Effective transverse radius of nucleon in high-energy elastic diffractive scattering

High-energy elastic diffraction of nucleons is considered in the framework of the simplest Regge-eikonal approximation. It is demonstrated explicitly that the effective transverse radius of nucleon in this nonperturbative regime is $\sim 0.2\div 0.3$ fm and much less than the transverse size of the diffractive interaction region.


Introduction
Elastic diffractive scattering of hadrons is one of the most interesting and important areas of high-energy hadron physics: in pp collisions the fraction of elastic diffraction events in the total number of events is very high (from more than 15% at the ISR to about 25% at the LHC). However, the main problem related to this sector of strong interaction physics is the fact that the characteristic distances for diffractive interaction of hadrons are of order 1 fm, so that perturbative QCD is inapplicable. Hence, one has to search for some approaches not related to pQCD directly, to provide at least qualitative description of the corresponding high-energy observables.
One of the most natural theoretical frameworks which helps to deal with the nonperturbative sector of hadron physics is Regge theory wherein interaction of hadrons is described in terms of exchanges by reggeons (off-mass-shell and off-spin-shell composite particles). In paper [1] a simple Regge-eikonal model for high-energy elastic diffraction of nucleons was examined. In this model, the standard eikonal representation of the nonflip scattering amplitude [2], (here s and t are the Mandelstam variables and b is the impact parameter), is exploited together with the single-reggeon-exchange approximation to the eikonal (Born amplitude) in the kinematic range s ≫ {m 2 p , |t|}: where s 0 = 1 GeV 2 and α P (t) is the Regge trajectory of pomeron (a C-even reggeon which absolutely dominates over other reggeons at the SPS, Tevatron, and LHC energies). More detailes, regarding the Regge-eikonal approach, can be found in [2] or, partly, in the Appendix. In [1] the unknown functions α P (t) and Γ P (t) are considered independent and treated with the help of some simple test parametrizations which are fitted to the data. Formally, the used parametrizations have provided a satisfactory description of the available experimental data at √ s > 500 GeV and −t < 2 GeV 2 . However, a question emerges about possible correlation between the behavior of α P (t) and Γ P (t) at low t, since both the functions have a strong impact on the t-behavior of the scattering amplitude. Such a correlation exists and can be taken into account explicitly. As a consequence, it becomes possible to extract from the data a valuable information concerning the effective transverse size of nucleon in the nonperturbative regime of high-energy diffractive scattering.

Structure of the pomeron Regge residue
First of all, we should note that in the framework of the Regge-eikonal approach the pomeron exchange contribution into the eikonal of nucleon-nucleon elastic scattering appears as where α ′ P (t) originates from the pomeron propagator (see the Appendix) and, in general, the factor 2 −α P (t) πα ′ P (t) is not related to the pomeron-nucleon coupling. In literature this factor is usually included into the corresponding Regge residue [2]: g 2 P (t)2 −α P (t) πα ′ P (t) ≡ Γ P 2 (t). If to consider g P (t) as a nontrivial unknown function of t, then expressions (2) and (3) are equivalent from the standpoint of description of data. However, the replacement g P (t) → Γ P (t) degrades the physical transparency of the model since the shortened form (2) ignores the evident correlation between the behavior of the pomeron Regge residue and the pomeron Regge trajectory. Moreover, if the t-dependence of g P (t) is weak at low t, then usage of (2) instead of (3) leads to loss of physical information.
As the quantity g P (0) can be considered as an effective "pomeron charge", so the ratio g P (t)/g P (0) should be considered as the transverse "pomeron form factor" of nucleon. Hence, possible weak t-dependence of this quantity at low t could be interpreted as the effective transverse (quasi-)pointlikeness of nucleon in the high-energy diffractive scattering regime.
As well, let us remind that although QCD itself does not predict the behavior of α P (t) in the diffraction domain (0 < −t < 2 GeV 2 ), this function is expected to satisfy the following conditions [2,3]: The first condition originates from the dispersion relations for Regge trajectories (if not more than one subtraction is needed), and the second one follows from the natural presumption that at high values of the transferred momentum the exchange by pomeron turns, due to asymptotic freedom, into the exchange by 2 noninteracting gluons which can be considered in the same way as the exchange by 2 photons. At high energies such Born amplitudes behave as ∼ s 1 [4]. Thus, if the effective transverse size of nucleon is small, then, in view of restrictions (4), the t-dependence of eikonal (3) is determined mainly by the Herglotz function α P (t). Consequently, the quality of description of the differential cross-section dσ dt = |T (s,t)| 2 16πs 2 becomes extremely sensitive to the quantitative behavior of α P (t) at low negative t. Such a sensitivity implies that the procedure of fitting α P (t) and g P (0) to experimental angular distributions in a wide enough kinematic range could be considered as implicit extraction of these quantities from the data.

Fitting to the experimental data
Let us reconsider the model from [1], having singled out the factor 2 −α P (t) πα ′ P (t) in the Regge residue, as in (3): The results of fitting α P (t) and g P (t) to the experimental differential cross-sections at √ s > 500 GeV and 0.005 GeV 2 < −t < 2 GeV 2 [5] are presented in Tabs. 1, 2, and 3 and Fig. 1.
The deviation of the model predictions from the pp elastic scattering data in the dip region at √ s = 62.5 GeV [6] can be explained by the noticeable contribution of secondary reggeons into the real part of the eikonal. Detailed discussion of this matter can be found in [1].  The D0 data [7] were not included into the fitting procedure, since they have a normalization uncertainty about 14.4%. If to multiply them by factor 0.92, the description quality becomes much better (see Tab. 4). The same can be said regarding the unrenormalized CDF data at √ s = 1800 GeV [8] which are inconsistent with the E-710 data.

Discussion
Now let us analyze the produced outcomes.
The description of the nucleon-nucleon diffractive pattern in the considered kinematic range is satisfactory. The replacement a g = 0.23 GeV −2 → a g = 0 which implies neglection of the nucleon size disfigures the differential cross-sections (see the dashed lines in Fig. 1). However, this distortion decreases with energy and becomes not catastrofic already at the LHC energies. 108.6 ± 6.9 28.2 ± 2.5 22.5 ± 1.8 14000 109.9 ± 7.1 28.6 ± 2.5 22.6 ± 1.8 32000 124.9 ± 9.7 34.1 ± 3.7 24.2 ± 1.9 100000 148.0 ± 14.1 42.8 ± 5.6 26.6 ± 2.2 Table 3: Predictions for the pp total and elastic cross-sections and the forward logarithmic slope of the corresponding differential cross-sections.  Table 4: The quality of description of the data not included into the fitting procedure.
It is a consequence of the fact that although we did not fix the nucleon transverse size, it has turned out rather small. Indeed, the form factor g P (t)/g P (0) = (1 − a g t) −2 corresponds to a certain effective transverse distribution in the impact parameter representation: f (b) = (4πa 3 g ) −1 bK 1 (b/a g ), where K 1 (x) is the modified Bessel function. The effective transverse size of nucleon obtained via average over this distribution, √ < b 2 > ∼ 0.3 fm, is noticeably smaller than the effective transverse size of the diffractive interaction region (in the considered interval of the collision energy √ 2B > 1 fm, B is the forward logarithmic slope of dσ/dt which increases with energy). Such a difference could be interpreted as if pomeron was coupled to a very small zone inside nucleon. Hence, we come to the main conclusion: • The quantitative evolution of the nucleon-nucleon elastic diffractive scattering observables at ultrahigh energies is determined mainly by the behavior of the pomeron Regge trajectory α P (t) and the value of the effective "pomeron charge" g P (0) of nucleon and rather weakly depends on the nucleon shape.
The examined single-reggeon-exchange eikonal approximation is expected to be valid at least in the interval 0.2 TeV ≤ √ s ≤ 14 TeV. Therefore, the forthcoming TOTEM measurements (as well as desirable analogous measurements at the RHIC) could decrease the uncertainties of the model degrees of freedom and, thus, improve the phenomenological estimations as for the nucleon effective transverse size, so for the pomeron Regge trajectory and the pomeron charge of nucleon.

Appendix. Regge approximation for Born amplitude
The eikonal representation (1) itself implies just a replacement of the unknown function of two variables, T (s, t), by another one, δ(s, t).
The key assumption is that the eikonal is proportional (with high accuracy) to some effective relativistic quasi-potential of two-hadron interaction. According to the Van Hove interpretation of such a quasi-potential as the "sum" over all single-meson exchanges in the t-channel [9], the eikonal can be represented as is the propagator of spin-j meson particle, J (f,j,m j ) α 1 ...α j are the corresponding meson currents of interacting hadrons (index f denotes the sort of the hadron), ∆ is the transferred 4-momentum, p 1 and p 2 are the 4-momenta of the incoming particles, symbol m j denotes the summing over all of spin-j mesons with different masses (which, in what further, will be transformed into the summing over reggeons).
Obviously, in the kinematic range (p 1 + p 2 ) 2 ≡ s ≫ {p 2 1,2 , ∆ 2 , (p 1,2 ∆)} the eikonal can be approximated by the following expression: , and F (f,j,m j ) is the structure function at the tensor structure p α 1 ...p α j in the current J (f,j,m j ) α 1 ...α j (p, ∆). Certainly, the symmetric tensors J (f,j,m j ) α 1 ...α j should be transverse with respect to ∆ α k and traceless, for the unambiguous interpretation of j as the massive meson spin, though these transversality and tracelessness are not used in the leading approximation. Now let us introduce the single-meson-exchange amplitudes of definite signature: If m 2 j and h (j,m j ) (t) at even and odd j are the values of some analytic functions which are holomorphic at Re j > − 1 2 and behave as O(e k|j| ), k < π, at j → ∞, then, under the Carlson theorem [10], the unilocal analytic continuation of (A.3) into the region of complex j is possible (the Regge hypothesis [2]). We denote these functions by m 2 ± (j) and h ± (t, j, m 2 ± (j)), respectively. Via the Sommerfeld-Watson transform [2,11], we replace the sum over j in (A.3) by the integral over the contour C encircling the real positive half-axis on the complex j-plane, including the point j = 0, in such a way that the half-axis is on the right: where s 0 is some scale determined a priori (for example, s 0 = 1 GeV 2 ) and related directly to the factors g (i) n (t) which should be interpreted as the effective couplings of reggeons to the colliding particles. ξ ± (α) are the so-called reggeon signature factors: ξ + (α) = i + tg π(α−1) 2 and ξ − (α) = i − ctg π(α−1) 2 .
The last formula (which is valid, as well, for inelastic scattering 2 → 2 and for reactions with off-shell particles), together with the eikonal representation (1) of the scattering amplitude, is the essence of the Regge-eikonal approach [2].