Longitudinal structure function $F_{L}$ at low $Q^{2}$ and low $x$ with model for higher twist: an update

Knowledge of both $F_{2}$ and $F_{L}$ structure functions, from the photoproduction to the deep inelastic region is needed in the calculations of QED radiative corrections to the data from the Deep Inelastic Scattering (DIS) process, l+p $\rightarrow$ l^′+X (l, l^′ are leptons), Ref. [1]. This knowledge is also essential in verifying the sum rules, e.g. the Gottfried sum rule or (in case of the spin-dependent function $g_{1}$) the Bjorken sum rule [2], a fundamental relation of Quantum ChromoDynamics (QCD). Thus the electroproduction structure functions, $F_{2}$, $F_{L}$ and $g_{1}$ in the full kinematic region are indispensable in the data analysis, especially in the context of the future DIS facilities like the Electron Ion Collider (EIC), currently planned in the US [3]. In this paper we shall consider the spin-independent longitudinal structure function $F_{L}$.

The structure functions depend on two variables: $x$ and $Q^{2}$, conventionally defined as $x=Q^{2}/(2p\cdot q)$ and $Q^{2}=-q^{2}$ where $q$ and $p$ denote the four momentum transfer between the incident and scattered leptons and the four momentum of the target proton respectively; these four-vectors and the four-momentum $p_{l}$ of the incident lepton, define the inelasticity $y$ as $y=p_{l}\cdot p/(q\cdot p)$. Unlike $F_{2}$ and $g_{1}$, the experimental data for $F_{L}$ are rather scarce for the low $Q^{2}$ values and thus an extrapolation to this region needs to be performed with a physically motivated model with least number of free parameters. The model should also contain an extrapolation to the region of low $x$, as data both from fixed-target and colliders correlate these two kinematic regions. Such model for $F_{L}$ was proposed some time ago [4]. It was based on the $k_{T}$-factorization formula which involves the unintegrated gluon density with their transverse momentum ($k_{T}$) dependence. The $k_{T}$-factorization was derived in the high-energy limit of $s\gg|t|$, where $s$ is the centre-of-mass energy squared in the scattering process and $t$ is the four-momentum transfer in the $t$-channel. It was derived for processes like heavy quark production in hadron-hadron collisions as well as for DIS. In principle the unintegrated gluon distribution should be obtained from the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation, which resums the large logarithms of energy (or small $x$). In the model used in Ref.[4] the unintegrated gluon distribution function was constructed from the collinear integrated gluon density through the logarithmic derivative over the scale dependence. This effectively neglected higher order small $x$ contributions in the gluon anomalous dimension; these contributions are expected to be significant only at extremely low values of $x$.

The $k_{T}$-factorization formula was then extrapolated to the low $Q^{2}$ region by introducing the cutoff on the low quark transverse momenta. This region is dominated by the soft physics with a higher twist contribution to $F_{L}$ vanishing at large $Q^{2}$. In the model it was treated phenomenologically and its normalization was determined from the (non-perturbative) part of the structure function $F_{2}$. The model embodied the kinematic constraint $F_{L}\sim Q^{4}$ in the limit $Q^{2}\rightarrow 0$ for fixed $2p\cdot q$. It thus contained only physically motivated parameters.

In this paper we revisit the model of Ref. [4] for the extrapolation of $F_{L}(x,Q^{2})$ to the region of low values of $Q^{2}$ at low $x$. We have included the updated parametrizations of the Parton Distribution Functions (PDFs) for the quark and gluon densities in the proton, checked the sensitivity of the results to the assumed quark masses and tested the gluon distributions supplemented with the Sudakov form-factor.

The calculations have been compared with high $Q^{2}$ measurements of $F_{L}$ by HERA and with low $Q^{2}$ data by SLAC and JLab. Here one has to be aware that most of the SLAC and JLab data are in the regime which only marginally overlaps with the region of applicability of the model, as they correspond to rather high values of $x\lower 2.71246pt\hbox{$\;\stackrel{{\scriptstyle\textstyle>}}{{\sim}}\;$}0.1$ .

The outline of the paper is as follows. In the next section we discuss the general properties of the longitudinal structure function, in Sec. 3 the $k_{T}$-factorization is introduced, while in Sec. 4 the details of the model for the higher twist are given. Numerical results are presented in Sec. 5. Finally, in Sec. 6 we state our conclusions.

In the ‘naive’ quark-parton model the structure function $F_{L}(x,Q^{2})$ vanishes, as a consequence of quarks having spin 1/2. More precisely, $F_{L}$ vanishes when the transverse momenta of the quarks are limited. At large $Q^{2}$, $F_{L}$ is proportional to ($\langle m_{q}^{2}\rangle+\langle\kappa_{T}^{2}\rangle)/Q^{2}$, where $m_{q}$ is the quark mass and $\kappa_{T}$, the quark transverse momentum, is by definition limited in the naive parton model. This remains approximately valid in the leading logarithmic order in the collinear approximation. At higher logarithmic orders the point-like QCD interactions (gluon emissions) allow for the average transverse momenta $\langle\kappa_{T}\rangle$ to grow with increasing $Q^{2}$. Thus at higher orders of perturbation theory $F_{L}(x,Q^{2})$ acquires a leading twist contribution. In the limit of small $x$, the longitudinal structure function is driven by the gluons through the $g\rightarrow q\overline{q}$ transition and thus permits a direct measurement of the gluon density in the nucleon.

In the limit $Q^{2}\rightarrow 0$ the structure function $F_{L}$ has to vanish as $Q^{4}$ (for fixed $2p\cdot q$) which eliminates potential singularities at $Q^{2}=0$ of the hadronic tensor $W^{\mu\nu}$. It also reflects the vanishing of the cross section $\sigma_{L}\sim F_{L}/Q^{2}$ in the real photoproduction limit.

The longitudinal structure function is theoretically fairly well understood at high $Q^{2}$, thanks to the framework of perturbative QCD. On the contrary, very little is known about its extrapolation towards the region of low $Q^{2}$; it is possible that it contains large contributions from higher twists there.

The calculated $F_{L}$ is shown in Figs 2 and 3 as function of $Q^{2}$ in bins of $x$ and as function of $x$ in bins of $Q^{2}$, respectively. Two sets of the quark and gluon PDFs, both at the leading order accuracy, were used: the GRV94LO [31] (used previously in Ref. [4]) and CT14LO [32]. In the region of our interest, low $x$ and low $Q^{2}$ values the results practically do not depend on that choice. We have also verified that changing from LO to NLO PDFs has negligible impact on our results in the non-perturbative region. Therefore we stick to the LO choice of the PDFs. In what follows the CT14LO parametrization²²2The more modern distributions, e.g. CT18 do not contain the LO PDFs. will be used.

Apart from the approach displayed in Eq. (9) we also used the unintegrated gluon density with the Sudakov form-factor, Eq. (10) and found negligible differences between these two approaches. Besides the dominant (at small $x$) photon-gluon fusion $\gamma^{*}+g\rightarrow q+\bar{q}$ mechanism, we have also included a contribution from quarks, $\gamma^{*}+q(\bar{q})\rightarrow q^{\prime}(\bar{q}^{\prime})+g$ , taking into account threshold effects, for details see Ref.[4].

In the calculations we took into account the contributions from u, d, s and c quarks of masses 0.35, 0.35, 0.5 and 1.5 GeV respectively ($m_{q,h}$ masses). For comparison lower quark masses equal to 0.14 GeV for u, d, s quarks were also employed ($m_{q,l}$). The parameter $\kappa_{0T}^{\prime 2}$ was set to 0.8 GeV². This value was varied in the interval 0.5 – 1.2 GeV² which resulted in changing the $F_{L}$ by at most 10%.

In Ref. [4] we also have performed calculations within the on-shell approximation, which corresponded to setting the gluon transverse momentum to zero in the photon-gluon amplitude, Eq. (2) and restricting the integration over $k_{T}^{2}$ to $k_{T}^{2}\ll Q^{2}$. We found that the differences between off-shell and on-shell calculations are small, particularly in the low $Q^{2}$ region. Therefore in this update we only stick to the off-shell calculation.

The (small) $x$-dependence is weak at low $Q^{2}$ and slightly growing with decreasing $x$ while the $F_{L}$ at low $Q^{2}$ and low $x$ is very small, less than 0.005 (for $Q^{2}\lesssim 0.1$ GeV²) and strongly decreasing with decreasing $Q^{2}$. Contributions to $F_{L}$ from the quarks, the perturbative part and the higher twist are illustrated in Figs 4 - 5 and clearly visualize the interplay of different mechanisms in building the $F_{L}$ in our model. We note that the perturbative mechanism contributes very little in the low $Q^{2}$ region.

In Fig. 6 we compare the calculations with measurements. Unfortunately the latter are very scarce in the non-perturbative region, see Fig. 1 in Ref.[33] where a compilation of $F_{L}$ measurements is given. Only results from the JLab E99-118 [34], E94-110 [35], E00-002 [33], SLAC E140X [36] and SLAC GLOBAL analysis [37] extend to the edge of the low $x$ non-perturbative region. There our model with standard light quark masses clearly underestimates the data (broken line in Fig. 6). On the other hand, results for the set of low masses of the light quarks (u,d,s), $m_{q,l}=0.14$ GeV which were used in the dipole model calculations [23] seem to reproduce the measurements well (continuous line in Fig. 6). The strong dependence of the results on the assumed quark masses, is illustrated in Fig. 7 for two kinematic $(x,Q^{2})$ points.

Finally, the H1 Collaboration measurements of $F_{L}$ in the perturbative region at HERA, [5] are very well reproduced, see Fig. 8.

In this paper a parametrization of the proton longitudinal structure function $F_{L}$ at low $Q^{2}$ and low $x$ is presented. It is a revisit and update of the parametrization modelled in Ref.[4] and based on the photon-gluon fusion treated in the $k_{T}$ factorization, suitably extrapolated to the region of low $Q^{2}$. There is a severe need to know $F_{L}$ in this region, down to $Q^{2}=0$, in the electroproduction data analysis , e.g. in the QED radiative corrections. Unfortunately there are practically no $F_{L}$ measurements in this region. This leaves the modelling constrained only at the upper end of the validity interval, as (scarce) measurements are limited to $Q^{2}\lower 2.71246pt\hbox{$\;\stackrel{{\scriptstyle\textstyle>}}{{\sim}}\;$}$ 0.1 GeV² and $x\lower 2.71246pt\hbox{$\;\stackrel{{\scriptstyle\textstyle>}}{{\sim}}\;$}$ 0.1, the latter being the limit of applicability of our model.

The model fulfills the kinematic constraint $F_{L}\sim Q^{4}$ at $Q^{2}\rightarrow 0$. It also contains a higher twist term, which can be interpreted as a soft Pomeron exchange. This term comes from the low transverse momenta of the quarks in the quark box diagram. The coupling of the soft Pomeron to the external virtual photon is modeled by a constant, which is fixed from the non-perturbative term of the transverse structure function. Thus our model for the higher twist in principle does not have any free parameters.

As in the previous version of the model we have used the prescription for the unintegrated gluon density as the logarithmic derivative of the standard integrated gluon density. The unintegrated gluon density which includes the Sudakov form-factor was also employed as an alternative. Compared to the old version of the model, we have used newer parton distributions. The updated parton distributions did not affect the results in the regime of low $x$ and low $Q^{2}$ in any significant way.

The structure function $F_{L}$ turns out to be very small at $Q^{2}\lower 2.71246pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}$ 0.1 GeV², essentially independent of $x$ at low $x$ and low $Q^{2}$ and practically insensitive to the input parton distributions and perturbative accuracy (LO or NLO) of the PDFs. The model of $F_{L}$ underestimates the JLab and SLAC measurements, performed at low $Q^{2}$ and $x\sim 0.1$ unless the light quark masses are lowered down to 0.14 GeV, as in the dipole model, which brings the calculations to a very good agreement with the data. These data are however very scarce. On the other hand the model reproduces well the H1 measurements taken in the perturbative region ($Q^{2}\lower 2.71246pt\hbox{$\;\stackrel{{\scriptstyle\textstyle>}}{{\sim}}\;$}$ 1.5 GeV²). In conclusion, we believe that the presented model will be useful at several stages of the data analysis, e.g. evaluation of the QED radiative corrections at the Electron Ion Collider.

We thank Fredrick Olness and Wojciech Słomiński for discussions; we express our gratitude to Eric Christy, Cynthia Keppel, Peter Monaghan, Ioana Niculescu, and Katarzyna Wichmann for supplying us with and discussing the measurements. AMS is supported by the U.S. Department of Energy grant No. DE-SC-0002145 and in part by National Science Centre in Poland, grant 2019/33/B/ST2/02588. The code for calculating $F_{L}(x,Q^{2})$ is available upon request from the authors.