Color dipole cross section and inelastic structure function

Instead of starting from a theoretically motivated form of the color dipole cross section in the dipole picture of deep inelastic scattering, we start with a parametrization of the deep inelastic structure function for electromagnetic scattering with protons, and then extract the color dipole cross section. Using the parametrizations of $F_2(\xi=x \ {\rm or}\ W^2,Q^2)$ by Donnachie-Landshoff and Block et al., we find the dipole cross section from an approximate form of the presumed dipole cross section convoluted with the perturbative photon wave function for virtual photon splitting into a color dipole with massless quarks. The color dipole cross section determined this way reproduces the original structure function within about 10\% for $0.1$ GeV$^2\leq Q^2\leq 10$ GeV$^2$. We discuss the large and small form of the dipole cross section and compare with other parameterizations.


I. INTRODUCTION
Of broad interest in particle and astroparticle physics is the hadronic cross section. The inclusive hadronic cross section is measured over a range of energies in laboratory experiments [1], and in cosmic ray experiments, for example, the recent proton-air cross section measured at √ s = 57 TeV by the Pierre Auger Observatory [2]. The hadronic cross section is important for the analysis of a range of experiments and it is an essential input to many particle shower simulations, particularly for the forward production of hadrons. In the context of astroparticle physics, the hadronic cross section is needed to calculate the atmospheric flux of leptons from cosmic ray interactions with air nuclei [3,4].
The structure of hadrons is also probed by weak and electromagnetic interactions. Laboratory experiments in electron scattering and neutrino scattering with hadronic targets measure cross sections up to √ s ≃ 200 GeV [1], while neutrino telescopes and gamma ray telescopes are sensitive to even higher energies. For purely hadronic interactions and electroweak inelastic scattering with hadrons, in many cases, one is interested in kinematic regimes far different from the experimentally measured regimes. This requires additional theoretical input to the modeling of the hadronic structure. Hadronic structure is most easily studied with deep inelastic electromagnetic scattering. Deep inelastic electromagnetic scattering is characterized by structure functions F i that depend on kinematic variables Bjorkenx and momentum transfer Q. In the parton model, the structure functions are written in terms of parton * ysjeong@cskim.yonsei.ac.kr † cskim@yonsei.ac.kr ‡ mary-hall-reno@uiowa.edu distribution functions with an evolution in ln Q 2 governed by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [5][6][7]. When the Bjorken-x range becomes very small, DGLAP evolution fails. In parton language, the failure is due to gluon recombination which introduces nonlinear effects into the evolution equations [8,9]. More generally, unitarity is ultimately violated with the DGLAP approach.
An alternative approach which models small x saturation effects is the color dipole model [10,11]. The dipole model for DIS has virtual photons fluctuate to qq dipoles which then scatter with the hadronic target via the dipole cross sectionσ. In this approach, the structure function F i comes from the convolution of the virtual photon wave function squared (for splitting into the color dipole) |Ψ (f ) L,T (r, α; Q 2 )| 2 with the dipole cross sectionσ(r, x), integrated over the spatial splitting of the dipole r and the longitudinal momentum fraction α of the quark in the dipole. The dipole picture can be useful in many contexts, for example, ultrahigh energy neutrino scattering where one may approach the unitarity limit, and high energy hadronic production of charm, where small x and Q ∼ m c are relevant kinematic variables.
The usual starting point in the color dipole model is to postulate a functional form of the dipole cross section guided by theoretical or phenomenological considerations. Given this parametrization ofσ, hadronic processes as well as DIS scattering can be used to determine free parameters inσ. There are a number of models for the dipole cross section [12][13][14][15][16][17][18][19][20][21]. The Golec-Biernat-Wüstoff model [12] explicitly includes saturation and an effect called geometric scaling. Geometric scaling describes the behavior of DIS observables at low x, for example σ γ * p , depends on a scaling variable T = Q 2 R 2 0 (x), independent of W 2 , the square of the total energy in the γ * p collision [13]. Other authors, including Soyez [16] have done more elaborate fits to DIS data for dipole cross section parameterizations guided by theory that also in-clude geometric scaling [14,15].
Our goal here is to examine how a parametrization of F 2 translates to a correspondingσ. This phenomenological approach is the inverse of the usual starting point of σ. Eventually, one hopes to explore how different treatments of the small x behavior of F 2 guided by unitarity considerations translate to the dipole cross section and vice versa. This paper is a first step in this phenomenological program.
We take advantage of approximate relations between F 2 andσ as discussed by Ewerz, von Manteuffel and Nachtmann in Ref. [37]. As noted by Ewerz et al. [37], the convolution of the photon wave function and the dipole cross section is simplified in the m q = 0 case. We use Fourier transforms to factorize the convolution for m q = 0 to find an approximateσ 0 from an inverse Fourier transform. This dipole cross section is a starting point for the dipole treatment when m q = 0.
In the next section, we review the relevant formulas for the structure function F 2 in terms of the dipole cross section and photon wave functions. We show the convenient variables introduced in Ref. [37] make the problem amenable to inversion by Fourier transforms in the case of m q = 0. In Sec. III, we discuss the determination ofσ 0 in toy model with a single massless unit charged quark. Fourier transforms require knowing the structure function at both low and high Q 2 . For a starting point, we use the Donnachie-Landshoff parametrization [22] of F 2 which is defined for the full range of Q 2 , even if it does not describe DIS data for the full Q 2 range. In Sec. IV, we look at the more physical case with 5 massive quarks. We find that with a simple normalization constant, the dipole cross section solution to the toy model does quite well in reproducing the Donnachie-Landshoff structure function. In Sec. V, we compare our results with the GBW [12] and Soyez [16] dipole cross section parameterizations. We compare our numerical results with the small Q and large Q approximations of Ref. [37]. We conclude in Sec. VI.

II. DIPOLE FORMALISM AND THE DIPOLE CROSS SECTION
The dipole formula for deep inelastic electromagnetic scattering involves the photon wave function and the dipole cross section. Our focus is on the electromagnetic structure function F 2 (x, Q 2 ) in deep inelastic scattering where ep → eX involves the subprocess γ * p → X. In terms of the subprocess cross sections, the dipole picture has [10] in terms of the virtual photon-proton cross section The photon wave functions are written in terms of the Bessel functions K 0 and K 1 , and they depend on the quark electric charge e f = 2/3, −1/3 and mass m f through 14 GeV, m c = 1.4 GeV and m b = 4.5 GeV, however, when m f = 0, the photon wave functions depend on Q only through z = Qr, a considerable simplification. The electromagnetic fine-structure constant is labeled α e and N c = 3 is the number of colors in eqs. (3)(4).
The dipole cross section in eq. (2) depends on the dipole transverse size r and ξ. Ewerz et al. in Ref. [37] consider ξ = x and ξ = W 2 . While theoretically ξ = W 2 is more justifiable, the choice of ξ = x allows the dipole model to match experimental data over a wider range of Q 2 [37]. There are a number of models for the dipole cross section that depend on x rather than W 2 [12,[14][15][16][17]. Since our aim is to invert parameterizations of F 2 to determine the dipole cross section, our choice of ξ is determined by how F 2 is parameterized. For our discussion here, Ewerz et al. in Ref. [37] discuss the convolution formula of eq. (2) and consider limiting regimes for the dipole cross section. They have conveniently rewritten the convolution in terms of smooth functions with properties amenable to treatment with Fourier transforms. First, the integral over dα and the angular integral in d 2 r in eq. (2) are performed to write Here, we include the electric charge in the definition of h, different from the convention in Ref. [37]. With z = Qr and further definitions and defining z 0 ≡ Q 0 r 0 , For a single, massless quark of unit charge, a "toy model," this can be rewritten [37] F where with e f in h(z 0 e τ , 0) set to unity. The dipole cross section labeled with a subscript "0" represents the N f = 1, m f = 0 case with e f = 1. Fig. 1 shows κ(τ ) versus τ , a function independent of Q for the massless case where we have set z 0 = 2.4 so that κ(τ ) is nearly symmetric around τ = 0. Ewerz et al. in Ref. [37] exploit the peaked nature of κ(τ ) at τ = 0 to derive approximate relations betweenσ and ∂F 2 /∂t for both the large Q 2 and small Q 2 regime. Our approach here is to use the smooth, nearly symmetric form of κ(τ ) for the massless quark case to deconvolute the integral in Eq. (2).
Given F (x, t) in Eq. (9) for a wide range of t (namely, a wide range of Q), S(x, t ′ ) can be found by taking the Fourier transform to factorize the convolution, then inverting the Fourier transform. In the next section, we perform this procedure numerically where we use F 2 (x, Q 2 ) parameterized by Donnachie and Landshoff (1993) [22]. Even though this parametrization has a limited range of applicability in its description of the experimental data (e.g., 0.1 GeV 2 < Q 2 < 10 GeV 2 ), we use this simple form to demonstrate the procedure and to interpret the results forσ.

III. MASSLESS, UNIT CHARGE QUARK EXAMPLE
As a starting point for using the Fourier transform to extractσ from a parametrization of F 2 , we consider the toy model of a massless quark with unit charge.
Using the usual formulas for the Fourier transform and its inverse, for a completely symmetric function κ(τ ), b κ (k) = 0. We approximate b κ (k) ≃ 0 in what follows. We find that the dipole cross section we obtain, when convoluted with the photon wave function and appropriately normalized, is a good approximation to the parameterized structure function. Because in the massless case κ(τ ) depends only on τ , the convolution integral factorizes with Fourier transformed and simplifies to the following when b κ = 0: The Fourier transform of κ is straightforward. For the Fourier transform of F , as noted above, we need F 2 (x, Q 2 = Q 2 0 e 2t ) for the full range of Q 2 , in principle between Q 2 = 0 → ∞. The Donnachie-Landshoff parametrization [22] meets the requirement of being defined for all Q 2 , even if not valid for the full range, where A = 0.324 a = 0.5616 GeV 2 α = 1.0808 B = 0.098 b = 0.01114 GeV 2 . β = 0.5475 This is based on a fit to NMC data [38] for Q 2 < 10 GeV 2 . Our comparison will be restricted to the Q 2 range between 0.1 − 10 GeV 2 . We have chosen Q 0 = 0.82 GeV so that F (x, t) has a maximum at t = 0. While it is not completely symmetric, F (x, t) has a similar shape for t less than and greater than zero, for small x.  The resulting dipole cross sectionσ 0 from the Fourier transform and inversion of F and κ, for m f = 0 with a unit charged quark, is shown in Fig. 2. Three values of x are shown, x = 10 −3 , 10 −4 and 10 −5 in increasing order. The dipole cross section can be parameterized aŝ  Table I. The x dependence follows directly from the x dependence of F 2 in the Donnachie-Landshoff parametrization, where for this range of x and Q 2 > 0.1 GeV 2 , the first term in Eq. (15) dominates. Fig. 3 shows the result for F 2 when the dipole cross section shown in Fig. 2 is convoluted with the massless unit-charged quark wave function. It matches well with the original parametrization up to Q ≃ 2 − 3 GeV. At Q 2 = 0.1 GeV 2 , the dipole formula withσ 0 is about 7% larger than the Donnachie-Landshoff parametrization of F 2 , while for Q 2 = 10 GeV 2 , the dipole formula is about 11% below the D-L parametrization. We do not get an exact match to F 2 in our inversion because we have made the approximation b κ (k) = 0 and some numerical approximations in our integration at large k. Nevertheless, the procedure works reasonably well. As we show below, when m f = 0, this approximate σ 0 is sufficient.

IV. MASSIVE QUARKS
For massive quarks, eq. (2) involves a sum over flavors and the introduction of an r dependence in h(z = Qr, m f r). We will rewrite m f r = m f z/Q. Fig. 4 shows a comparison of κ for m f = 0 (dashed line) and κ for five flavors with m f = 0 and Q = 1 GeV.To findσ, we take as our starting pointσ 0 . Since the γ * p center of mass energy squared W 2 ≃ Q 2 /x, for x < 5 × 10 −3 we have , so we use all five flavors in the massive case. The toy modelσ 0 must be normalized to account for the electric charge and the sum over flavors with massive quarks. We findσ Here, N σ is the factor that indicates the effect of the charge and the masses of quarks, and this is expressed as (19) where η m {u,d,s} = 0.9292, η mc = 0.0376, and η m b = 7.959 × 10 −4 . These values of η account for the effects of the quark masses in the photon wave functions. Usingσ 0 , we found η m f from convolutingσ 0 with h(z, m f z/GeV) compared to with h(z, 0), so the normalization factor depends on the input F 2 which in turn determinesσ 0 .
With the inclusion of masses for five quark flavors, we show F 2 (x, Q 2 ) versus Q 2 for x = 10 −3 , 10 −4 , 10 −5 in Fig. 5. The agreement between the original Donnachie-Landshoff parametrization shown by the solid lines and the structure function evaluated usingσ is quite good for the range of validity of the DL parametrization.
For reference, we illustrate the range of r most relevant for the evaluation of F 2 . Fig. 6 shows the ratio for F 2 (x, Q 2 ) = F 2 (x, Q 2 , r max → ∞) evaluated by eq. (2). When Q 2 = 10 GeV 2 , approximately 10% of the structure function comes from lowσ at low r, r < ∼ 1 GeV −1 , and approximately 10% comes from r > ∼ 4 GeV −1 . For Q 2 = 0.1 GeV 2 , 80% of the evaluation of F 2 (x, Q 2 ) comes from r ∼ 2 − 5 GeV −1 . Thus, the bulk of the evaluation of F 2 for the range of Q 2 of interest for the Donnachie-Landshoff parametrization is in the range of r ∼ 1 − 5 GeV −1 where the dipole cross section makes the transition from the low r to large r forms.

V. DISCUSSION
The parameters for fits to the dipole cross section extracted from the Donnachie-Landshoff F 2 structure functions are shown in Table 1. To first approximation, a simple power law at low r and high r is approximatelŷ The rising behavior at low r is characteristic of other parameterizations of the dipole cross section, but the falling behavior is not. In this section, we compareσ with two examples of dipole cross sections, the Golec-Biernat and Wüsthoff cross section [12] and the Soyez parametriza- tion [16] of the dipole cross section based on the Iancu-Itakura-Munier form [15]. The Golec-Biernat and Wüsthoff color dipole cross section is [12] σ GBW = σ GBW where σ GBW 0 = 23 mb, λ = 0.29, x 0 = 3 × 10 −4 and r c = 1 GeV −1 . This is often written in terms of an x dependent saturation scale Q s (x) = Q 0 (x 0 /x) λ/2 where Q 0 = 1/r c = 1 GeV. The approximate functional form incorporates the observed geometric scaling behavior and depends only on the one combined variable T r = rQ s . The limiting behavior at small r is thereforeσ GBW ≃ σ GBW 0 (T r /2) 2 , while at large r,σ GBW → σ GBW 0 . In Ref. [16], Soyez writes the color dipole cross section in terms of the scaling variable T r , the saturation scale Q s (x) and where Y = ln(1/x). In this parametrization, N 0 = 0.7, γ s = 0.738, λ = 0.220, x 0 = 1.63 × 10 −5 and σ S 0 = 27.3 mb. The "effective anomalous dimension" in eq. (24) is with κ = 9.94 [39]. For r = 0.3 GeV −1 and x = 10 −4 , γ eff ≃ 0.84, soσ ∼ r 1.68 at small r for this value of x, a slower rise with r than either the GBW dipole or ourσ. The low r and large r approximate forms of the dipole cross section and their relation to F 2 are discussed in Ref. [37]. Our dipole cross section does not precisely match these limiting forms. For small r, Ewerz et al. have shown that [37], This relation relies on approximations including that σ(r, x)/r 2 is slowly varying for small r and on the approximate relation that Qr ≃ z 0 . At large Q 2 (equivalent to small r), Q 2 ∂F 2 /∂Q 2 ∼ 1/Q 2 , soσ ∼ r 2 /Q 2 ∼ r 4 /z 2 0 . This dependence on r is contrary to the assumption that σ ∼ r 2 as r → 0, e.g., the low r form of the GBW dipole cross section. The dipole cross section we obtain scales approximately asσ ∼ r 3.34 around r ∼ 1 GeV −1 . Given thatσ(r, x)/r 2 is not slowly varying for small r, one would not expect a precise agreement between our low r parametrization and Eq. (26).
For large r, the small Q 2 regime, the approximate behavior ofσ is related to the logarithmic derivative of the virtual photon cross section. To get the correct physical behavior of σ γ * p for Q 2 → 0, one needs to fix W 2 ≃ Q 2 /x in σ γ * p ∝ F 2 /Q 2 . We therefore write, The Donnachie-Landshoff parametrization of F 2 together with this large r approximation leads toσ ∼ r −2 . The dipole cross section we determined hasσ ∼ r −2.62 for r ≃ 5 GeV −1 . Eq. (27) also relies onσ being a slowly varying function of r at large r, so the fact that our approximate power law behavior deviates fromσ ∼ r −2 is again not surprising. As remarked in Ref. [37], the falling dipole cross section as a function of large r is correct when one uses the standard perturbative photon wave function for all Q 2 , even for very low Q 2 . The GBW dipole hasσ → σ c , a constant value. With the standard perturbative photon wave function, the GBW form of the dipole cross section yields a virtual photon-proton cross section σ T (W 2 , Q 2 ) + σ L (W 2 , Q 2 ) that does not have the correct Q 2 → 0 behavior without a modification of the low Q 2 photon wave function. Our approach here is to use the standard perturbative wave function even at the lowest Q 2 values. The result is thatσ(r, x) decreases at large r.

VI. CONCLUSIONS
In summary, we have used the simplified form of the convolution formula in the color dipole formalism with m f = 0 to extract the dipole cross section from a parametrization of the structure function F 2 (x, Q 2 ). As a demonstration of the method, we have used the Donnachie-Landshoff parametrization because its simple form illustrates some of the limiting behavior discussed in Ref. [37]. We find that consistent with the Ewerz et al. discussion in Ref. [37], unless the perturbative photon wave function is modified for low Q 2 , the dipole cross section should fall as r becomes large, contrary to what is assumed in many models. The low and high r relationships between F 2 and dipole cross section discussed in Ref. [37] are only approximate according to our dipole extraction.
Our approach is complementary to much of the current literature in which a theoretical form of dipole cross section is postulated and its parameters fit to F 2 data. Our dipole formulas in eqs. (16)(17)(18) reproduce the Donnachie-Landshoff results for F 2 (x, Q 2 ) for a range of x and for 0.1 GeV 2 < Q 2 < 10 GeV 2 . The dipole does not exhibit geometric scaling in T r because this parametrization of F 2 does not exhibit that same scaling behavior in T . Work is in progress to extractσ from a range of other parameterizations of F 2 . This is not completely straightforward because many parameterizations of F 2 do not per-mit the required Fourier integration from Q 2 = 0 → ∞. In particular, low Q extrapolations are required. Ultimately, one would like to find a range of dipole cross sections that represent a range of postulated x and Q dependencies in parameterizations of F 2 (x, Q 2 ), e.g., as in Refs. [23,29,30].