{'fraction_non_alphanumeric': 0.06067468374199594, 'fraction_numerical': 0.029439325316258005, 'mean_word_length': 3.8401360544217686, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Ultrahigh-energy neutrino-nucleon deep-inelastic scattering and the Froissart bound

We present a simple formula for the total cross section sigma^{nu N} of neutral- and charged-current deep-inelastic scattering of ultrahigh-energy neutrinos on isoscalar nuclear targets, which is proportional to the structure function F_2^{nu N}(M_V^2/s, M_V^2), where M_V is the intermediate-boson mass and s is the square of the center-of-mass energy. The coefficient in the front of F_2^{nu N}(x, Q^2) depends on the asymptotic low-x behavior of F_2^{nu N}. It contains an additional ln(s) term if F_2^{nu N} scales with a power of ln(1/x). Hence, an asymptotic low-x behavior F_2^{nu N} propto ln^2(1/x), which is frequently assumed in the literature, already leads to a violation of the Froissart bound on sigma^{nu N}.

For more than a decade, large experiments have been searching for ultrahigh-energy (UHE) cosmic neutrinos (ν), with energies E ν > 10 6 GeV, using detectors scanning for events in large volumes of water, ice, the Earth's atmosphere, and the lunar regolith [1]. While no clear indication of such an event has yet been reported, experimental bounds on UHE-neutrino fluxes could be established, which, put together, now cover energies way up to 10 12 GeV and start to constrain scenarios of astrophysical interest. Since these limits directly depend on the total cross section σ νN (E ν ) of UHE-neutrino deep-inelastic scattering (DIS) off nucleons (N ), it is an urgent task to provide reliable theoretical predictions for the latter in the asymptotic high-E ν regime, which lies far beyond the one explored by laboratory-based νN DIS experiments and corresponds to asymptotically low values of Bjorken's scaling variable x. This requires extrapolation over several orders of magnitude in E ν , for which various approaches exist [2][3][4][5][6][7][8]. These are based on successful descriptions of the terrestrial data within the framework of perturbative QCD and frequently impose the Froissart bound [9] on σ νN . According to the latter, unitarity and analyticity limit the total cross section of a scattering process not to grow faster with energy than ln 2 s.
In this Letter, we derive a general formula for σ νN that is remarkably concise and correctly accounts for the asymptotic high-energy behavior making it perfectly suitable for UHE-neutrino phenomenology. It is proportional to the DIS structure function F νN 2 (x, Q 2 ), which has a well-known representation in terms of parton distribution functions (PDFs) within the parton model (PM) of QCD, with x and the typical energy scale Q appropriately defined in terms of E ν and M V (V = W, Z). To be on the conservative side, we assume for the time being, as in Refs. [2][3][4][5][6][7][8], that the available experimental data on DIS allow for an extrapolation to very high (low) values of E ν (x) using an appropriate parameterization of F νN 2 . If the latter rises too steeply as x → 0, then possible new QCD phenomena, such as gluon saturation or recombination, color glass condensates, multiple pomeron exchanges, etc., are expected to enter the stage as a cure at x values below those currently probed by DIS experiments (for a review, see Ref. [10]). We shall return to this issue below, considering two popular models of screening [11,12].
Specifically, we consider the charged-current (CC) and neutral-current (NC) DIS processes, respectively, where N = (p + n)/2 denotes an isoscalar nucleon target of mass M , X collects the unobserved part of the final state, and the four-momentum assignments are indicated in parentheses, and introduce the familiar kinematic variables where q = k − k ′ . In the target rest frame, we have s = M (2E ν + M ) and xy = Q 2 /(2M E ν ). In the kinematic regime of interest here, the inclusive spin-averaged double-differential cross sections of processes (1) are, to very good approximation, given by [3] where i = CC, NC, G F is Fermi's constant, and K(y) = 2 − 2y + y 2 . In the so-called wee parton picture appropriate for the low-x regime [7], we have K CC = 1 and K NC = 1/2−x w +(10/9)x 2 w , where x w = sin 2 θ w , with θ w being the weak mixing angle. Using x w = 0.231 [13], we have K NC = 0.328. The contributions due to the structure functions F νN L and F νN 3 to the r.h.s. of Eq. (3) are negligibly small in our applications; F νN L tends to zero as Q 2 rises [14], and F νN 3 essentially refers to valence partons, which hardly contribute in the low-x regime.
Detailed inspection of the available ℓN DIS data (see, e.g., Fig. 1 for ep data from HERA I [15]) suggests that, in the limit x → 0, F ℓN 2 exhibits a singular behavior of the form F ℓN , where δ is a small positive number andF ℓN 2 diverges less strongly than any power of x, i.e.,F ℓN 2 (x, Q 2 )/x −λ → 0 as x → 0 for any positive number λ. Assuming a symmetric quark sea, as is appropriate for the low-x regime, we have F νN 2 (x, Q 2 ) = (18/5)F ℓN 2 (x, Q 2 ), so that the low-x behavior ofF ℓN 2 carries over toF νN 2 . Imposing the lower cut-off Q 2 0 on Q 2 , the total cross sections of processes (1) are evaluated as The inner integral on the r.h.s. of Eq. (5) can be rewritten as the Mellin convolution Hence, Eq. (5) becomes Because the . This is our master formula. Further simplification depends on the actual size of δ, and we distinguish two cases. (1) If δ is not too small, so thatx δ ≪ const, then the lower limitx of the inner i ci × 10 3 δi × 10 2 a1i × 10 2 a2i × 10 3 bi × 10 2 0 189. 4 10   (14), (15), (18), and (20).
integral on the r.h.s. of Eq. (5) may be put to zero, so thatM becomes independent ofx and Q 2 . (2) On the other hand, if δ ≪ 1, then we havẽ We note thatδ is determined by the asymptotic low-x behavior ofF νN Now we apply Eq. (10) to the three most popular types of F ℓN 2 parameterization, namely the standard PM representation implemented with up-to-date proton PDFs [15,18,19], a modification of the impressively simplistic log-log form proposed by Haidt (H) [20], and the more sophisticated form recently introduced by Berger, Block, and Tan (BBT) [6]. While the Q 2 dependence of the PM representation of F νN 2 is governed by the DGLAP evolution, those of the heuristic H and BBT forms are directly determined by global fits to experimental data covering a wide Q 2 range. In the low-x regime, the PM parameterization of F ℓN 2 may be well approximated by the following ansatz: with where it is understood that Q 2 is taken in units of GeV 2 . To suppress higher-twist effects, impose the cut Q 2 > 3.5 GeV 2 . Fitting Eqs. (13)- (15) to the result for F ℓN 2 evaluated at next-to-leading order (NLO) with the HERAPDF1.0 [15] proton PDFs, we obtain the values of c i and δ i collected in Table I. From Eq. (15) and Table I, we glean that so that Eq. (10) is to be used with Eq. (11). Using the MSTW [18] and CT10 [19] PDFs, we obtain δ PM (M 2 V ) ≈ 0.35 and 0.38, respectively. The resulting high-E ν behavior σ νN i (E ν ) ∝x −δPM(M 2 V ) is in good agreement with other studies [4].
For the reader's convenience, we recollect here the BBT parameterization of F ℓN 2 appropriate for the range x < x P = 0.11 [8] relevant for our applications [21]. It reads [6][7][8]: where A 0 = F P /(1 − x P ), with F P = 0.413 [8], and with the values of a ij listed in Table I [ 8]. Here, Eq. (10) is to be used with Eq. (12), and we find From Eqs. (17) and (19), we glean that, in the highenergy limit s → ∞, F ℓN 2,BBT (x, M 2 V ) ∝ ln 2 s and 1/δ BBT (x, M 2 V ) ∝ ln s. This leads us to the important observation that σ νN BBT ∝ ln 3 s, which manifestly violates the Froissart bound [9] in contrast to what is stated in Refs. [6][7][8]. This violation of the Froissart bound is attributed to the presence of the ln 2 x term in Eq. (17). On the other hand, if F ℓN 2 just rises linearly in ln x as x → 0, then σ νN i ∝ ln 2 s in accordance with the Froissart bound.
In fact, this is the case for the original H ansatz [20], B ln(x 0 /x) ln(1 + Q 2 /Q 2 0 ), which contains just three fit parameters. To enable the fit quality to be improved, we introduce three more parameters by writing Eq. (10) is again to be used with Eq. (12), and we obtain  [15] with the cuts x < 0.01 and Q 2 > 1.5 GeV 2 , we obtain x 0 = 0.05791, Q 2 0 = 2.578 GeV 2 , B 0 = 0.1697, and the values of b i listed in Table I Looking at Fig. 1, we observe that our fit (solid lines) indeed yields a surprisingly good description of the experimental data over the full x and Q 2 ranges considered. The approximation works particularly well for low x and large Q 2 values and is thus likely to allow for a  Fig. 1.
We now consider νN DIS with UHE neutrinos. For the sake of brevity, we focus our attention on CC DIS. The corresponding NC results may be obtained by substituting K CC → K NC and M W → M Z in our formulas. In order to determine the range of validity of our master formula (10) for σ νN CC , we compare it with the exact formula (5), which requires two-dimensional numerical integration, for the PM, BBT, and H cases considered above. In each case, we find excellent agreement for E ν values of order 10 7 GeV and above, which corresponds to x values of order 10 −3 and below in F ℓN 2 . This is illustrated for the BBT case in Fig. 2, where the approximate evaluation of Eq. (10) with Eqs. (12) and (19) is compared with the exact one of Eq. (5) with Eqs. (17) and (18) (dashed line). The large-E ν approximation may be somewhat improved by evaluatingδ(x, Q 2 ) by one-dimensional integration via Eq. (7) instead of using Eq. (19) (dotted line).
The PM and H results for σ νN CC evaluated from our master formula (10), with Eqs. (11) and (16) in the PM case and with Eqs. (12) and (21) in the H case, are also displayed in Fig. 2. Comparing them with the corresponding BBT result, we observe that all three predictions agree relatively well in the range 10 7 GeV E ν 10 9 GeV, where the high-E ν approximation is already working and the respective F ℓN 2 parameterizations are still con- strained by the HERA data. However, these three predictions steadily diverge as E ν further increases until they differ by 1-2 orders of magnitude at typical UHE values of E ν , reflecting the different low-x behaviors of the respective parameterizations of F ℓN 2 . In summary, we derived a novel concise relationship, given by Eqs. (10)- (12), between the total cross section σ νN i (E ν ) of CC and NC νN DIS in the high-E ν limit and the structure function F ℓN 2 (x, Q 2 ) in the low-x limit. It is particularly useful for applications to UHE-neutrino physics providing reliable predictions in a very quick and convenient way as it does. Being given in terms of a closed analytic formula, it also allows one to unambiguously determine if σ νN i resulting from a given functional form of F ℓN 2 satisfies the Froissart bound [9] or not, while this is hardly possible using the numerical solution of the two-dimensional integral in Eq. (5). Specifically, if F ℓN 2 exhibits a low-x behavior ∝ ln p (1/x), which corresponds to a high-s behavior ∝ ln p s in Eq. (10), then the coef-ficientM in that equation produces an additional factor ∝ ln s, so that the Froissart bound is violated for p > 1. In fact, this is the case for the BBT [6][7][8] parameterization of F ℓN 2 , for which p = 2. On the other hand, the H [20] one is characterized by p = 1, so that the Froissart bound is satisfied. This motivated us to update the analysis of Ref. [20] by fitting our improved ansatz (20) to the recent combination of the complete H1 and ZEUS data on F ℓN 2 from HERA I [15].
For completeness, we also performed a fit to the PM result for F ℓN 2 evaluated at NLO with an up-to-date set of proton PDFs, namely the HERAPDF1.0 one obtained by H1 and ZEUS by fitting their own data [15], and presented the resulting prediction for σ νN i . As expected, the low-x behavior of the PM result for F ℓN 2 is too singular for σ νN i to satisfy the Froissart bound. It is likely that the inclusion of nonlinear terms, such as screening corrections generated by gluon saturation or recombination, in the evolution equations will cure this problem [10]. In fact, considering the Ayala-Gay-Ducati-Levin [11] and the generalized Golec-Biernat-Wüsthoff (GBW) [12] models of saturation, where, due to their specific gluon densities, F ℓN 2 ∝ Q 2 ln p (1/x) with p = 1 and p = 0, respectively, we obtain σ νN i ∝ ln p+2 s, where the second additional logarithm arises from the Q 2 integration in Eq. (9). Thus, saturation strongly modifies the powerlike perturbative asymptotics of total cross sections and has the potential to restore the Froissart bound, as in the case of the GWB model. Future measurements of νN DIS with UHE neutrinos will eventually provide direct access to the low-x asymptotic behavior of F ℓN