F_L in Deep Inelastic Scattering

• F_L is the longitudinal structure function in DIS, defined as the hadronic response to longitudinally polarized virtual photons and a direct manifestation of QCD dynamics.
• It is extracted via Rosenbluth separations and analyzed through factorization, gluon convolution, and double-logarithmic resummation techniques.
• F_L serves as a sensitive probe for gluon dynamics at low Bjorken x, influencing precision measurements and heavy-quark DIS analyses.

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 (Tvaskis et al., 2016, 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 $Q^2\to 0$ (Rezaei et al., 2014, Badelek et al., 2022).

1. Definition and kinematic role

The standard inclusive DIS reduced cross section at $Q^2\ll M_Z^2$ may be written as

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$0

so the longitudinal contribution is isolated by the high-$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$1 term (0805.2809, Jiménez-López et al., 2024). In fixed-target notation, the same physics is expressed through

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$2

with

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$3

so $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$4 is directly tied to longitudinal virtual-photon absorption (Tvaskis et al., 2016).

A complementary decomposition, especially common in heavy-quark leptoproduction, uses transverse and longitudinal photon structure functions: $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$5 This basis makes explicit that $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$6 mixes transverse and longitudinal response, whereas $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$7 isolates the longitudinal channel (Ivanov, 2016). In inclusive DIS one also has

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$8

so $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$9 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 $y^2$0 is not obtained from a single cross-section measurement at fixed $y^2$1. A Rosenbluth-type variation of $y^2$2, achieved either by changing beam energies or by combining measurements at different $y^2$3, is required to extract the slope in $y^2$4 or $y^2$5 (0805.2809, Tvaskis et al., 2016).

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 $y^2$6, the gluonic contribution may be written as

$y^2$7

with a perturbatively calculable gluonic DIS kernel $y^2$8 (Rezaei et al., 2014). In the non-singlet formulation,

$y^2$9

which makes the coefficient-function content of $F_L$0 explicit (Basdew-Sharma et al., 2022).

A distinctive perturbative feature is that massless $F_L$1 has no Born term. In the notation $F_L$2, the longitudinal coefficient function begins as

$F_L$3

so the leading contribution is already radiative (Basdew-Sharma et al., 2022). This is the operator-level expression of the familiar statement that $F_L$4 in the naive quark-parton model and becomes nonzero only through QCD dynamics (0805.2809).

The massless unpolarized Wilson coefficients for $F_L$5 have been calculated through three loops in the $F_L$6 scheme, including the non-singlet quark, pure-singlet quark, and gluon channels (Blümlein et al., 2022). Beyond this, the flavour non-singlet $F_L$7 and $F_L$8 contributions to the four-loop coefficient function have been obtained, with the new $F_L$9 terms reported to be numerically much larger than the previously known leading large-$x$0 $x$1 pieces (Basdew-Sharma et al., 2022). This establishes $x$2 as a precision perturbative observable rather than merely a qualitative probe.

The perturbative organization is especially consequential because $x$3 is strongly gluon sensitive at low $x$4. A plausible implication is that higher-order coefficient-function control and reliable PDF evolution are both indispensable when $x$5 is used as a direct constraint on the gluon density.

3. Small-$x$6 dynamics, gluon sensitivity, and competing frameworks

At small $x$7, one widely used ansatz is a Regge-like gluon distribution

$x$8

with $x$9 corresponding to a hard Pomeron. In an NLO analysis built on this assumption, the gluon density can be eliminated between $F_L$0 and the charm structure function $F_L$1, yielding

$F_L$2

so $F_L$3 can be inferred from measured charm data in the small-$F_L$4, high-$F_L$5 regime (Rezaei et al., 2014). In that treatment, the hard-Pomeron form gives a good description of both $F_L$6 and the charm structure functions.

A different small-$F_L$7 approach uses double-logarithmic resummation. Starting from

$F_L$8

the analysis identifies a $F_L$9 enhancement first appearing in the $Q^2$0 contribution to the auxiliary amplitude $Q^2$1, and combines it with all-order resummed double logarithms to obtain the asymptotic behavior

$Q^2$2

with $Q^2$3 for fixed coupling and $Q^2$4 for running coupling (Ermolaev et al., 2020). The explicit conclusion is that the power-like rise of $Q^2$5 at small $Q^2$6 is a synergic effect of the $Q^2$7-order $Q^2$8 factor and the steep $Q^2$9-dependence of the totally resummed double logarithmic contributions.

In $F_L\propto Q^4$0-factorization, $F_L\propto Q^4$1 is represented as

$F_L\propto Q^4$2

and therefore becomes directly sensitive to the transverse-momentum-dependent gluon density (V. et al., 2023). HERA $F_L\propto Q^4$3 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 $F_L\propto Q^4$4 (V. et al., 2023).

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

$F_L\propto Q^4$5

and HERA averages of $F_L\propto Q^4$6 were found to lie close to this limit for $F_L\propto Q^4$7 (Ewerz et al., 2012). A later comment argued that the stronger claim of a model-independent “rigorous” CDP test actually depends on the additional assumption

$F_L\propto Q^4$8

and reinterpreted the same HERA comparison as compatible with the relation $F_L\propto Q^4$9 in a different CDP argument (Schildknecht, 2012). The disagreement is therefore not about the numerical relevance of $Q^2\to 0$0, 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 $Q^2\to 0$1 alone. The basic definitions are

$Q^2\to 0$2

with

$Q^2\to 0$3

throughout (Ivanov, 2012). This reformulation matters because large radiative corrections to $Q^2\to 0$4 and $Q^2\to 0$5 separately can cancel in the ratio.

The key heavy-quark statement is that, contrary to the transverse structure function $Q^2\to 0$6, the longitudinal one $Q^2\to 0$7 does not contain leading mass logarithms of the type $Q^2\to 0$8 at both LO and NLO (Ivanov, 2016). In a comparison of FFNS with ACOT($Q^2\to 0$9) VFNS, this implies that charm-density resummation enhances $Q^2\ll M_Z^2$0 but leaves $Q^2\ll M_Z^2$1 comparatively unaffected, driving the Callan–Gross ratio downward. Quantitatively, NLO FFNS corrections to $Q^2\ll M_Z^2$2 are reported as $Q^2\ll M_Z^2$3 for $Q^2\ll M_Z^2$4 and $Q^2\ll M_Z^2$5, whereas ACOT($Q^2\ll M_Z^2$6) decreases the LO FFNS prediction for $Q^2\ll M_Z^2$7 by about $Q^2\ll M_Z^2$8 for practically all $Q^2\ll M_Z^2$9 (Ivanov, 2016). This makes $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$00 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 $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$01 are less than $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$02 for $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$03, and the hadron-level low-$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$04 predictions for $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$05 and $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$06 are stable under DGLAP evolution of the gluon density (Ivanov, 2012). Because the reduced cross section may be written as

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$07

stable analytic approximations to $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$08 simplify the extraction of $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$09 and $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$10 from HERA data (Ivanov, 2012).

This suggests a broader methodological point: in the heavy-quark sector, $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$11 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 $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$12 data with positron beam energy $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$13 GeV and proton beam energies $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$14, $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$15, and $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$16 GeV, covering

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$17

for the extracted $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$18 points (0805.2809). Rosenbluth-type fits to $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$19 versus $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$20 produced positive $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$21 values, for example

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$22

and

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$23

with overall agreement with higher-order QCD calculations based on H1, MSTW, and CTEQ PDFs (0805.2809). The measurement established that $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$24 is clearly positive in the explored low-$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$25 region and consistent with a sizeable low-$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$26 gluon density.

At the opposite end of the kinematic plane, Jefferson Lab Hall C experiment E00-002 performed separated measurements of $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$27 for hydrogen and deuterium in the region roughly

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$28

using Rosenbluth separations (Tvaskis et al., 2016). 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 $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$29 above the resonance region, the paper found

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$30

which was interpreted as evidence either for an unexpected proton–neutron difference in $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$31 or for a suppression of the longitudinal response in the deuteron (Tvaskis et al., 2016).

World data have also been recast into moments. The lowest three longitudinal Nachtmann moments were extracted over

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$32

using global $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$33 and $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$34 data, including modern HERA and Jefferson Lab measurements (Monaghan et al., 2012). 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 $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$35 and/or a larger gluon distribution at high $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$36 (Monaghan et al., 2012).

The diffractive analogue $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$37 has also been measured directly at HERA. In diffractive DIS,

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$38

and the extracted $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$39 was found to be nonzero, with five points more than $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$40 above zero (Collaboration, 2011). The diffractive longitudinal-to-transverse ratio,

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$41

implied a relative longitudinal contribution larger than in inclusive DIS, with

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$42

on average (Collaboration, 2011).

6. Low-$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$43 modeling, future precision, and notation

A dedicated low-$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$44, low-$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$45 model updates the older photon–gluon fusion description by extrapolating $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$46-factorization toward the photoproduction region while enforcing

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$47

and adding a higher-twist term that vanishes as $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$48 (Badelek et al., 2022). In that framework,

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$49

with $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$50 obtained from off-shell photon–gluon fusion with exact gluon kinematics and $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$51 associated with the low-transverse-momentum region (Badelek et al., 2022). The updated model reproduces H1 data well in the perturbative region, but the sparse low-$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$52 SLAC and Jefferson Lab data are underestimated unless lower light-quark masses are used, notably $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$53 rather than $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$54–$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$55 (Badelek et al., 2022).

Prospective EIC measurements indicate a sharp improvement in direct $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$56 determination. Using Rosenbluth extractions from inclusive pseudodata, a default five-energy scenario was studied with

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$57

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$58

and at least three usable $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$59 points were required for each extraction (Jiménez-López et al., 2024). In the optimistic $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$60 inter-energy systematic scenario, the absolute $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$61 uncertainty reaches about $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$62 over a wide kinematic range; reducing the luminosity to $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$63 per energy has little impact, indicating that the measurement is already systematically limited (Jiménez-López et al., 2024). The projected EIC coverage is described as complementary to both fixed-target and HERA data and extends down to approximately

$$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$64

for $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$65 (Jiménez-López et al., 2024).

This trajectory reinforces the status of $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$66 as a direct, approximately linear probe of the gluon density in global analyses (Jiménez-López et al., 2024). It also clarifies a notational point: while $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$67 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 $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$68 in magnetic-superconductor vortex dynamics and as the Jacobian matrix $$F_L(x,Q^2)=F_2(x,Q^2)\left(1+\frac{4M^2x^2}{Q^2}\right)-2xF_1(x,Q^2),$$69 of harmonic coordinates in stochastic homogenization (Bulaevskii et al., 2012, Otto et al., 2024).