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Published by Institute of Physics Publishing for SISSA Received: June Revised: July Accepted: July Published: July 15, 17, 18, 30, 2007 2007 2007 2007 Nonlinearity of Regge trajectories in the scattering region Institute for High Energy Physics, Pobeda str. 1, 142281 Protvino, Russia E-mail: [email protected], [email protected] Abstract: The nonlinearity of Regge trajectories at real negative values of the argument is discussed as their general QCD-inspired property. The processes of elastic diffractive √ scattering p+p → p+p and p̄+p → p̄+p at collision energies s > 23 GeV and transferred momenta squared 0.005 GeV 2 < −t < 3 GeV 2 are considered in the framework of the Regge-eikonal model [3]. By comparison of phenomenological estimates with available experimental data on angular distributions it is demonstrated that in this kinematical range the data can be satisfactorily described as if taking into account only three nonlinear Regge trajectories with vacuum quantum numbers (“soft” pomeron, C-even f2 /a2 -reggeon and C-odd ω/ρ-reggeon). It is also shown that their nonlinearity is essential and not to be ignored. The correspondence of the Kwiecinski q q̄-pole [26] to the secondary reggeons and the relevance of the Kirschner-Lipatov “hard” pomeron pole [25] to elastic diffraction are discussed. Keywords: Phenomenological Models, Nonperturbative Effects. c SISSA 2007 ° http://jhep.sissa.it/archive/papers/jhep072007083 /jhep072007083 .pdf JHEP07(2007)083 Anton A. Godizov and Vladimir A. Petrov Contents 1. Introduction 1 2. The QCD pomeron 4 3. The model 6 4. The fitting results 8 5. Conclusions 12 Our goal is to substantiate that nonlinearity of leading Regge trajectories at negative values of the argument (nonlinearity of trajectories in the resonance energy region is discussed in [1, 2]) following from some general requirements is essential and not to be neglected under considering strong interaction phenomena in the framework of Regge-based models. We will demonstrate this on the example of high energy elastic (anti)pp-scattering and will exploit for this purpose the Regge-eikonal model [3]. We do not intend to compete with [4 – 11] and other authors in getting the lowest value of χ2 /d.o.f. over all available data on angular distributions at high energies. The nonlinearity of the leading trajectory was demonstrated in the experiment on measurement of single diffraction cross-sections [12]. Also, the use of the effective nonlinear dipole pomeron trajectory provided the successful description of existing data on highenergy elastic (anti)pp-scattering [5], on photoproduction of vector mesons [13, 14] and deeply virtual Compton scattering [15]. Note also that authors [4, 16] as well as many others insist on the linearity of some leading Regge trajectories in the Euclidean domain at least at −2 GeV 2 < t < 0. The first phenomenological argument is a natural desire to continue Chew-Frautschi plots to the region of negative values of the argument. But all poles corresponding to resonances are situated on different unphysical sheets since they possess not only the mass but also the nonzero width. Hence, appropriate points on any Chew-Frautschi plot pertain to different branches of the corresponding analytic function and straight continuation to the region below the lowest threshold is not correct from the analytical point of view (although it can be used, for example, for rough estimation of the intercept value). We have to conclude that hadron spectroscopy does not provide absolutely reliable grounds for determining the behavior of Regge trajectories in the Euclidean domain. The second “evidence” of the linearity originates from the data on exchange processes π − +p → π 0 +n and π − +p → η+n in –1– JHEP07(2007)083 1. Introduction 1.3 rHtL 1.2 1.1 -t,GeV 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 0.8 0.7 0.6 Figure 1: Function r(t) extracted from the data on exchange processes π − + p → π 0 + n and π − + p → η + n at collision energies 11.0 and 19.4 GeV in the Born approximation. √ √ For any reaction and any different collision energies s1 and s2 the function α(t) must be the same (if the Born approximation works) and, hence, the function r(t) ≡ (1 − 0 0 απ +n (t))/(1 − αη+n (t)) (where απ +n (t) and αη+n (t) are the trajectories from angular distributions for prosesses π − + p → π 0 + n and π − + p → η + n correspondingly at collision √ √ energies s1 = 11.0 GeV and s2 = 19.4 GeV ) must be strictly equal to unity at any argument value. However, this is not evident after extraction this function from the data on exchange reactions (figure 1). The only conclusion which an unprejudiced person can draw looking at this picture is that the Born approximation seems invalid at transferred momenta −t > 0.2 GeV 2 and, consequently, the linearity of the ρ/a2 -trajectory at higher scattering angles is not guaranteed. So, linear parametrizations for Regge trajectories have no phenomenological advantages over nonlinear ones. Also note that the best fit to the trajectory extracted from the earlier data on π − + p → η + n [19] under exploiting the Born approximation was obtained for nonlinear parametrization [20, 21] α(t) = α(∞) + [α(0) − α(∞)]2 α(0) − α(∞) − tα′ (0) (1.2) with α(∞) = 0. Our viewpoint is based on the conviction that QCD is the fundamental theory of strong interaction. This rather general requirement imposes restrictions on the behavior of Regge trajectories in the range of the perturbative QCD validity since at large transferred –2– JHEP07(2007)083 the Born approximation [17]. Since within this approach only two of the leading reggeons (ρ and a2 ) give contribution to the amplitude and we can consider these trajectories approximately equal (as a consequence of the weak degeneracy following from the hadron spectroscopy data) one could try to extract them directly from the corresponding highenergy angular distributions [18]. The appropriate formula is ! ¯ ¯ ¶ Ã µ dσ ¯¯ dσ ¯¯ 1 s1 −1 − ln . (1.1) ln ln α(t) = 1 + 2 s2 dt ¯s=s1 dt ¯s=s2 momenta exchanges by single reggeons must turn into exchanges by colour-singlet parton combinations. If we assume that the reggeon exchange giving the leading contribution to the eikonal at high energies (pomeron) turns into some multi-gluon exchange in the perturbative range we will come to [22 – 25] lim αgg...g (t) = 1 . t→−∞ (1.3) In the case of the quark-antiquark pair (f2 -reggeon, ω-reggeon etc.) one obtains [26] r √ √ 8 αs ( −t) + o(α1/2 αq̄q (t) = (1.4) s ( −t)) 3π where Φ(x) is an arbitrary function analytic in the region defined by the analyticity in t, s (µ) −1 . For any quantum field µ is the renormalization scale, K ′ (αs ) = 1/β(αs ) ≡ (µ2 ∂α∂µ 2 ) √ model with asymptotic freedom (1.5) leads in the perturbative sector (µ = −t, t → −∞) to lim f (t) = const . (1.6) t→−∞ √ Actually, this corresponds to the existence of the free field limit as αs ( −t) → 0. In the theory of potential scattering the squared effective radius of interaction corresponding to the reggeon exchange [20, 29] R2 ∼ α′ (t)(2α(t) + 1) → 0 at t → −∞, α(t) → const. This purely quantum mechanical result is in agreement with the property of asymptotic freedom at short distances (in the case of linear trajectory α(t) (α′ (t) > 0) R2 → −∞ at t → −∞). As a consequence of (1.6) the essential nonlinearity of the Regge trajectories takes place in the range −∞ < t < 0. This nonlinearity is their fundamental property. Besides, we assume that Im α(t + i0) ≥ 0 increases slowly enough at t → +∞ (for example, not faster than Ct ln−1−ǫ t, ǫ > 0) so that the dispersion relations with not more than one subtraction take place, i.e. Z t +∞ Im α(t′ + i0) ′ dt , (1.7) α(t) = α0 + π tT t′ (t′ − t) 1 The universality of the asymptotic vanishing at t = −∞ of meson trajectories seems to contradict the existence of (pseudo)scalars. In fact, if απ (t) behaved like (1.4) at t → −∞ it could not be monotonic function of t. Possible way to preserve the monotony is to suppose that at t → −∞ the trajectory tends to some negative value. Similar behavior takes place, e.g., in the phenomenological studies of the trajectories containing heavy quarkonia [27]. To what extent is it possible in QCD remains unclear. –3– JHEP07(2007)083 where αs (µ) ≡ gs2 (µ)/4π is the running coupling.1 This property of Regge poles (tending to constant at t → −∞) is quite general and follows from the fact of their invariance relative to the renormalization group transformations (the requirement of renorm-invariance is well-grounded by observability of bound states and resonances). The general solution of the renorm-group differential equation for the renorm-invariant quantity f ( µt2 , αs (µ)) is (in the case of massless fields) of the form [28] µ ¶ µ ¶ t t K(αs (µ)) f , αs (µ) = Φ e (1.5) µ2 µ2 and Im α(t + i0) ≥ 0 at t ≥ tT > 0 (we would like to point out that these assumptions are strictly fulfilled in the theory of perturbations and the theory of potential scattering [30]). In this case we obtain [20, 30] dn α(t) > 0 (t < tT , n = 1, 2, 3, . . .). dtn (1.8) The t-channel unitarity implies some special t-dependence of the trajectories near the threshold [30] Im α(t) ∼ (t − tT )α(tT )+1/2 (t ≥ tT ) . 2. The QCD pomeron Regge trajectories which are tightly related to the hadron spectroscopy deal, generally, with confinement of quarks and gluons. Thus, without a great progress in the solution of this outstanding problem the QCD theory of Regge trajectories is still in its primordial stage. It is natural that in the absence of a regular non-perturbative technique some progress is limited by perturbative calculations. In the literature the wide attention is being paid to so-called “BFKL pomeron” (or other “BFKL reggeons”) [33]. In this approach one strives to formulate some kind of Bethe-Salpeter equation for the gluon-gluon scattering amplitude (Green function) in which “t-channel” gluons are actually gluon Regge trajectories (“reggeized gluons”) that have to be preliminary calculated from another Bethe-Salpeter –4– JHEP07(2007)083 As was argued in [31] this must have observable effects. However, in this paper we do not take this requirement into account, considering such effects as a kind of “fine structure” which is beyond the accuracy level we adopted. It is shown in [32] that the assumption of the linearity of Regge trajectories leads to the situation when the diffractive pattern at high energy (anti)proton-proton elastic scattering can be described in the framework of the Regge-eikonal approach only after introduction of several vacuum reggeons giving contributions to the eikonal (one needs not less than three C-even poles with intercepts higher than unity). In other Regge-based models one also need more than one reggeon with intercept higher than unity (see, for example, [5, 6, 11]). The use of nonlinear trajectories allows us to curtail the number of reggeons essential for an acceptable description of the data on high energy elastic nucleonnucleon scattering. Namely, using only one trajectory with vacuum quantum numbers and intercept higher than unity (“soft” pomeron) and two secondary reggeons with intercepts lower than unity (C-even f2 /a2 -reggeon and C-odd ω/ρ-reggeon) we will be able to describe √ available experimental data in a wide kinematical range s > 23 GeV , 0.005 GeV 2 < −t < 3 GeV 2 (at −t > 3 GeV 2 we can not ignore the contributions from other leading vacuum reggeons — C-even “hard” pomeron(s) and C-odd odderon(s)). The very fact that we have managed to satisfactorily reproduce the diffractive pattern within this rather wide kinematical range in the framework of such a simple phenomenological scheme is quite encouraging and points to the agreement between general QCD-related theoretical conclusions and experiment. This expression cannot be accepted as a true value of the pomeron intercept as it depends on the renormalization scheme and arbitrary renormalizaton scale µ via αs . As was argued recently [36] within the limit of massless quark fields the true intercept of any Regge trajectory has to be strictly independent of the QCD coupling constant. These curcumstances enforce us (while waiting for further theoretical progress) to try some purely phenomenological ansatz for the pomeron trajectory which retains but one feature of (2.1): it tends to 1 at |t| high enough (see relation (1.3)). –5– JHEP07(2007)083 equation for the colour-octet t-channel. This trick can make an impression that, say, the pomeron trajectory is not just a 2-gluon exchange (with account of interaction between these two gluons via “ladder rung exchanges”) but contains many-gluon configurations in the t-channel as well. However, the very “reggeized gluons” are related to no more than two gluons each (since they result from the the BS-type equation with 2-gluon irreducible kernels). That is one deals with no more than 4 usual gluons in the t-channel. The use of fixed number of gluon exchanges seems to be justified at |t| large enough, where the elementary short-distance structure of reggeons has to show up. As to small values of t it is clear that due to genuinely strong interaction one cannot limit the problem by any fixed number of exchanged partons. Certainly, one can argue about “valence” gluons but these have to be essentially non-perturbative and different from the “slightly reggeized” gluons mentioned above. It is even quite probable that the corpuscular language in this quasi-classical region seases to be adequate and undulatory gluonic fields are more relevant. In spite of such a little bit gloomy landscape one still can try to account (when describing the scattering data in terms of Regge exchanges) for such a distinctive QCD prediction as asymptotic constancy of Regge trajectories in deeply Euclidean region, −t → ∞ (in other words, the nonlinearity of the pomeron and secondary reggeon trajectories which was already mentioned in the previous section). Via solving the Bethe-Salpeter-like equation for the gluon-gluon scattering amplitude it was obtained in [25] that " ¶1/3 µ ¶2/3 # µ √ √ 3/4 7ζ(3) 12 ln 2 , t → −∞ . αs ( −t) 1 − α2/3 αhard (t) = 1 + P s ( −t) π 2 ln 2 11−2/3 nf (2.1) 2 The second term in brackets is approximately equal to 0.09 at t = −MZ when αs (MZ ) ≈ 0.118 (here MZ ≈ 91.2 GeV is the Z-boson mass and nf = 5) and such an expansion is justified at this scale (at t = −(2.5 GeV )2 , αs (2.5 GeV ) ≈ 0.3 this term is about 0.17). Even at t = −MZ2 the value of αhard (−MZ2 ) ≈ 1.28 is quite high (αhard (−6 GeV 2 ) ≈ P P 1.66). If one assumes the monotony of αhard (t) the intercept, αhard (0), has to lie even P P higher. The last statement agrees with a rough estimate of the lower bound for the BFKL pomeron intercept value αhard (0) − 1 ≥ 0.3 obtained in [34]. Such a value can be hardly P relevant for the data on elastic diffraction. Some time ago a NLO result was obtained for the pomeron intercept [35] µ ¶ 12 ln 2 20 αP (0) = 1 + αs (µ) 1 − αs (µ) . (2.2) π π ΑPHtL 3 2.5 2 1.5 1 0.5 -3 -2 -1 1 2 3 t Figure 2: The qualitative behavior of function αP (t) from (3.2). 3. The model ¶ µ ¶αP (t) s π(αP (t) − 1) + (3.1) βP (t) δ(s, t) = δP (s, t) + δ+ (s, t) ∓ δ− (s, t) = i+tg 2 s0 µ ¶ µ ¶α+ (t) µ ¶ µ ¶α− (t) π(α+ (t)−1) π(α− (t)−1) s s + i + tg ∓ i−ctg β+ (t) β− (t) 2 s0 2 s0 µ where s0 ≡ 1 GeV 2 , αP (t) and βP (t) are the trajectory and the residue of the “soft” pomeron, α+ (t) and α− (t) are the trajectories of secondary reggeons (i.e. we assume that α+ (t) ≈ αf2 (t) ≈ αa2 (t) and α− (t) ≈ αω (t) ≈ αρ (t) due to the isospin symmetry of the quark flavors), β+ (t) ≡ βf2 (t) + βa2 (t) and β− (t) ≡ βω (t) + βρ (t), the sign “–” (“+”) before δ− (t) corresponds to the particle-on-particle (particle-on-antiparticle) scattering. Our phenomenological ansatz for the “soft” pomeron pole function is (see figure 2) h ³ π ´i . (3.2) αP (t) = 1 + p1 1 − p2 t arctg(p3 − p2 t) − 2 It bears the above-mentioned characteristic feature, i.e. tends to 1 at t → −∞. At large and positive t (3.2) mimicrizes the “stringy” behavior, i.e. grows linearly with t. We do not give a great significance to this, though. Moreover, such a behavior, if taken seriously, leads to complex singularities of αP (t) and one has to provide a special care to save the amplitude from causality (analyticity) violations [37].2 2 Earlier, in [38] there was proposed a similar form for the pomeron trajectory αP (t) = 1 + arctg st1 . This function also flattens at t → −∞ but in [38] s1 was chosen large enough for the linear approximation was valid in the diffraction region. As a consequence, limt→−∞ αP (t) = 1 − α′P (0)s1 π2 < 0. So, our approach (3.2) is quite different because (as it will be shown below) asymptotic relation (1.3) together with the requirement of the monotony of αP (t) results in a strong nonlinearity of the pomeron trajectory. α′P (0)s1 –6– JHEP07(2007)083 Since our primary goal is not to get the best phenomenological description of all available data on elastic diffraction at high energies but to find a phenomenological confirmation of such a macroscopic phenomenon as following from QCD nonlinear behavior of Regge trajectories in the scattering region we choose the eikonal to have the simplest form (ignoring the contribution of “hard” pomeron(s), odderon(s) etc.) Residues are chosen as βP (t) = BP ebP t (1+d1 t+d2 t2 +d3 t3 +d4 t4 ) , β+ (t) = B+ eb+ t , β− (t) = B− eb− t . (3.3) Our parametrization for the secondary trajectories contains the QCD-inspired expressions α+ (t) = where µ ¶1/2 8 p γ( −t + c+ ) , 3π 4π γ(µ) ≡ 11 − 23 nf α− (t) = Ã 1 2 µ ln Λ 2 + µ ¶1/2 8 p γ( −t + c− ) 3π 1 1− µ2 Λ2 ! (3.4) (3.5) to the coordinate representation, using the eikonal representation of the scattering amplitude e2iδ(s,b) − 1 (3.7) T (s, b) = 2i through inverse Fourier-Bessel transformation Z ∞ √ T (s, t) = 4πs db2 J0 (b −t)T (s, b) (3.8) 0 obtain its value in the momentum representation (during numerical calculating integrals from (3.6), (3.8) we change upper limits of integration to 4 GeV 2 and 400 GeV −2 ≈ (4 F m)2 correspondingly) and substitute it into the expression for the differential cross-section dσ |T (s, t)|2 = . dt 16πs2 (3.9) In the conclusion of this section we must note that our consideration of the high-|t| behavior provides the explanation of their nonlinearity in the framework of QCD but we –7– JHEP07(2007)083 is the so-called one-loop analytical QCD effective coupling constant [39], nf = 3 is the number of quark flavors taken into account, Λ = Λ(3) = 0.346 GeV is the QCD dimensional parameter (the value was taken from [40]) and c+ , c− > 0 are free phenomenological parameters. This analytic approximation for the secondary reggeons is obtained in the following way. We take an expression for the q q̄ Regge pole in the perturbative sector [26] derived via solving the Bethe-Salpeter-like equation in the range of the perturbative QCD validity and then replace the perturbative effective coupling constant to expression (3.5) obtained in the framework of the dispersive approach [39]. Further, the introduction of the free parameters c+ and c− is the simplest phenomenological way to take into account the disparity between α+ (t) and α− (t) and not to spoil the asymptotic behavior of these trajectories in the perturbative sector (−t ≫ 1 GeV 2 ). To obtain angular distributions we substitute (3.2), (3.3), (3.4) into (3.1), proceed via Fourier-Bessel transformation Z ∞ √ 1 d(−t)J0 (b −t)δ(s, t) δ(s, b) = (3.6) 16πs 0 p1 p2 p3 BP bP d1 d2 d3 d4 αP (0) α′P (0) pomeron 0.123 1.58 GeV −2 0.15 43.5 2.4 GeV −2 0.43 GeV −2 0.39 GeV −4 0.051 GeV −6 0.035 GeV −8 1.123 0.28 GeV −2 c+ f2 /a2 -reggeon 0.1 GeV 2 c− ω/ρ-reggeon 0.9 GeV 2 B+ b+ 153 4.7 GeV −2 B− b− 46 5.6 GeV −2 α+ (0) α′+ (0) 0.78 0.63 GeV −2 α− (0) α′− (0) 0.64 0.07 GeV −2 do not intend to say that our approach is capable to give the description of cross-sections at fixed angles. Our concrete parametrizations for the unknown functions αP (t), α+ (t), α− (t), βP (t), β+ (t), β− (t) in the eikonal (3.1) are purely phenomenological quantitative approximations valid in the soft diffraction region (only relations (1.3), (1.4), (1.8) have physical sense). Nonetheless, the QCD asymptotics of the trajectories (tending to constants at t → −∞) are fairly compatible with, e.g. “quark counting rules” [24]. The dual amplitude with Mandelstam analyticity model (DAMA) with logarithmic trajectories [41] provides an alternative way to reproduce the QCD-type behavior at fixed angles. 4. The fitting results Turn to the description of experimental data. The results of fitting over data on angular √ distributions in the kinematical region s > 23 GeV , 0.005 GeV 2 < −t < 3 GeV 2 [42]3 are represented in table 1, 2 and figure 3, 4. If we compare angular distributions obtained using parametrization (3.2), (3.3), (3.4) with those obtained using the same values of free parameters but with replacement of the nonlinear pomeron trajectory to its linear approximation (only two first terms in the Taylor expansion) we will disclose a huge difference between the corresponding results (see figure 3, 4) which is a consequence of the fact that the simultaneous fulfilment of the conditions α′P (t) > 0 at t ≤ 0, limt→−∞ αP (t) = 1 together with the phenomenological estimates α′P (0) > 0.2 GeV −2 , αP (0) < 0.15 makes the approximation αP (t) = αP (0) + α′P (0)t for the pomeron trajectory invalid in the region −t > 0.8 GeV 2 , i.e. the effect of the nonlinearity turns out strong even at low −t (see figure 5, 6). Here we must emphasize that the linear approximations to our nonlinear trajectories differ from linear trajectories used by other authors. In the most of papers quoted in table 3 linear Regge trajectories were 3 For calculation of electromagnetic correstions to the scattering amplitude we used the recipe by R. Cahn [43] (see also [44]). –8– JHEP07(2007)083 Table 1: Parameters obtained by fitting to the data. dΣ , mbGeV-2 dt 1. ´ 1012 p p ® p p 1. ´ 108 31 GeVH1010L 53 GeVH108L 10000 546 GeVH104L 1 630 GeVH102L 0.0001 1800 GeV -t, GeV2 0 0.5 1 1.5 2 2.5 3 Figure 3: Differential p̄ p → p̄ p cross-sections for the cases of nonlinear (solid lines) and linearly approximated (dotted lines) “soft” pomeron trajectory. successfully applied to the phenomenological description of the data but QCD asymptotic relations (1.3), (1.4) were ignored. In figure 7 the predictions for the total cross-section dependence on the center-of-mass energy are shown. In particular, σtot (200 GeV ) ≈ 52 mb, σtot (14 T eV ) ≈ 111 mb. The √ noticeable disagreement with the experimental p̄ p data at s < 13 GeV points to the fact that at such energies the simplest phenomenological scheme (3.1) is inapplicable for satisfactory quantitative description of the data and the contribution of f2′ - and φ-reggeons must be taken into account. Since the corresponding terms in the imaginary part of the eikonal have equal (opposite) signs for the p̄ p (p p) scattering the discrepancy between the model curve and the p̄ p data turns out larger than for the p p data. Also, in this figure by comparison of the imaginary part of the forward amplitude with the same from the Born approximation it is demonstrated that we can not neglect absorptive corrections which significantly reduce the amplitude value. Figure 8 represents the dependence of ρ-parameter on the center-of-mass energy. In the last two figures the “soft” pomeron and secondary trajectories and their slopes as functions of the transferred momentum squared are presented. Point out that although –9– JHEP07(2007)083 62 GeVH106L dΣ , mbGeV-2 dt 1. ´ 1013 p p ® p p 1. ´ 109 23.5 GeVH1010L 100000. 30.7 GeVH108L 10 62.5 GeVH104L 200 GeVH102L 0.001 14000 GeV -t, GeV2 0 0.5 1 1.5 2 2.5 3 Figure 4: Differential p p → p p cross-sections for the cases of nonlinear (solid lines) and linearly approximated (dotted lines) “soft” pomeron trajectory. Set of data √ s = 23 GeV (p p) √ s = 31 GeV (p p) √ s = 53 GeV (p p) √ s = 62 GeV (p p) √ s = 31 GeV (p̄ p) √ s = 53 GeV (p̄ p) √ s = 62 GeV (p̄ p) √ s = 546 GeV (p̄ p) √ s = 630 GeV (p̄ p) √ s = 1800 GeV (p̄ p) Total Number of points 124 154 85 107 38 60 40 181 19 50 858 χ2 280 467 423 409 108 336 156 352 78 129 2738 Table 2: The quality of description of data on angular distributions. – 10 – JHEP07(2007)083 53 GeVH106L ΑP HtL 1 0.8 Α+ HtL Α- HtL 0.4 t, GeV2 -2 -1.5 -1 -0.5 0 Figure 5: Approximate “soft” pomeron and secondary trajectories obtained in the fitting procedure lin (dashed lines: αlin f (t) = 0.69 + 0.81 t and αω (t) = 0.44 + 0.92 t are the continuations of the ChewFrautschi plots corresponding to f2 -reggeon and ω-reggeon and αlin P (t) = 1.1 + 0.25 t is the linear “soft” pomeron trajectory usually used in literature). Ref. [4] [5] [6] [7] [8] [9] [10] [11] [32] χ2 /d.o.f. Not presented 2.0 2.4 Not presented 4.3 Not presented 2.8 1.5 2.6 kinematical range √ 23 GeV ≤ s ≤ 546 GeV √ 53 GeV ≤ s ≤ 630 GeV , 0 < −t ≤ 5 GeV 2 √ 19 GeV ≤ s ≤ 1800 GeV , 0.1 GeV 2 ≤ −t ≤ 14 GeV 2 √ 23 GeV ≤ s ≤ 546 GeV √ 23 GeV ≤ s ≤ 1800 GeV , 0 < −t ≤ 6 GeV 2 √ 546 GeV ≤ s ≤ 1800 GeV √ 23 GeV ≤ s ≤ 1800 GeV , 0.01 GeV 2 ≤ −t ≤ 14 GeV 2 √ 6 GeV ≤ s ≤ 1800 GeV , 0.1 GeV 2 ≤ −t ≤ 6 GeV 2 √ 23 GeV ≤ s ≤ 1800 GeV , 0.01 GeV 2 ≤ −t ≤ 14 GeV 2 Table 3: Some information on χ2 /d.o.f. over the data on elastic nucleon-nucleon scattering obtained by other authors. – 11 – JHEP07(2007)083 0.6 0.6 0.5 0.4 aP HtL 0.2 0.1 a- HtL t, GeV2 -1 -0.8 -0.6 -0.4 -0.2 0 Figure 6: The slopes (in GeV −2 ) of the Regge trajectories from the previous figure, aR (t) ≡ α′R (t). the intercepts of secondary reggeons in our scheme are higher than those ones from the rough linear approximation based on the appropriate resonance data the corresponding slopes turn out noticeably smaller than in the linear scheme. So, our approximations (3.4) to the true secondary trajectories in the scattering region may be laced in a smooth and monotonous manner with the corresponding Chew-Frautschi plots in the resonance region without any artificial behavior at low positive values of the argument (see figure 5). 5. Conclusions Now we can discuss the correspondence of the Kwiecinski q q̄-pole [26] to the secondary reggeons and the relevance of the Kirschner-Lipatov “hard” pomeron pole [25] to elastic diffraction at accessible energies and small scattering angles. It is evident that the Kwiecinski asymptotic form (1.4) continued analytically in some way to the non-perturbative sector corresponds to the phenomenological secondary Regge trajectories giving a noticeable contribution to the observed elastic diffraction cross-sections. In other words, there exists a simple analytical way to connect their asymptotic perturbative behavior (1.4) with the Regge phenomenology in the non-perturbative sector. – 12 – JHEP07(2007)083 a+ HtL 0.3 200 Σtot, mb 150 100 50 p p p p !!! s , GeV 30 10 100 1000 10000 Figure 7: Total cross-sections (the eikonalized amplitudes — solid lines, the Born amplitudes — dashed lines) for high-energy nucleon-nucleon scattering as functions of center-of-mass energy (experimental data were taken from Particle Physics Data System http://wwwppds.ihep.su:8001/ppds.html). With the Kirschner-Lipatov “hard” pomeron the situation is quite different (see section “The QCD pomeron”). Trying to apply the Kirschner-Lipatov pole to phenomenology we have an alternative: either to insist on the correspondence between the “soft” pomeron trajectory and the Kirschner-Lipatov pole and so to accept that the trajectory is not monotonous in the Euclidean domain (in this case we come to the situation when we can not exploit its expression in form (2.1) for soft diffraction at any energies) or to assume that these poles are different ones (and so to presume that the Kirschner-Lipatov pole contribution to the eikonal is suppressed in the residue in the non-perturbative range of the argument value — it must dominate in the diffraction sector at ultra-high energies (higher than 1.8 T eV ) and also in the perturbative sector at accessible energies). The latter variant is more preferable from the analytical point of view but in both cases we are not able to use (2.1) in the Regge phenomenology of diffraction phenomena at accessible energies. As to (2.2) it gives even less reasons to use it, as explained above. The following picture seems to us reasonable. At “very high” (−t) we deal with pure gluon exchanges, then, – 13 – JHEP07(2007)083 70 0.2 Re T Hs, 0L Ρ = Im T Hs, 0L 0.1 p p 0 p p -0.1 -0.3 !!! s , GeV 10 100 1000 10000 Figure 8: The forward amplitude real part to imaginary part ratio as a function of center-of-mass energy (experimental data were taken from Particle Physics Data System http://wwwppds.ihep.su:8001/ppds.html). with (−t) diminishing, we come to “partially collectivized” exchanges in the form of “hard pomerons” (2.1). The latter could be seen in gluon-gluon elastic scattering with colorless exchanges (“Mueller-Navelet jets”) with high (−t). The hadron diffraction at (−t) ≤ 3 GeV 2 is dominated by “soft” or non-perturbative pomerons (reggeons) which cannot be thought as composed of definite number of partons. Unfortunately, QCD-literature has no much to say in this case. So, in the framework of the minimal Regge-eikonal model with using those parametrizations for Regge trajectories in which their asymptotic properties and fundamental nonlinearity are taken into account (1) it was shown that the diffractive pattern for the elastic p̄ p and p p scattering at √ energies 23 GeV < s < 2 T eV and scattering angles −t < 3 GeV 2 is mainly formed by contribution of three Regge trajectories, (2) it was demonstrated (from both theoretical and phenomenological points of view) that we can not ignore the nonlinearity of the trajectories in the considered kinematical range, – 14 – JHEP07(2007)083 -0.2 (3) the relevance of the Kwiecinski q q̄-pole to the secondary reggeons and the impossibility to apply the Kirschner-Lipatov trajectory to elastic diffraction at accessible energies was argued. Acknowledgments The authors are indebted to V.V. Ezhela, A.V. Prokudin, R.A. Ryutin, S.M. Troshin and A.K. Likhoded for helpful discussions and useful criticism. 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