---
title: F_L in Deep Inelastic Scattering
url: https://www.emergentmind.com/topics/f_l
type: topic
---

# F_L in Deep Inelastic Scattering

In deep-inelastic scattering (DIS), \(F_L(x,Q^2)\) denotes the longitudinal structure function, i.e. the hadronic response to longitudinally polarized virtual photons. It is related to the standard unpolarized structure functions by
\[
F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),
\]
enters the reduced cross section with a coefficient proportional to \(y^2\), and vanishes in the naive quark-parton model, so a nonzero \(F_L\) is intrinsically a QCD effect [1606.02614][0805.2809]. At low Bjorken \(x\), \(F_L\) is especially sensitive to gluon dynamics, whereas at low \(Q^2\) current conservation implies the kinematic constraint \(F_L\propto Q^4\) as \(Q^2\to 0\) [1402.0163][2202.04223].

## 1. Definition and kinematic role

The standard inclusive DIS reduced cross section at \(Q^2\ll M_Z^2\) may be written as
\[
\sigma_r(x,Q^2,y)=F_2(x,Q^2)-\frac{y^2}{Y_+}F_L(x,Q^2),\qquad Y_+=1+(1-y)^2,
\]
so the longitudinal contribution is isolated by the high-\(y\) term [0805.2809][2412.16123]. In fixed-target notation, the same physics is expressed through
\[
\frac{1}{\Gamma}\frac{d^2\sigma}{d\Omega\, dE'}=\sigma_T+\varepsilon \sigma_L,
\]
with
\[
F_L(x,Q^2)=\frac{Q^2}{4\pi^2\alpha}(1-x)\sigma_L,
\qquad
R(x,Q^2)=\frac{\sigma_L}{\sigma_T}=\frac{F_L}{2xF_1},
\]
so \(F_L\) is directly tied to longitudinal virtual-photon absorption [1606.02614].

A complementary decomposition, especially common in heavy-quark leptoproduction, uses transverse and longitudinal photon structure functions:
\[
F_2(x,Q^2)=2x\bigl(F_T+F_L\bigr),\qquad R(x,Q^2)=\frac{F_L}{F_T}.
\]
This basis makes explicit that \(F_2\) mixes transverse and longitudinal response, whereas \(F_L\) isolates the longitudinal channel [1604.04310]. In inclusive DIS one also has
\[
0\le F_L\le F_2,
\qquad
R=\frac{\sigma_L}{\sigma_T}=\frac{F_L}{F_2-F_L},
\]
so \(F_L\) can be viewed either as an independent structure function or as the numerator of a longitudinal-to-transverse ratio [0805.2809].

The experimental consequence of these formulae is that \(F_L\) is not obtained from a single cross-section measurement at fixed \((x,Q^2)\). A Rosenbluth-type variation of \(y\), achieved either by changing beam energies or by combining measurements at different \(\varepsilon\), is required to extract the slope in \(y^2/Y_+\) or \(\varepsilon\) [0805.2809][1606.02614].

## 2. Factorization and perturbative structure

In QCD factorization, DIS structure functions are convolutions of coefficient functions with parton distributions. For the longitudinal channel at small \(x\), the gluonic contribution may be written as
\[
F_L^g(x,Q^2)=K^G\!\left(\frac{x}{y},Q^2\right)\otimes G(y,Q^2),
\]
with a perturbatively calculable gluonic DIS kernel \(K^G\) [1402.0163]. In the non-singlet formulation,
\[
{\cal F}_{L,\rm ns}(x,Q^2)=\big[\,{\cal C}_{L,\rm ns}\otimes q_{\rm ns}\,\big](x,Q^2),
\qquad
{\cal F}_L=\frac{1}{x}(F_2-2xF_1),
\]
which makes the coefficient-function content of \(F_L\) explicit [2211.16485].

A distinctive perturbative feature is that massless \(F_L\) has no Born term. In the notation \(a_s=\alpha_s/(4\pi)\), the longitudinal coefficient function begins as
\[
{\cal C}_{L,\rm ns}(x,Q^2)=\sum_{n=1}^\infty a_s^{\,n}\,c_{L,\rm ns}^{(n)}(x),
\]
so the leading contribution is already radiative [2211.16485]. This is the operator-level expression of the familiar statement that \(F_L=0\) in the naive quark-parton model and becomes nonzero only through QCD dynamics [0805.2809].

The massless unpolarized Wilson coefficients for \(F_L\) have been calculated through three loops in the \(\overline{\mathrm{MS}}\) scheme, including the non-singlet quark, pure-singlet quark, and gluon channels [2208.14325]. Beyond this, the flavour non-singlet \(n_f^2\) and \(n_f^3\) contributions to the four-loop coefficient function have been obtained, with the new \(n_f^2\) terms reported to be numerically much larger than the previously known leading large-\(n_f\) \(n_f^3\) pieces [2211.16485]. This establishes \(F_L\) as a precision perturbative observable rather than merely a qualitative probe.

The perturbative organization is especially consequential because \(F_L\) is strongly gluon sensitive at low \(x\). A plausible implication is that higher-order coefficient-function control and reliable PDF evolution are both indispensable when \(F_L\) is used as a direct constraint on the gluon density.

## 3. Small-\(x\) dynamics, gluon sensitivity, and competing frameworks

At small \(x\), one widely used ansatz is a Regge-like gluon distribution
\[
G(x,Q^2)\xrightarrow[x\to 0]{} x^{-\delta},
\]
with \(\delta\simeq 0.5\) corresponding to a hard Pomeron. In an NLO analysis built on this assumption, the gluon density can be eliminated between \(F_L\) and the charm structure function \(F_2^c\), yielding
\[
F_L^g(x,Q^2)= \frac{[K^G(1-z,Q^2)\otimes (1-z)^\delta]}
{[C_{g,2}^c(1-z,Q^2)\otimes (1-z)^\delta]}\,F_2^c(x,Q^2),
\]
so \(F_L\) can be inferred from measured charm data in the small-\(x\), high-\(Q^2\) regime [1402.0163]. In that treatment, the hard-Pomeron form gives a good description of both \(F_L\) and the charm structure functions.

A different small-\(x\) approach uses double-logarithmic resummation. Starting from
\[
F_L=F_2-2xF_1=4x^2 B,
\]
the analysis identifies a \(1/x\) enhancement first appearing in the \(\alpha_s^2\) contribution to the auxiliary amplitude \(B\), and combines it with all-order resummed double logarithms to obtain the asymptotic behavior
\[
F_L \sim x^{-\Delta^{(DL)}},
\]
with \(\Delta^{(DL)}_{fix}=0.29\) for fixed coupling and \(\Delta^{(DL)}=0.07\) for running coupling [2012.10633]. The explicit conclusion is that the power-like rise of \(F_L\) at small \(x\) is a synergic effect of the \(\alpha_s^2\)-order \(1/x\) factor and the steep \(x\)-dependence of the totally resummed double logarithmic contributions.

In \(k_T\)-factorization, \(F_L\) is represented as
\[
F_L(x,Q^2)=\int_x^1\frac{dz}{z}\int d{\mathbf k}_T^2
\sum e_f^2\hat C^g_L(x/z,Q^2,m_f^2,{\mathbf k}_T^2)\,
f_g(z,{\mathbf k}_T^2,\mu^2),
\]
and therefore becomes directly sensitive to the transverse-momentum-dependent gluon density [2301.09967]. HERA \(F_L\) data were found to discriminate between two CCFM-evolved TMD gluons, with the LLM'2022 density giving a better description than JH'2013 set 2, especially at low \(Q^2\) [2301.09967].

The color-dipole picture has generated a more explicit controversy. Within the standard dipole picture, using only the positivity of the dipole cross section, one obtains the bound
\[
\frac{F_L}{F_2}\le 0.27139,
\]
and HERA averages of \(F_L/F_2\) were found to lie close to this limit for \(3.5\,\mathrm{GeV}^2\le Q^2\le 20\,\mathrm{GeV}^2\) [1201.6296]. A later comment argued that the stronger claim of a model-independent “rigorous” CDP test actually depends on the additional assumption
\[
\sigma_{(q\bar q)p}(r_\perp,z(1-z),W^2)=\sigma_{(q\bar q)p}(r_\perp,W^2),
\]
and reinterpreted the same HERA comparison as compatible with the relation \(F_L=0.27\,F_2\) in a different CDP argument [1204.5647]. The disagreement is therefore not about the numerical relevance of \(F_L/F_2\), but about which assumptions are intrinsic to the dipole framework.

## 4. Heavy-quark DIS and stable longitudinal ratios

In heavy-quark leptoproduction, the most useful longitudinal observables are often ratios rather than \(F_L\) alone. The basic definitions are
\[
R(x,Q^2)=\frac{F_L}{F_T},
\qquad
R_2(x,Q^2)=2x\frac{F_L}{F_2}=\frac{R}{1+R},
\]
with
\[
F_2(x,Q^2)=2x(F_T+F_L)
\]
throughout [1212.3783]. This reformulation matters because large radiative corrections to \(F_T\) and \(F_L\) separately can cancel in the ratio.

The key heavy-quark statement is that, contrary to the transverse structure function \(F_T(x,Q^2)\), the longitudinal one \(F_L(x,Q^2)\) does not contain leading mass logarithms of the type \(\alpha_s\ln(Q^2/m^2)\) at both LO and NLO [1604.04310]. In a comparison of FFNS with ACOT(\(\chi\)) VFNS, this implies that charm-density resummation enhances \(F_T\) but leaves \(F_L\) comparatively unaffected, driving the Callan–Gross ratio downward. Quantitatively, NLO FFNS corrections to \(R\) are reported as \(\lesssim 15\%\) for \(x\sim 10^{-3}\text{--}10^{-1}\) and \(Q^2/m^2<10^4\), whereas ACOT(\(\chi\)) decreases the LO FFNS prediction for \(R\) by about \(50\%\) for practically all \(Q^2/m^2>10\) [1604.04310]. This makes \(R\) a direct probe of charm density.

A related fixed-order analysis reached a similar conclusion from a different angle: in heavy-quark DIS, NLO contributions to \(R(x,Q^2)\) are less than \(10\%\) for \(x>10^{-4}\), and the hadron-level low-\(x\) predictions for \(R\) and \(R_2\) are stable under DGLAP evolution of the gluon density [1212.3783]. Because the reduced cross section may be written as
\[
\tilde{\sigma}(x,Q^2)=F_2(x,Q^2)\left[1-\frac{y^2}{1+(1-y)^2}R_2(x,Q^2)\right],
\]
stable analytic approximations to \(R_2\) simplify the extraction of \(F_2^c\) and \(F_2^b\) from HERA data [1212.3783].

This suggests a broader methodological point: in the heavy-quark sector, \(F_L\) is often most robust when embedded in ratios that suppress radiative and parametric instabilities while retaining strong sensitivity to gluon and heavy-flavor dynamics.

## 5. Experimental determination and world data

The first direct HERA measurement of the proton longitudinal structure function used inclusive \(e^+p\) data with positron beam energy \(27.5\) GeV and proton beam energies \(920\), \(575\), and \(460\) GeV, covering
\[
12 \le Q^2 \le 90~\mathrm{GeV}^2,
\qquad
0.00024 \le x \le 0.0036
\]
for the extracted \(F_L\) points [0805.2809]. Rosenbluth-type fits to \(\sigma_r\) versus \(y^2/Y_+\) produced positive \(F_L\) values, for example
\[
F_L=0.22\pm0.11\quad \text{at }Q^2=12~\mathrm{GeV}^2,\ x=0.00028,
\]
and
\[
F_L=0.38\pm0.10\quad \text{at }Q^2=25~\mathrm{GeV}^2,\ x=0.00062,
\]
with overall agreement with higher-order QCD calculations based on H1, MSTW, and CTEQ PDFs [0805.2809]. The measurement established that \(F_L\) is clearly positive in the explored low-\(x\) region and consistent with a sizeable low-\(x\) gluon density.

At the opposite end of the kinematic plane, Jefferson Lab Hall C experiment E00-002 performed separated measurements of \(F_L\) for hydrogen and deuterium in the region roughly
\[
0.2 \lesssim Q^2 \lesssim 1.1~\mathrm{GeV}^2,
\qquad
0.02 \lesssim x \lesssim 0.6
\]
using Rosenbluth separations [1606.02614]. The proton values were positive and typically of order a few hundredths to a few tenths, while the deuteron results were systematically smaller. Averaged over \(x\) above the resonance region, the paper found
\[
R_D-R_H = -0.042 \pm 0.018 \qquad (Q^2<5~\mathrm{GeV}^2),
\]
which was interpreted as evidence either for an unexpected proton–neutron difference in \(R\) or for a suppression of the longitudinal response in the deuteron [1606.02614].

World data have also been recast into moments. The lowest three longitudinal Nachtmann moments were extracted over
\[
Q^2 = 0.75 \text{ to } 45.0~(\mathrm{GeV}/c)^2,
\]
using global \(F_L\) and \(F_2\) data, including modern HERA and Jefferson Lab measurements [1209.4542]. Those moments were found to be underestimated by leading-twist structure-function parameterizations, especially for the higher moments, suggesting either significant higher-twist effects in \(F_L\) and/or a larger gluon distribution at high \(x\) [1209.4542].

The diffractive analogue \(F_L^{D(3)}\) has also been measured directly at HERA. In diffractive DIS,
\[
\sigma_r^{D(3)}=F_2^{D(3)}-\frac{y^2}{Y_+}F_L^{D(3)},
\]
and the extracted \(F_L^D\) was found to be nonzero, with five points more than \(3\sigma\) above zero [1107.3420]. The diffractive longitudinal-to-transverse ratio,
\[
R^D=\frac{F_L^D}{F_2^D-F_L^D},
\]
implied a relative longitudinal contribution larger than in inclusive DIS, with
\[
R^D/R = 2.8 \pm 1.1
\]
on average [1107.3420].

## 6. Low-\(Q^2\) modeling, future precision, and notation

A dedicated low-\(x\), low-\(Q^2\) model updates the older photon–gluon fusion description by extrapolating \(k_T\)-factorization toward the photoproduction region while enforcing
\[
F_L\sim Q^4 \qquad (Q^2\to 0)
\]
and adding a higher-twist term that vanishes as \(Q^2\to\infty\) [2202.04223]. In that framework,
\[
F_L=F_L^{LT}+F_L^{HT},
\]
with \(F_L^{LT}\) obtained from off-shell photon–gluon fusion with exact gluon kinematics and \(F_L^{HT}\) associated with the low-transverse-momentum region [2202.04223]. The updated model reproduces H1 data well in the perturbative region, but the sparse low-\(Q^2\) SLAC and Jefferson Lab data are underestimated unless lower light-quark masses are used, notably \(m_u=m_d=m_s=0.14~\mathrm{GeV}\) rather than \(0.35\)–\(0.5~\mathrm{GeV}\) [2202.04223].

Prospective EIC measurements indicate a sharp improvement in direct \(F_L\) determination. Using Rosenbluth extractions from inclusive pseudodata, a default five-energy scenario was studied with
\[
(18,275,141~{\rm GeV},15.4~{\rm fb}^{-1}),\ 
(10,275,105~{\rm GeV},100~{\rm fb}^{-1}),\ 
(10,100,63~{\rm GeV},79.0~{\rm fb}^{-1}),
\]
\[
(5,100,45~{\rm GeV},61.0~{\rm fb}^{-1}),\ 
(5,41,29~{\rm GeV},4.4~{\rm fb}^{-1}),
\]
and at least three usable \(\sqrt{s}\) points were required for each extraction [2412.16123]. In the optimistic \(1\%\) inter-energy systematic scenario, the absolute \(F_L\) uncertainty reaches about \(0.05\) over a wide kinematic range; reducing the luminosity to \(1~{\rm fb}^{-1}\) per energy has little impact, indicating that the measurement is already systematically limited [2412.16123]. The projected EIC coverage is described as complementary to both fixed-target and HERA data and extends down to approximately
\[
x\sim 10^{-4}
\]
for \(Q^2>1~{\rm GeV}^2\) [2412.16123].

This trajectory reinforces the status of \(F_L\) as a direct, approximately linear probe of the gluon density in global analyses [2412.16123]. It also clarifies a notational point: while \(F_L\) conventionally denotes the longitudinal DIS structure function in high-energy scattering, the same typography appears in unrelated contexts, for example as the ac Lorentz driving force \(F_L(t)=F_{\rm ac}\sin(\omega t)\) in magnetic-superconductor vortex dynamics and as the Jacobian matrix \(F_L=\nabla u_L\) of harmonic coordinates in stochastic homogenization [1210.3820][2404.13641].

Source: https://www.emergentmind.com/topics/f_l