On Ultrahigh-energy Neutrino Scattering
aa r X i v : . [ h e p - ph ] A ug On Ultrahigh-energy Neutrino ScatteringMasaaki Kuroda
Institute of Physics, Meijigakuin UniversityYokohama, Japan
Dieter Schildknecht
Fakult¨at f¨ur Physik, Universit¨at BielefeldD-33501 Bielefeld, GermanyandMax-Planck Institute f¨ur Physik (Werner-Heisenberg-Institut),F¨ohringer Ring 6, D-80805, M¨unchen, Germany
Abstract
We predict the neutrino-nucleon cross section at ultrahigh energies relevantin connection with the search for high-energy cosmic neutrinos. Our inves-tigation, employing the color-dipole picture, among other things allows usto quantitatively determine which fraction of the ultrahigh-energy neutrino-nucleon cross section stems from the saturation versus the color-transparencyregion. We disagree with various results in the literature that predict a strongsuppression of the neutrino-nucleon cross section at neutrino energies above E ∼ = 10 GeV . Suppression in the sense of a diminished increase of theneutrino-nucleon cross section with energy only starts to occur at neutrinoenergies beyond E ∼ = 10 GeV . 1nitiated by the experimental search for cosmic neutrinos of energies largerthan E ≃ GeV , the theoretical investigation of the neutrino-nucleon in-teraction at ultrahigh energies received much attention recently. Predictionsrequire a considerable extension of the theory of neutrino-nucleon deep inelas-tic scattering (DIS) into a kinematic domain beyond the one where resultsfrom experimental tests are available at present. Different theoretical ap-proaches have been employed ranging from conventional linear evolution ofnucleon parton distributions to the investigation of possible non-linear effectsconjectured to becoming relevant in the ultrahigh-energy domain.In the present note, we consider neutrino scattering in the framework ofthe color dipole picture (CDP) . The CDP is uniquely suited for a treat-ment of ultrahigh-energy neutrino scattering. Extrapolating the results fromelectron-proton scattering at HERA, we expect the total neutrino-nucleoncross section at ultrahigh energies to be dominantly due to the kinematicrange of x ≪ . x bj ≡ x ∼ = Q /W . This is thedomain of validity of the CDP.In particular, we shall focus on the question of color transparency versussaturation. Does the total neutrino-nucleon cross section at ultrahigh ener-gies dominantly originate from the region of large values of the low-x scalingvariable [4, 5], η ( W , Q ) = ( Q + m )Λ sat ( W ) , (1)namely η ( W , Q ) ≫ η ( W , Q ) ≪ sat ( W ) denotes the “saturation scale” that increases with the γ ∗ ( Z , W ± ) p center-of-mass energy squared, W , as ( W ) C , where C ≃ . sat ( W ) approximately rangesfrom 2GeV < ∼ Λ sat ( W ) < ∼ . The γ ∗ ( Z , W ± ) virtual four-momentasquared in (1) is denoted by q = − Q , and m ≃ . (for light quarks).Compare Fig. 1 for the ( Q , W ) plane with the line of η ( W , Q ) = 1. Compare refs. 16-24 in [1] Compare e.g. refs. 2-8 in [2] Compare ref. [3] for recent reviews on the CDP and an extensive list of references. Q , W ) plane showing the line η ( W , Q ) = 1 that separatesthe saturation region from the color-transparency region.The charged-current neutrino-nucleon cross section we shall concentrateon, as a function of the neutrino energy, E , is given by (e.g. [6]) σ νN ( E ) = Z sQ min dQ Z Q s dx xs ∂ σ∂x∂y , (2)where ∂ σ∂x∂y = G F s π M W Q + M W ! σ r ( x, Q ) , (3)and σ r ( x, Q ) in (3) denotes the “reduced cross section” σ r ( x, Q ) = 1 + (1 − y ) F ν ( x, Q ) − y F νL ( x, Q ) + y (1 − y xF ν ( x, Q ) . (4)In standard notation, s denotes the neutrino-nucleon center-of-mass energysquared, s = 2 M p E + M p ∼ = 2 M p E, (5)with M p being the nucleon mass, q = − Q is the four-momentum squaredtransferred from the neutrino to the W ± boson of mass M W , and G F is theFermi coupling. The Bjorken variable is given by x = Q qP = Q W + Q − M p ∼ = Q W , (6)3here the approximate equality in (6) is valid in the relevant range of x ≪ . W ± boson, y , isgiven by y = Q M p Ex ∼ = W s . (7)For the subsequent discussion, it will be useful to replace the integrationover dx in (2) by an integration over W , rewriting (2) as σ νN ( E ) = G F π Z s − M p Q min. dQ M W Q + M W ! Z s − Q M p dW W σ r ( x, Q ) . (8)Due to the vector-boson propagator, contributions to the total cross sectionfor Q ≫ M W are strongly suppressed, and with W ≤ s and s in theultrahigh energy range, s ≫ M W , we expect the cross section to dominantlyoriginate from x ≈ Q /W ≪ . F ν ( x, Q ) in (8) according to (4). For small values of x < ∼ .
1, DIS of electrons and neutrinos on nucleons,in terms of, respectively, the γ ∗ p and the ( W ± , Z ) p forward scattering am-plitude, proceeds via scattering of long-lived massive hadronic fluctuations, γ ∗ ( Z ) → q ¯ q and W − → ¯ ud etc., that undergo diffractive forward scatteringon the nucleon (CDP) [3].For the flavor-symmetric ( q ¯ q ) N interaction at x ≪ .
1, the neutrino-nucleon structure function, F νN ( x, Q ), and the electromagnetic structurefunction, F eN ( x, Q ), are related by (1 /n f ) F νN ( x, Q ) = (1 / P q Q q ) F eN ( x, Q ),or F νN ,L ( x, Q ) = n f P n f q Q q F eN ,L ( x, Q ) , (9)where n f denotes the number of actively contributing quark flavors, and Q q the quark charge, and n f / P q Q q = 18 / n f = 4 flavors of quarks. As aconsequence of the proportionality (9), the total neutrino-nucleon cross sec-tion (8) may be predicted by inserting the electromagnetic structure functioninto (4).The electromagnetic structure function, F ep ( x, Q ), is related to the total The contribution due to F νL ( x, Q ) turned out to be less than 6 %, compare thediscussion in connection with Table 4 below. The contribution from the structure function F ( x, Q ) in (4), that is due to valence-quark interactions, can be ignored. σ γ ∗ p ( W , Q ), by F ep ( x, Q ) = Q π α σ γ ∗ p ( W , Q ) . (10)In the CDP, as a consequence [4, 7] of the interaction of the color dipole withthe gluon field in the nucleon, the photoabsorption cross section becomes afunction of the low- x scaling variable, η ( W , Q ), σ γ ∗ p ( W , Q ) = σ γ ∗ p ( η ( W , Q )) ∼ σ ( ∞ ) ( ln η ( W ,Q ) , for η ( W , Q ) ≪ , η ( W ,Q ) , for η ( W , Q ) ≫ , (11)where the cross section σ ( ∞ ) ≡ σ ( ∞ ) ( W ) is of hadronic size, and, at most,it depends weakly on W . Both, the dependence on the single variable η ( W , Q ) (for σ ( ∞ ) ∼ = const. ) in (11), and the specific functional form ofthis dependence, are general consequences [4, 7] of the color-gauge-invariantinteraction of a ( q ¯ q ) dipole with the color field in the nucleon. Any spe-cific ansatz for a parameterization of the dipole-nucleon cross section has toprovide an interpolation between the ln (1 /η ( W , Q )) and the 1 / η ( W , Q )dependence in (11). It is well known [4], compare Fig. 2, that the depen- η -2 -1
10 1 10 b ) µ ( * p γ σ -3 -2 -1 ZEUSH1EMCE665BCDMSNMC(W=275) σ (W=10) σ Figure 2: The theoretical prediction [4, 7] for the photoabsorption crosssection σ γ ∗ p ( η ( W , Q )) compared with the experimental data on DIS.dence (11) on the single variable η ( W , Q ) is fulfilled by the experimentaldata with σ ( ∞ ) ∼ = const. in the HERA energy range. The saturation scale is The low- x approximation is used for the factor in front of σ γ ∗ p ( W , Q ) in (10). sat ( W ) = C W ! C , C = 0 . GeV , C ∼ = 0 . . (12)The value of the exponent C ∼ = 0 .
29 is fixed [7] by requiring consistency ofthe CDP with the pQCD-improved parton model.We return to neutrino scattering. Employing relation (9), we replace theneutrino structure function, F ν ( x, Q ), in (4) by the electromagnetic one, F ep ( x, Q ), or rather by the photoabsorption cross section, compare (10).The neutrino-nucleon total cross section (8) becomes σ νN ( E ) = G F M W π α n f P q Q q R s − M p Q Min dQ Q ( Q + M W ) × R s − Q M p dW W (1 + (1 − y ) ) σ γ ∗ p ( η ( W , Q )) . (13)We first of all look at the ratio r ( E ) = σ νN ( E ) η ( W ,Q ) < σ νN ( E ) . (14)In (14), σ νN ( E ) η ( W ,Q ) < denotes that part of the total neutrino-nucleoncross section in (13) that originates from contributions from the saturationregion of η ( W , Q ) < η ( W , Q ) < Q , W ) integrationdomain in (13). According to (1) and (12), the restriction of η ( W , Q ) < Q Max. ≥ Q ≥ Q Min = Λ sat ( M p ) − m , and Q Max ≫ m ) upon employing W Max = s − Q , yields W ≥ W ( Q ) Min = Q + m C ! C ,Q ≤ Q Max = Λ sat ( s ) (cid:16) − C Λ sat ( s ) s + o ( Λ sat ( s ) s ) (cid:17) . (15)From (15), for the ultrahigh-energy corresponding to s = 10 GeV , with(12), one finds Q < Q Max = Λ sat ( s ) = 3 . × GeV ≪ s . We observe thateven for s = 10 GeV , the range of Q < Q Max covered under restriction (15)is smaller than the W ± mass squared, M W ≈ . × GeV , that determinesthe maximum of the Q -dependent factor in (13). We accordingly expect asmall value of r ( E ) ≪ We restrict ourselves to the dominant term F ν ( x, Q ) in (4), ignoring F L ( x, Q ) and F ( x, Q ). r ( E ) in (14) is evaluated in two steps. In a first step, we onlyrely on the very general low-x scaling restrictions for σ γ ∗ p ( η ( W , Q )) in (11)with (12) and derive an upper bound on r ( E ) < ¯ r ( E ) on r ( E ). In a secondstep, we introduce a concrete representation for σ γ ∗ p ( η ( W , Q )) in the CDPthat smoothly interpolates the regions of η ( W , Q ) < η ( W , Q ) > r ( E ) in (14), upon substituting (13) and taking into account(15), becomes r ( E ) = R Q Max ( s ) Q Min dQ Q ( Q + M W ) R s − Q W ( Q ) Min dW W (1 + (1 − y ) ) σ γ ∗ p ( η ( W , Q )) R s − M p Q Min dQ Q ( Q + M W ) R s − Q M p dW W (1 + (1 − y ) ) σ γ ∗ p ( η ( W , Q )) . (16)Using the scaling behaviour (11) for η ( W , Q ) < η ( W , Q ) >
1, wederive an upper limit, r ( E ) < ¯ r ( E ) , (17)on the ratio r ( E ) in (16). Appropriately substituting the behaviour (11) of σ γ ∗ p ( η ( W , Q )) into (16), and simplifying by putting y = 0 in the numeratorand y = 1 in the denominator, an upper bound on r ( E ) reads ¯ r ( E ) = 2 R Q Max ( s ) Q Min dQ Q ( Q + M W ) R s − Q W ( Q ) Min dW W ln η ( W ,Q ) R s − M p Q Min dQ Q ( Q + M W ) R s − Q M p dW W η ( W ,Q ) . (18)For Λ sat ( s ) < M W ≪ s , one finds that the numerator in (18) is approximatelygiven by N ( E ) = 12 12 C Λ sat ( s ) M W ! + o ( Λ sat ( s ) M W ! ) . (19)The denominator in (18) becomes D ( E ) = 12 C Λ sat ( s ) M W ! o ( M W s log M W s ) ! . (20)Inserting (19) and (20) into (18), we find the upper bound on r ( E ), r ( E ) < ¯ r ( E ) = 12 Λ sat ( s ) M W . (21)Numerical values of ¯ r ( E ), using (12), are given in Table 1, together withthe results for r ( E ) resulting from an explicit expression for σ γ ∗ p ( η ( W , Q ))from the CDP to be discussed below. In the denominator of (18), we inserted the 1 / η ( W , Q ) dependence only valid for η ( W , Q ) >
1. We explicitly checked that the enlargement of the cross section as aconsequence of this approximation amounts to only a few percent in the energy range upto E ∼ GeV under consideration. (GeV) ¯ r ( E ) r ( E ) | Table3 r ( E ) | Table4 . × − . × − . × − . × − . × − . × − . × − . × − . × − Table 1: The upper bound, ¯ r ( E ) > r ( E ), on the fraction of the to-tal neutrino-nucleon cross section originating from the saturation region of η ( W , Q ) <
1. The results for ¯ r ( E ) in the second column are based on(21) with (12). The results for r ( E ) | Table 3 are based on evaluating (16)upon substitution of (22) with (25). The results for r ( E ) | Table 4 are based onevaluating (16) upon substitution of (29) with (25).According to (21) and Table 1, the fraction of the total neutrino-nucleoncross section arising from the saturation region is strongly suppressed. Thesaturation region contributes less than a few percent, except for extremelyultrahigh energies of order E ≃ GeV.We turn to an evaluation of the neutrino-nucleon cross section based onan explicit form of σ γ ∗ p ( η ( W , Q )) in the CDP.The CDP leads to a remarkably simple form of the photoabsorption crosssection that moreover can be represented by a closed expression, [4, 7] σ γ ∗ p ( W , Q ) = σ γ ∗ p ( η ( W , Q )) + O m Λ ( W ) ! == αR e + e − π σ ( ∞ ) ( W ) I ( η ( W , Q )) + O m Λ ( W ) ! , (22)where I ( η ( W , Q )) = 1 q η ( W , Q ) ln q η ( W , Q ) + 1 q η ( W , Q ) − ∼ = (23) ∼ = ln η ( W ,Q ) + O ( η ln η ) , for η ( W , Q ) → m Λ ( W ) , η ( W ,Q ) + O (cid:16) η (cid:17) , for η ( W , Q ) → ∞ , and R e + e − = 3 X q Q q . (24)Comparing (22) and (23) with (11), one notes that (22) smoothly interpolatesthe regions of η ( W , Q ) ≪ η ( W , Q ) ≫ We note that the closed form for the photoabsorption cross section in (22) with (23)contains the simplifying assumption of “helicity independence” leading to F epL = 0 . F ep rather than F epL = 0 . F ep . This simplifying approximation is unimportant in the presentcontext. Compare refs. [7, 8] for the refinement that implies the result F epL = 0 . F ep that is consistent with the HERA experimental observations. σ ( ∞ ) ( W ) in(22) is determined by consistency of σ γ ∗ p ( W , Q ) with Regge behavior [4, 9]in the photoproduction limit of σ γp ( W ) = σ γ ∗ p ( W , Q = 0), and alterna-tively, by consistency with the double-logarithmic fit to photoproduction bythe Particle Data Group, σ ( ∞ ) ( W ) = 3 πR e + e − α Λ sat ( W ) m σ Reggeγp ( W ) ,σ P DGγp ( W ) . (25)The fits to photoproduction, compare refs. [4], [9] and [10] (in units of mb ,with W in GeV ) are explicitly given by σ ( a ) γp ( W ) = 0 . W ) . + 0 . W ) − . , (26) σ ( b ) γp ( W ) = 0 . W ) . + 0 . W ) − . σ ( c ) γp ( W ) = 0 . .
71 + πM ln W ( M p + M ) ! + 0 . ( M p + M ) W ! . , where M p stands for the proton mass and M = 2 . σ γ ∗ p ( W , Q ) ∼ (ln W )( W ) C in the color-transparencyregion (for σ ( ∞ ) ( W ) ∼ σ P DGγp ( W ) / ln Λ sat ( W ) m ) of η ( W , Q ) > σ γ ∗ p ( W , Q ) ∼ (ln W ) , once the saturation limit of η ( W , Q ) < upon substitution of the photoabsorption cross section from(22) with Λ sat ( W ) from (12), m = 0 . , and σ ( ∞ ) ( W ) determinedby (25) and (26). The results in Table 2 for σ ( b ) νN ( E ) and σ ( c ) νN ( E ) based on σ ( ∞ ) ( W ) from the Regge fit (b) and the PDG fit (c), respectively, coincide ingood approximation. The enhancement of the cross section σ ( a ) νN ( E ) relativeto σ ( b,c ) νN ( E ) is a consequence of the stronger increase of the Pomeron contribu-tion (( W ) . versus ( W ) . ) in σ ( ∞ ) ( W ) originating from (26). At thehighest energy under consideration, E = 10 GeV, the enhancement reaches The CDP contains the limit of Q →
0, such that Q Min may be put to Q Min = 0in (13). The actual dependence on Q Min is negligible, as long as < ∼ Q Min < ∼ M p . Wealso note tht the replacement of the lower limit W ≥ M p by W ≥ const M p for e.g.const ≤
20 leads to an insignificant change of the neutrino cross section. σ ( a ) νN σ ( b ) νN σ ( c ) νN σ ( a,b,c ) νN [ cm ],from the CDP as a function of the neutrino energy, E [GeV]. Compare textfor details.a factor of about 1.5. Concerning the energy dependence, by comparingneighboring results in Table 2 for E ≥ GeV, one notes an increase (only)slightly stronger than expected from the proportionality to Λ sat ( s ) ∼ s C inthe estimate (20). This is a consequence of the energy dependence (25) of σ ( ∞ ) = σ ( ∞ ) ( W ) ignored in (20).We return to the question of the relative contribution to the neutrinocross section from the saturation region relative to the color-transparencyregion. We subdivide the neutrino cross section into the sum σ ( c ) νN ( E ) = σ ( c ) νN ( E ) η ( W ,Q ) < + σ ( c ) νN ( E ) η ( W ,Q ) > . (27)The results are shown in Table 3. From Table 3, one finds that the fraction E σ ( c ) νN η > η < σ ( c ) νN ( E ) [ cm ] as a function of E [ GeV ] from the color transparency( η ( W , Q ) >
1) and the saturation ( η ( W , Q ) <
1) region compared withthe full cross section, σ ( c ) νN ( E ) taken from Table 2.of the total cross section originating from the saturation region, r ( E ) in(14) and (16), increases from r ( E = 10 GeV) | T able ∼ = 1 . · − to r ( E =10 GeV) | T able ∼ = 1 . · − . The increase is consistent with the upper bound(21), compare Table 1. With increasing energy, there is a strong increase fromthe saturation region, but even at E = 10 GeV its contribution is of theorder of only 17%.The result that the dominant part of the neutrino-nucleon cross sectionis due to contributions from large values of η ( W , Q ) ≫ Q = 10 GeV ∼ = M W , and for W below10 ≤ GeV (or x ≤ . η ( W , Q ) reaches values of η ( W , Q ) ≤ η Max ( W , Q ) ∼ = 10 . For such large values of η ( W , Q ), aspreviously analysed [4, 7], the theoretical expression (22) for the photoab-sorption cross section must be corrected by elimination of contributions fromhigh-mass ( q ¯ q ) fluctuations, γ ∗ → q ¯ q , of mass M q ¯ q . The life time of high-massfluctuations in the rest frame of the nucleon becomes too short to be ableto actively contribute to the q ¯ q -color-dipole interaction. The restriction onthe q ¯ q mass, m ≤ M q ¯ q ≤ m ( W ) is taken care of by the energy-dependentupper bound, m ( W ), where m ( W ) = ξ Λ sat ( W ) , (28)and empirically ξ = 130 [7]. Employing the restriction (28) extends thevalidity of the CDP to high values of η ( W , Q ) ≫ ξ/η ( W , Q ). One obtains [7] σ γ ∗ p ( W , Q ) = αR e + e − π σ ( ∞ ) ( W ) I ( η ( W , Q )) × G L ξη ( W , Q ) ! + 2 G T ξη ( W , Q ) !! + O m Λ ( W ) ! (29)where13 G L ξη ( W , Q ) ! + 2 G T ξη ( W , Q ) !! =1 (cid:16) ξη ( W ,Q ) (cid:17) ξη ( W , Q ) ! + 2 ξη ( W , Q ) ! + ξη ( W , Q ) ! ∼ = ( , for η ( W , Q ) ≪ ξ = 130 ξη ( W ,Q ) , for η ( W , Q ) ≫ ξ = 130 ; (30)We note in passing that the theoretical prediction shown in Fig. 2 includes[7]the large- η ( W , Q ) correction (29) .In Table 4, third and fourth line, we present our final results for theneutrino-nucleon cross section based on substituting (29) into (13). The The photoabsorption cross section obtained from the simple closed expression (29)coincides within a (negative) deviation of up to approximately 25 % with the results shownin fig. 2 that are based on the more elaborate treatment in ref.[7], compare footnote 8
11 1.0E+04 1.0E+06 1.0E+08 1.0E+10 1.0E+12 1.0E+141.19E-34 1.69E-33 9.26E-33 4.29E-32 1.88E-31 7.77E-31 σ ( c ) νN σ ( c ) νN ( E )[ cm ], as a function ofthe neutrino energy E [ GeV ] upon imposing the restriction (28) on the massof actively contributing q ¯ q fluctuations (3rd and 4th line) compared with theresult from Table 3 (2nd line) that ignores the restriction (28). The resultsin the 3rd and 4th line are based on Λ sat ( W ) ∼ ( W ) C with C = 0 .
29 and C = 0 .
27, respectively.PDG result for σ ( ∞ ) ( W ) in (25) is used, and, for comparison, the resultfor σ ( c ) νN ( E ) from Table 2 ( i.e. σ ( c ) νN ( E ) without the restriction (28)) is againshown in the second line of Table 4. We explicitly verified that the addition in(13) of the contribution corresponding to the longitudinal structure functionaccording to (4) diminishes the neutrino cross section in Table 4 by less than6 % in the whole range of neutrino energies under consideration. In order todemonstrate the sensitivity under variation of the exponent C of the energydependence of the saturation scale, Λ sat ( W ) ∼ ( W ) C , in Table 4, we givethe neutrino-nucleon cross section for C = 0 .
29 and C = 0 .
27. Both valuesare consistent with the available experimental information on DIS. -36 -35 -34 -33 -32 -31 Figure 3: The effect on the neutrino-nucleon cross section of excluding inac-tive high-mass q ¯ q fluctuations.The results from Table 4 (2nd and 3rd line) are graphically represented12n Fig. 3. With increasing neutrino energy, the exclusion of inactive large-mass q ¯ q fluctuations by the restriction of M q ¯ q < m ( W ) = ξ Λ sat ( W ),where ξ = 130, becomes less important. Most of the contributions to theneutrino-nucleon cross section in the extreme ultrahigh-energy limit ( E ≃ GeV) are due to moderately large values of η ( W , Q ) that correspondto q ¯ q fluctuations of sufficiently long life time. Quantitatively, from Table 4,at E = 10 GeV the cross section is diminished by a factor of 0.32, while at E = 10 GeV, this factor is equal to 0.89. This effect is also seen in theratio r ( E ) in Table 1. At E = 10 GeV, the ratio r ( E ) exceeds the crudeestimate of ¯ r ( E ) from (18). -36 -35 -34 -33 -32 -31 Figure 4: Comparison of the CDP prediction for the neutrino-nucleon crosssection, σ νN ( E )[ cm ], according to (13) with (29) and σ P DGγp ( W ) from (25),with the predictions from the pQCD-improved parton model. The bandof the prediction from the CDP illustrates the sensitivity of σ νN ( E ) undervariation of the exponent C in Λ sat ( W ) ∼ ( W ) C between C = 0 .
27 and C = 0 . . In Fig. 4, we compare our final results for the neutrino-nucleon cross sec-tion, σ νN ( E ) ≡ σ ( c ) νN ( E ) from Table 4, 3rd and 4th line, based on the CDP,with the ones obtained [1, 2] by employing the parton distributions from aconventional perturbative QCD (pQCD) analysis of DIS. Fig. 4 shows con-sistency of our CDP results with the ones from the pQCD-improved partonmodel. Our predictions are also consistent with the ones in ref. [11].13 series of recent papers [12] - [15] treats DIS at HERA energies and ultra-high-energy neutrino scattering by adopting an ansatz with an (ln W ) de-pendence of the underlying hadron-nucleon cross section. The ansatz is basedon the asymptotic behavior of strong-interaction cross sections as (ln W ) due to Heisenberg[16] and Froissart[17].The ansatz of F ep ( x, Q ) ∼ P n,m =0 , , a nm (ln Q ) n (ln(1 /x )) m , with sevenfree fit parameters [12] - [15], yields a successful representation of the HERAexperimental results for all x and Q in the region of x < ∼ .
1. The subsequentevaluation [12] - [15] of the neutrino-nucleon cross section with this ansatzfor F ep ( x, Q ), essentially according to (9) and (13), for E > ∼ GeV led to across section that is suppressed relative to pQCD results, and, consequently,also in comparison with our CDP predictions. Compare Fig. 5. -36 -35 -34 -33 -32 -31 Figure 5: A comparison of the results for the neutrino-nucleon cross sec-tion from the CDP according to fig. 4 with the results from the “Froissart-inspired” ansatz from [14].Since the CDP contains an (ln W ) dependence, compare e.g. the dis-cussion immediately following (26), the result of Fig. 5 may look like aninconsistency. The apparent inconsistency is resolved in fig. 6. Figure 6shows the prediction for the neutrino-nucleon cross section from the CDPfor an extended energy range up to E = 10 GeV . As seen in fig. 6, inconsistency with the (ln W ) dependence of σ γ ∗ p ( W , Q ) in the saturationregion of η ( W , Q ) <
1, also the CDP implies a decreasing growth of the14eutrino-nucleon cross section. In distinction from the prediction from the“Froissart-inspired” ansatz, the decreasing growth of the cross section in theCDP is shifted to energies above E ∼ = 10 GeV . -36 -35 -34 -33 -32 -31 -30 -29 -28 -27 Figure 6: The neutrino-nucleon total cross section, σ νN ( E ) ≡ σ ( c ) νN ( E ), fromthe CDP as a function of the neutrino energy E for the extended range ofenergies up to E = 10 GeV . For comparison, we also show that part ofthe cross section, σ νN ( E ) | η ( W ,Q ) < , that is obtained upon restricting thecontributions of σ γ ∗ p ( W , Q ) to the neutrino-nucleon cross section to thesaturation region of η ( W , Q ) < σ ( c ) νN ( E ) η ( W ,Q ) > in (27). In theultra-ultra-high-energy limit, the neutrino-nucleon cross section in (13) be-comes saturated by contributions from that region of the photoabsorptioncross section where the (ln( W )) dependence becomes dominant.We must conclude that the requirement of a “Froissart-like” ansatz forthe underlying hadron-nucleon cross section by itself does not imply a weakergrowth, compared with e.g. the pQCD prediction, for the neutrino-nucleoncross section above E = 10 GeV . It is the combination of the energy de-pendence for F ep ( x, Q ), contained in ln(1 /x ) and (ln(1 /x )) terms, with theseven-free-parameter fit to the ad hoc polynomial ln Q dependence of thecoefficients of the ln(1 /x ) and (ln(1 /x )) terms that leads to a suppressionabove E = 10 GeV. 15n the CDP, the Q dependence is uniquely fixed by the Q dependenceof the “photon-wave function”, i.e. the transition of the (virtual) photon to q ¯ q dipole states with subsequent propagation of these q ¯ q states of mass M q ¯ q .The interaction of the q ¯ q color dipoles is restricted by being a gauge-invariantinteraction with the gluon field in the nucleon.Taking into account the more detailed dynamics of the CDP, and themuch smaller number of free fit parameters, compared with the ln(1 /x ) and(ln(1 /x )) ansatz, we are thus led to disagree with the conclusion of an on-set of a suppression of the neutrino-nucleon cross section for E > ∼ GeV implied by the analysis [12] - [15] of the “Froissart-inspired” ansatz.A suppression, in the sense of a reduced growth of the total neutrino-nucleon cross section with increasing energy, is expected to occur, however,for neutrino energies beyond E = 10 GeV . Acknowledgement
Questions on the subject matter by Paolo Castorina and by participantsof the Oberwoelz symposium on Quantum Chromodynamics, History andProspects (Oberwoelz, Austria, September 3 – 8, 2012) are gratefully ac-knowledged.
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