Saturation in lepton and hadron induced reactions

Abstract

Possible saturation of the matter density in two different classes of reactions, those induced by hadrons and leptons are studied. They may have common dynamical origin and be of the same nature. 1 Hadron-induced reactions: The Black Disc Limit at the LHC? Unitarity in the impact parameter b representation reads: Ih(s, b) = |h(s, b)| +G(s, b), where h(s, b) is the elastic scattering amplitude at the center of mass energy √ s, Ih(s, b) is the profile function, representing the hadron opacity and G(s, b), called the inelastic overlap function, is the sum over all inelastic channel contributions. Integrated over b, the above equation reduces to a simple relation between the total, elastic and inelastic cross sections σtot(s) = σel(s) + σin(s). Unitarity imposes the absolute limit 0 ≤ |h(s, b)| ≤ Ih(s, b) ≤ 1, while the so-called black disc limit σel(s) = σin(s) = 1 2σtot(s), or Ih(s, b) = 1/2, is a particular realization of the optical model, namely it corresponds to the maximal absorption within the eikonal unitarization, when the scattering amplitude is approximated as h(s, b) = i 2 (1− exp [iω(s,b)] ), with a purely imaginary eikonal ω(s, b). Eikonal unitarization corresponds to a particular solution of the unitarity equation, with Rh(s, b) = 0, h(s, b) = 1 2 [ 1± √ 1− 4Gin(s, b) ] , the one with the minus sign. An alternative solution, that with a plus sign in front of the square root, is known and realized within the so-called U -matrix approach, where the unitarized amplitude is a ratio rather than an exponential typical of the eikonal approach: h(s, b) = U(s, b) 1− i U(s, b) , where U is the input Born term, the analogue of the eikonal ω. In the U -matrix approach, the scattering amplitude h(s, b) may exceed the black disc limit as the energy increases. The transition from a (central) black disc to a (peripheral) black ring, surrounding a gray disc, for the inelastic overlap function in the impact parameter space corresponds to the transition from shadowing to antishadowing. The impact parameter amplitude h(s, b) can be calculated either directly from the data (where, however, the real part of the amplitude was neglected) or by using a particular model that fits the data sufficiently well. In the dipole Pomeron (DP) model [1], logarithmically rising cross sections are produced with a Pomeron intercept equal to unity, thus respecting the Froissart-Martin bound. Apart from the conservative Froissart-Martin bound, any model should satisfy also schannel unitarity. We show that both the D-L and DP models are well below this limit and will remain so for long, in particular will so at the LHC. The elastic scattering amplitude corresponding to the exchange of a dipole Pomeron reads A(s, t) = d dα [ eG(α)(s/s0) α ] = e(s/s0) α[G′(α) + (L− iπ/2)G(α)] , where L ≡ ln s s0 and α ≡ α(t) is the Pomeron trajectory. The elastic amplitude in the impact parameter representation in our normalization is