The New F_L Measurement from HERA and the Dipole Model

From the new measurement of F_L at HERA we derive fixed-Q^2 averages. We compare these with bounds which are rigorous in the framework of the standard dipole picture. The bounds are sharpened by including information on the charm structure function F_2^(c). Within the experimental errors the bounds are respected by the data. But for 3.5 GeV^2<= Q^2<= 20 GeV^2 the central values of the data are close to and in some cases even above the bounds. Data on F_L/F_2 significantly exceeding the bounds would rule out the standard dipole picture at these kinematic points. We discuss, furthermore, how data respecting the bounds but coming close to them can give information on questions like colour transparency, saturation and the dependencies of the dipole-proton cross section on the energy and the dipole size.


Introduction
Recently new results for the structure functions F L and F 2 of deep inelastic electronand positron-proton scattering (DIS) have been published by the H1 Collaboration [1]. In this note we compare these results with predictions of the popular colour-dipole model of DIS. That is, we investigate if the data respect certain bounds for the ratios of structure functions. These bounds are rigorous predictions of the dipole model and rely only on the non-negativity of the dipole-proton cross section.
The kinematics of e ± p scattering is well known, see for instance [1,2]. The reaction is e ± (k) + p(p) −→ e ± (k ′ ) + X(p ′ ) (1) and we use the variables ( The measured structure functions F 2 and F L are related to the cross sections σ T and σ L for absorption of transversely or longitudinally polarised virtual photons by Here Hand's convention [3] for the virtual-photon flux factor is used and terms of order m 2 p /W 2 are neglected. For low to moderate values of Q 2 the dipole picture for DIS [4,5,6] is frequently used to describe the data. For various applications of the dipole model see for instance [7]- [27]. In [28,29] this dipole picture was thoroughly examined using functional methods of quantum field theory. In particular, the assumptions were spelled out which one has to make in order to arrive at the standard dipole-model formulae for σ T and σ L or, equivalently, F 2 and F L , see section 6 of [29]. In (4) w (q) T,L are the probability densities for the virtual photon γ * splitting into a quark-antiquark pair of flavour q and transverse separation r. Their standard expressions are given in Appendix A. An integration over the quark's longitudinal momentum is performed. The cross section for the qq pair scattering on the proton is denoted byσ (q) (r, ξ). This cross section depends on r and an energy variable ξ the choice of which is left open here. In [28,29,30,31,32] it was argued that the correct variable to choose is ξ = W . However, in the literature the energy variable used most frequently in the dipole cross section is ξ = x.
In the standard dipole model formulae (4) the densities w (q) T,L are known (see Appendix A) but the dipole-proton cross sectionsσ (q) have to be taken from a model. In the following we shall only use that they have to be non-negative, This alone allows to derive a non-trivial upper bound, valid in any dipole model, on the ratio see [29,30]. Equivalently, one can obtain a non-trivial upper bound on the ratio This bound can be substantially improved if information on the charm structure function F (c) 2 (x, Q 2 ) is included [31]. There is then an allowed domain, again valid in any dipole model, for the two-dimensional vector It is the purpose of this note to confront the dipole-model bounds on F L /F 2 and on the vector V (x, Q 2 ) with the new HERA results [1]. This is done in section 2. In section 3 we discuss the results, and we give a summary in section 4.

The dipole-model bounds and the data
We discuss first the bound for the ratio F L /F 2 of (7). For this we define where m q is the mass of the quark q. For the case of massless quarks, m q = 0, figure 1 shows 1 L )(r, Q 2 ) and g(Q, r, 0) as functions of r for three different values of Q = Q 2 (compare figure 10 of [29] for a similar plot of the function (w (q) Its behaviour for small and large r is as follows for m q = 0: For a derivation of these results and for the case m q = 0 see appendix A of [32]. For massless quarks the function g depends only on the dimensionless variable L )(r, Q 2 ) (left) and g(Q, r, m q ) (right) versus r, both for three fixed values of Q 2 and for quark mass m q = 0 ; see (4) and (9).
The functiong(z) has a maximum at It was shown in [31] that for all Q ≥ 0, r ≥ 0 and m q ≥ 0. Using then (5) the dipole-model formulae (4) lead to the bound We note that the bound (16) for F L /F 2 is equivalent to the bound for R (6) derived in [30,31], Data for F L and F 2 at the same kinematic points are presented in [1] for Q 2 values ranging from 1.5 to 45 GeV 2 . The data for the same Q 2 value span a small range of x and this range varies strongly with Q 2 ; see figure 12 of [1]. On the other hand, for all Q 2 bins the data are inside a narrow W interval 167 GeV − 232 GeV (18) with a mean value of about W 0 = 200 GeV. Therefore, in the following we find it more convenient to consider F L and F 2 as functions of W and Q 2 instead of x and Q 2 .  Since we do not expect any large variation of the ratio F L (W, Q 2 )/F 2 (W, Q 2 ) for fixed Q 2 within the W interval (18) of the measurement we have averaged the H1 data [1] for given Q 2 . Error weighted averages F L (W, Q 2 )/F 2 (W, Q 2 ) are calculated taking into account the total uncorrelated and correlated experimental uncertainties. The averages are confronted with the bound (16) in figure 2.
We note firstly, that electromagnetic gauge invariance requires for Q 2 → 0 at fixed W . The data indicate, indeed, a decrease of F L /F 2 for small Q 2 . Fitting F L /F 2 with a constant value, as done in [1], does not seem very plausible physically, in view of (19).
The second point to note is that the data in figure 2 are rather close to the upper bound (16) from the dipole model, especially so for The bound (16) on F L /F 2 can be improved if one takes into account that there is a non-vanishing contribution from charm quarks to F L and F 2 , see [31]. Specifically, considering massless u, d and s quarks, a massive c quark and neglecting b quarks we can derive certain allowed domains for the vector V (x, Q 2 ) (8) from the dipole model. Again these domains depend only on the known photon densities w (q) T,L , see (28)- (31), and on the non-negativity of the cross sectionsσ (q) , see (5). That is, for any dipole model with the standard photon probability densities w (q) T,L the vector V (x, Q 2 ) must be inside the appropriate allowed domain for the given Q 2 value. A detailed description of how these domains are obtained has been given in [31]. The allowed domains can be understood as correlated bounds for the ratios F L /F 2 and F  2 /F 2 is obtained using NLO QCD calculations provided by the OPENQCDRAD package [33], again with a charm pole mass of m c = 1.23 GeV. For this calculation the JR09FFNNLO parametrisation [34] of the proton parton density functions was used, which was found to describe preliminary HERA charm data [35] very well within the experimental correlated uncertainties of typically 3-9%. Here and in the following we do not consider the data point at Q 2 = 1.5 GeV 2 from [1] as it has an exceedingly large error.
The significance of the data points in relation to the bound can be seen more clearly

Discussion
We see from figures 2-4 that the data for F L /F 2 as derived from [1] come very close to the bounds which result from the dipole picture. We now discuss the meaning of this observation from the points of view of both, a dipole-model enthusiast, and a dipole-model sceptic, respectively.

Dipole-model enthusiast's view
The dipole-model enthusiast will say that within the errors of the data the bounds are respected. Furthermore, he can use the data to give qualitative arguments concerning the behaviour of the dipole-proton cross sections for small and large radii r. Let us assume power behaviour ofσ (q) (r, ξ) for r → 0 and r → ∞, Taking into account (10) we find that the integrals for F 2 and F L in (4) are convergent if a > 0 and b > 0 .
Of course, with the usual assumptions of colour transparency for small r, implying a = 2, and of saturation for the dipole-proton cross sections for large r, implying b = 2, the requirements (23) are satisfied. From the experimental findings of figures 2 to 4 we can now give qualitative arguments based on the data, that the exponents a and b in (22) cannot be too small. Indeed, for a small value of a the cross sectionsσ (q) (r, ξ) would decrease only slowly for r → 0 and this region of small r would contribute significantly in the integrals (4). But, as we see from the second plot in figure 1, the function g(Q, r, 0) is small there and this would lead to a small value for F L /F 2 , much below the bound (16), contrary to what is seen in the data. A similar argument applies to the exponent b in (23), considering the large r behaviour of g(Q, r, 0) in figure 1. Thus, the dipole-model enthusiast may hope that with more data it may even be possible to determine the exponents a and b from the data on F L /F 2 directly without making model assumptions forσ (q) (r, ξ).

Dipole-model sceptic's view
Let us now go over to the point of view of the dipole-model sceptic. He will note that some central values of the data for F L /F 2 in figure 3 are, in fact, above the corresponding bound. If any of the measured points with F L /F 2 > (F L /F 2 )| bound is confirmed, with corresponding small error, by further experiments then, as a clear consequence, the standard dipole picture would not be valid at this kinematic point. But what would be the consequences if the bound for F L /F 2 is not violated but saturated ? For the sake of the argument we shall now for a moment assume that the bound for F L /F 2 is reached in the Q 2 range (20). Clearly, this is not incompatible with the data, see figures 3 and 4. The consequence is that the dipole-proton cross sectionsσ (q) (r, ξ) in (4) should only contribute at that particular r values where the functions g(Q, r, 0) and g(Q, r, m c ) of (9) have their maximum. This is for both functions the case for r ≈ 0.51 fm We see this for g(Q, r, 0) from the second plot in figure 1 and this also holds for g(Q, r, m c ). Thus, the cross sectionsσ (q) (r, ξ) should be strongly peaked at these r values for the whole Q 2 interval (20), something like a δ function The corresponding r values range from 0.27 fm for Q 2 = 3.5 GeV 2 to 0.11 fm for Q 2 = 20 GeV 2 . With increasing Q the position of the delta function peak in (25) moves to smaller r values. As we have argued at length in [28,29,32], the correct energy variable in the dipole-proton cross sectionσ (q) (r, ξ) is ξ = W . Since the data on F L /F 2 is essentially at one value of W ≈ W 0 = 200 GeV (more precisely, in the narrow range (18) around W 0 ) we get from (25) a Q 2 dependence inσ (q) (r, W 0 ) which should not be there. The conclusion is that a saturation of the bound on F L /F 2 in a whole Q 2 interval as in (20) is incompatible with the dipole model and the dipole-proton cross sections having the correct functional dependenceσ (q) (r, W ).
With the -incorrect -choice of energy variable ξ = x inσ (q) (r, ξ) we get the following. Since the data on F L /F 2 is essentially at W = W 0 (namely in the narrow range (18) around W 0 ), we have from (2) Inserting this in (25) giveŝ Thus, there is in this case no immediate conflict with the functional dependencê σ (q) (r, x). But we note that as x decreases the peak of the cross sectionσ (q) (r, x) in (25) shifts to larger values of r. This is in contrast to what one finds in popular dipole models, like the one invented by Golec-Biernat and Wüsthoff [7]. There, one assumes a dipole-proton cross section saturating at large r with an x-dependent saturation scale. But in that model for decreasing values of x the cross sectionσ (q) (r, x) moves to smaller values of r, see figure 2 of [7]. This is in contradiction to what we found above in (27). The dipole-model sceptic could, furthermore, argue as follows. Since the bounds explored in the present paper are just more or less satisfied by the data it will certainly pay to explore further rigorous bounds which can be constructed using the methods of [31]. One could, for instance, consider correlated bounds on F L /F 2 at different Q 2 values. It remains to be explored if the dipole model survives such extended tests.

Summary
In this paper we have compared the recent data on F L /F 2 -to be precise: their fixed-Q 2 averages -with rigorous bounds derived in the framework of the dipole model. Within the experimental errors the bounds are satisfied. But the data is surprisingly close to the bounds for 3.5 GeV 2 ≤ Q 2 ≤ 20 GeV 2 . We have discussed the meaning of these findings from the points of view of both, the dipole-model enthusiast and the sceptic. The enthusiast will have to admit that the sceptic's arguments could give problems to the dipole picture if the central values of the data are confirmed with small errors by further experiments. The sceptic will have to concede that, given the errors of the data, δ functions for the cross sectionsσ (q) (r, ξ) as in (25) and (27) are not really necessary and that the widths of the distributions compatible with the data have to be explored. Thus, given the present data, we must leave it to the reader if he will join the camp of the enthusiast or that of the sceptic. More data with small errors would be needed to decide the issue. In any case we hope to have demonstrated in our paper that measurements of F L /F 2 give very valuable information on the dipole picture, its validity, and potentially on questions like colour transparency and saturation of the dipole-proton cross section. Thus, programs for future electron-and positron-proton scattering experiments (see for instance [36], [37]) certainly should foresee F L measurements as an important item on the list of physics topics.