appendix
A.1 General constraints
Let us reiterate here that the model with σ t ( s ) ∝ ln s isnot compatible with a linear pomeron trajectory having theintercept 1. Indeed, let us assume that α P ( t ) = 1 + α (cid:48) P t (36)and the partial wave amplitude has the form ϕ ( j, t ) = η ( j ) β ( j, t )[ j − − α (cid:48) P t ] n ≈ iβ (1 , t )[ j − − α (cid:48) P t ] n , (37)2 Evgenij Martynov, Basarab Nicolescu -10-5 0 5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 B pp ( t ) , B p a p ( t ) ( G e V - ) |t| (GeV ) FMO pp-7TeVFMO pp-13TeVFMO+C pp-7TeVFMO+C pp-13TeVFMO pap-1.96TeVFMO+C pap-1.96TeV
Fig. 15
Dependence on t of the slopes B ( s, t ) for pp scatter-ing at 7 and 13 TeV and for ¯ pp scattering at 1.96 TeV η ( j ) = 1 + ξe − iπj − sin πj . (38)For Pomeron (simple or double pole) and Froissaron sig-nature is positive, ξ = +1. -3 -2 -1
0 0.25 0.5 0.75 1 1.25 1.5 d σ / d t ( m b / G e V ) |t| (GeV ) pp 2.76 TeVpp 2.76 TeVpap1.8-1.96TeVpap-FMO, 1.96TeVpap-FMO, 2.76TeVpp-FMO, 2.76TeV Fig. 16 ¯ pp differential cross section at 1.18-1.96 TeV and pp differential cross section at 2.76 TeV In ( s, t )-representation amplitude ϕ ( j, t ) is transformed to a ( s, t ) = 12 πi (cid:90) djϕ ( j, t ) e ξj , ξ = ln( s/s ) . (39)Then, we have pomeron contribution at large s as a ( s, t ) ≈ − ˜ β ( t )[ln( − is/s )] n − ( − is/s ) α (cid:48) P t (40)where˜ β ( t ) = β ( t ) / sin( πα P ( t ) / . (41)If as usually ˜ β ( t ) = ˜ β exp( bt ) then we obtain σ t ( s ) ∝ ln n − s,σ el ( s ) ∝ s (cid:90) −∞ dt | a ( s, t ) | ∝ ln n − s. (42)According to the obvious inequality, σ el ( s ) ≤ σ t ( s ) (43)we have2 n − ≤ n − ⇒ n ≤ . (44)Thus we come to the conclusion that the a model with σ t ( s ) ∝ ln s (n=3) is incompatible with a linear pomerontrajectory . In other words the partial amplitude Eq. (37) with n = 3 is incorrect.If n = 1 we have a simple j -pole leading to constant totalcross section and vanishing at s → ∞ elastic cross section.However such a behaviour of the cross sections is not sup-ported by experimental data.If n = 2 we have the model of dipole pomeron ( σ t ( s ) ∝ ln( s )) and would like to emphasize that double j -pole is themaximal singularity of partial amplitude settled by unitaritybound (43) if its trajectory is linear at t ≈ TOTEM data for the pp total cross section exclude the dipole pomeron model whichis unable to describe with a reasonable χ the high values of σ pptot ( s ) at LHC energies.Thus, constructing the model leading to cross sectionwhich increases faster than ln( s ), we need to consider a morecomplicated case (we consider at the moment a region of small t and j ≈ ϕ + ( j, t ) = β ( j, t ) (cid:2) j − r ( − t ) /µ (cid:3) n ≈ iβ (1 , t ) (cid:2) j − r ( − t ) /µ (cid:3) n . (45)Making use of the same arguments as above, we obtain σ t ( s ) ∝ ln n − s, (46) σ el ( s ) ∝ ln n − − µ s and µ ≥ n − . (47)However in this case amplitude a ( s, t ) has a branch point at t = 0 which is forbidden by analyticity of amplitude a ( s, t ).dderon effects in the differential cross-sections at Tevatron and LHC energies 13A proper form of amplitude leading to t eff decreasingfaster than ln − s (it is necessary for σ t rising faster thanln s ) is the following ϕ + ( j, t ) = β ( j, t )[( j − m − rt ] n . (48)Now we have m branch points colliding at t = 0 in j -planeand creating the pole of order mn at j = 1 (but there is nobranch point in t at t = 0). At the same time t eff ∝ / ln m s and from σ el ∝ ln mn − − m s ≤ σ t ∝ ln mn − s ≤ ln s oneobtains (cid:26) mn ≤ m + 1 ,mn ≤ . (49)If σ el ∝ σ t then n = 1 + 1 /m . Furthermore, if σ t ∝ ln s then m = 1 and n = 2 which corresponds just to the dipolepomeron model. In the Froissaron (or tripole pomeron) model m = 2 and n = 3 /
2. It means that σ t ∝ ln s . A.2 Partial amplitudes
As it follows from Eq.(49) for the dominating at s → ∞ contribution in a Froissaron model with σ t ( s ) ∝ ln ( s ), i.e. n = 2, m = 3 /
2, we have to take (here and in what followswe used a more convenient notations ω = j − ω ± = r ± τ = r ± (cid:112) − t/t , t = 1GeV ). Then ϕ ± ( ω, t ) = η ± ( ω ) β ± ( ω, t )( ω + ω ± ) / = (cid:0) i (cid:1) e − iπω/ ˜ β ± ( ω, t )( ω + ω ± ) / (50)where η ± ( ω ) = 1 ∓ e − iπω sin πω . (51)For even signature˜ β + ( ω, t ) = β + ( ω, t ) / cos( ωπ/
2) (52)and for odd signature˜ β − ( ω, t ) = β − ( ω, t ) / sin( ωπ/ . (53)Now let us suppose that in agreement with the struc-ture of the singularity of φ ± ( ω, t ) at ω + ω ± = 0 thefunctions ˜ β ± ( ω, t ) depend on ω through the variable κ ± =( ω + ω ± ) / and it can be expanded in powers of κ ± φ ± ( ω, t ) = (cid:16) i (cid:17) e − iπω/ ˜ β ± ( t ) + κ ± ˜ β ± ( t ) + κ ± ˜ β ± ( t ) κ ± . (54)There are a different ways to add to partial amplitude ϕ ( j, t ) terms which at s → ∞ are small corrections (they canbe named as subasymptotic terms). t eff can be defined by behaviour of elastic scatteringamplitude at s → ∞ . If a ( s, t ) ≈ sf ( s ) F ( t/t eff ( s )) then σ el ( s ) ∝ | f ( s ) | (cid:82) −∞ dt | F ( t/t eff ) | = t eff | f ( s ) F (1) | . Thus we can expand the “residue” β ( ω, t ) in powers of ω (if β ( ω, t ) has not branch point in ω at ω = 0) or in powersof ( ω + ω ) / . Then, for the first case˜ β ( ω, t ) = ˜ β ( t ) + ω ˜ β ( t ) + ω ˜ β ( t ) , (55)and in the second case we have (just this case is explored inthe Section 4.2)˜ β ( ω, t ) = ˜ β ( t ) + ( ω + ω ) / ˜ β ( t ) + ( ω + ω ) ˜ β ( t ) . (56)Let us notice that the main terms in ϕ ( j, t ) ≡ ϕ ( ω, t )for both cases are coinciding having a pair of branch pointscolliding at ω = 0 ( t = 0) and generating a triple pole inpartial amplitude.Taking into account the table integrals ∞ (cid:90) dxx α − e − ωx J ν ( ω x ) = I αν ( ω, ω ) (57)where I ν +1 ν = (2 ω ) ν √ π Γ ( ν + 1 / ω + ω ) ν +1 / ,I ν +2 ν = 2 ω (2 ω ) ν √ π Γ ( ν + 3 / ω + ω ) ν +3 / , (58)one can find 1( ω + ω ) / = 1 ω ∞ (cid:82) dxxe − xω J ( ω x ) , (cid:82) C dω πi e ξω ( ω + ω ) / = J ( ω ξ ) ω ξ . (59)1 ω + ω = 1 ω ∞ (cid:82) dxe − xω sin( xω ) , (cid:82) C dω πi e ξω ω + ω = sin( ω ξ ) ω ξ . (60)1( ω + ω ) / = ∞ (cid:82) dxe − xω J ( ω x ) , (cid:82) C dω πi e ξω ( ω + ω ) / = J ( ω ξ ) . (61) References
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