Uhe Neutrinos: Fusing Gluons Within Diffraction Cone
Currently available estimates of the gluon-fusion effect in ultra-high energy neutrino-nucleon interactions as well as in DIS on protons suffer from uncertainty in defining the scattering profile function . Indeed, the area, , in the impact parameter space populated with interacting gluons varies by a factor of from one analysis to another. To get rid of uncertainties we specify the dipole-nucleon partial-wave amplitude which meets the restrictions imposed by both the total dipole-nucleon cross section and the small angle elastic scattering amplitude. The area becomes a well defined quantity proportional to the diffraction cone slope. We solve numerically the non-linear color dipole BFKL equation and evaluate the UHE neutrino-nucleon total cross section. Our finding is that the saturation is a rather weak effect, , up to GeV.
Practical needs of the neutrino astrophysics  inspired many papers on the cross section for the scattering of Ultra-High Energy (UHE) neutrinos on nucleons and nuclei. The question of interest is the interplay of unitarity constraints on and the evolution of QCD parton densities. The UHE neutrinos probe the gluon density in the target nucleon at very small values of Bjorken . The BFKL  distribution of gluons grows fast to smaller , , where, phenomenologically, . Hence, the neutrino-nucleon cross sections violating the Froissart bound.
The original idea of Ref. developed further in Ref. was that an overlap in transverse space and recombination of partons leads to a slow down of the growth of the parton density and finally to the saturation of parton densities. Quantitative QCD analysis of the non-linear effects in terms of the gluon density was initiated by Ref. . More recently, different derivations of the non-linear BFKL equation were presented in . The strength of the saturation effect is usually estimated as
where stands for some gluon interaction cross section. The radius of the area in the impact parameter plane, , within which the interacting gluons are expected to be distributed, varies considerably, from GeV in  down to GeV in . Besides, the area is assumed to be independent of . However, because of the BFKL diffusion property, under certain conditions, acquires the Regge contribution .
In this communication we specify the dipole-nucleon partial-wave amplitude which meets the restrictions imposed by both the total dipole-nucleon cross section and the small angle elastic scattering amplitude. The area becomes a well defined quantity proportional to the diffraction cone slope. We solve numerically the non-linear color dipole BFKL equation and evaluate the UHE neutrino-nucleon total cross section. Our finding is that the saturation is a rather weak effect, , up to GeV. This result differs from predictions found in extensive literature on the subject [7, 10].
2 The partial-wave amplitude and diffraction cone slope
Generalization of the color dipole BFKL approach developed in  to the equation for diffraction slope proceeds as follows [12, 13]. In the impact-parameter representation the imaginary part of the elastic dipole-nucleon amplitude reads
and the dipole cross section is The diffraction slope for the forward cone is
where is the profile function and is the impact parameter defined with respect to the center of the - dipole. In the state, the and dipoles have the impact parameter . Then ,
where and is the radial light cone wave function of the dipole with the Yukawa screening of infrared gluons 
Here are the - and - separations in the two-dimensional impact parameter plane for dipoles generated by the - color dipole source, is the - separation and is the modified Bessel function. At and in the approximation, the scaling BFKL equation  is obtained.
The BFKL dipole cross section sums the Leading-Log multi-gluon production cross sections. Consequently, a realistic boundary condition for the BFKL dynamics is the two-gluon exchange Born amplitude at . The running QCD coupling must be taken at the shortest relevant distance and in the numerical analysis an infrared freezing has been imposed. In Ref. it was found that incorporation of the asymptotic freedom into the BFKL equation splits the cut in the complex -plane into a series of isolated BFKL-Regge poles. Also, in  it was shown that breaking of scale invariance by a running supplemented by the finite gluon propagation radius , changes the nature of the BFKL pomeron from a fixed cut to a series of moving poles with the finite Regge slope of the pomeron trajectory The preferred choice fm gives GeV, and leads to a very good description of the data on the proton structure function and the diffraction cone slope  at small .
In  the diffraction slope in the elastic scattering of the dipole on the nucleon (with the gluon-probed radius ) was presented in a very symmetric form
3 Non-linear effects
The partial-wave amplitude in the impact parameter space reads
Shown in Fig. 1 is the ratio which quantifies the strength of non-linear effects in the cross section. Here represents the observable charged current cross section obtained from the numerical solution of Eq.( ‣ 3). The cross section derives from the solution of Eq. ( ‣ 3) with the non-linear term turned off . The cross section calculated for GeV and GeV is shown in Fig. 2, where is identical to . We conclude, that the non-linear effect is rather weak, at GeV. This result differs considerably from earlier predictions.
We thank all organizers of Low- Meeting 2011 for their warm hospitality. We are indebted to N.N. Nikolaev for useful comments. This work was supported in part by the RFBR grants 09-02-00732, 11-02-00441 and the DFG grant 436 RUS 113/991/0-1.
-  F. Halzen, IceCube, AIP Conf.Proc. 1304 (2010) 268-282.
-  E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Sov.Phys. JETP 44, 443 (1976); 45, 199 (1977); Ya.Ya. Balitskii and L.N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978).
-  O. V. Kancheli, Sov. Phys. JETP Lett. 18, 274 (1973); Pisma Zh. Eksp. Teor. Fiz. 18, 465 (1973).
-  N. N. Nikolaev and V. I. Zakharov, Phys. Lett. B 55, 397 (1975); V. I. Zakharov and N. N. Nikolaev, Sov. J. Nucl. Phys. 21, 227 (1975).
-  L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rep. 100, 1 (1983); A.H. Mueller, J. Qiu, Nucl. Phys. B 268, 427 (1986).
-  I. Balitsky, Nucl. Phys. B 463 99 (1996); Phys. Rev. D 60, 014020 (1999); Yu.V. Kovchegov, Phys. Rev. D 60, 034008 (1999); Phys. Rev. D 61, 074018 (2000); M. Braun, Eur. Phys. J. C 16 337 (2000). Phys. Rev. D 61, 074018 (2000).
-  K. Kutak, J. Kwiecinski, Eur.Phys.J. C 29, 521 (2003).
-  J. Bartels, E. Gotsman, E. Levin and U. Maor, Phys. Lett. B 556, 114 (2003); Phys. Rev. D 68, 054008 (2003).
-  N.N. Nikolaev, B.G. Zakharov and V.R. Zoller, JETP Lett. 66, 137 (1997).
-  M.V.T. Machado, Phys.Rev. D 70, 053008 (2004); E.M. Henley and J. Jalilian-Marian, Phys.Rev. D 73, 094004 (2006); R. Fiore, L.L. Jenkovszky, A.V. Kotikov, F. Paccanoni and A. Papa, Phys.Rev. D 73, 053012 (2006); N. Armesto, C. Merino, G. Parente and E. Zas, Phys. Rev. D 77, 013001 (2008).
-  N.N. Nikolaev, B.G. Zakharov and V.R. Zoller, JETP Lett. 59, 6 (1994); Phys. Lett. B 328, 486 (1994); J. Exp. Theor. Phys. 78, 806 (1994); N.N. Nikolaev and B.G. Zakharov, J. Exp. Theor. Phys. 78, 598 (1994); Z. Phys. C 64, 631 (1994).
-  N.N. Nikolaev, B.G. Zakharov and V.R. Zoller, JETP Lett. 60, 694 (1994).
-  N.N. Nikolaev, B.G. Zakharov and V.R. Zoller, Phys. Lett. B 366, 337 (1996).
-  L.N.Lipatov, Sov. Phys. JETP 63, 904 (1986).
-  R. Fiore, N.N. Nikolaev and V.R. Zoller, JETP Lett. 90, 319 (2009); N.N. Nikolaev and V.R. Zoller, Phys. Atom. Nucl. 73, 672 (2010); J. Nemchik, N.N. Nikolaev, E. Predazzi, B.G. Zakharov and V.R. Zoller, JETP 86, 1054 (1998).