Two-Component Lateral Distribution Function

• Two-component LDF is a reconstruction framework that explicitly decomposes ground-level signals into electromagnetic and muonic components for precise air-shower analysis.
• It employs additive modeling and probabilistic likelihood methods to simultaneously fit distinct signal contributions and improve energy and muon proxy estimations.
• This approach aids in distinguishing true dual-component models from single-parameterizations and provides diagnostic insights into hadronic-interaction modeling.

Searching arXiv for the most relevant papers on two-component lateral distribution functions in air-shower physics. A two-component lateral distribution function (LDF) is a reconstruction framework in which the measured lateral signal of an extensive air shower is represented as the superposition of two distinct contributions rather than by a single radial profile. In the literature summarized here, the term has a strict meaning only in a subset of cases. In IceTop analyses, it denotes an explicit decomposition of the detector signal into electromagnetic and muonic lateral distributions, fitted simultaneously on an event-by-event basis (Weyrauch et al., 2023, Weyrauch, 13 Jun 2025, Weyrauch, 16 Sep 2025). In radio-emission studies, a related but physically different two-component picture arises from the coherent sum of geomagnetic and charge-excess emission mechanisms, whose interference shapes the observed ground pattern (Vries et al., 2010). By contrast, several papers on radio, Cherenkov-light, charged-particle, and muon lateral profiles use single analytic parameterizations with multiple fit parameters or separate fits to different particle classes, but do not define a true two-component LDF in the mathematical sense [(Apel et al., 2013); (Al-Zubaidi et al., 2021); (Al-Rubaiee et al., 2011); (Fadhel et al., 2022); (Supanitsky et al., 3 Oct 2025)].

1. Concept and scope

A lateral distribution function describes the radial dependence of a shower observable at ground level, typically as a function of lateral distance from the shower axis. Depending on the detector and observable, the quantity may be an electric-field amplitude, Cherenkov photon density, charged-particle density, or muon density. The defining distinction of a two-component LDF is not merely increased parametric flexibility, but an explicit separation of the measured signal into two physically or observationally distinct contributions.

This distinction is essential because several studies employ one-piece forms that can reproduce both near-axis and far-axis behavior without introducing additive components. For example, the LOPES radio analysis uses a single-component exponential LDF,

$$\epsilon (R) \simeq \epsilon_{100} \exp\!\left(-(R - 100\,\mathrm{m}) / R_{0}\right),$$

and interprets only the slope as composition-sensitive; the paper explicitly does not fit the radio signal as a sum of two spatial components (Apel et al., 2013). Likewise, the Yakutsk Cherenkov analysis adopts a four-parameter Lorentzian-like profile,

$$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$

or equivalently

$Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$

which remains a single compact representation rather than a two-term decomposition (Al-Zubaidi et al., 2021).

The same caveat applies to other parameterizations. The Tunka-25 Cherenkov work introduces a single analytic approximation with four energy-dependent parameters (Al-Rubaiee et al., 2011); the AIRES-based charged-particle study uses a sigmoidal/logistic parameterization for separate $e^\pm$ and $\mu^\pm$ density curves but not an additive two-component LDF (Fadhel et al., 2022); the Auger muon paper reconstructs a single KASCADE-Grande-like muon LDF using two detector readout modes, binary and ADC, but those are two acquisition channels, not two LDF components (Supanitsky et al., 3 Oct 2025).

This suggests that the phrase “two-component LDF” should be reserved for cases where the model contains two explicit lateral terms whose sum defines the expected detector signal, or where the total field is treated as the coherent sum of two physically distinct emission contributions.

2. Explicit electromagnetic–muon decomposition in IceTop

The clearest explicit realization of a two-component LDF appears in IceTop reconstruction. The motivation is that IceTop tanks do not have dedicated muon detectors, yet the tank signals are mixtures of electromagnetic shower particles and low-energy muons. A one-component LDF is sufficient for standard shower geometry and energy reconstruction, but not for event-by-event inference of the low-energy muon content (Weyrauch et al., 2023, Weyrauch, 13 Jun 2025, Weyrauch, 16 Sep 2025).

In this framework, the total signal at lateral distance $r$ is written as

$S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$

The electromagnetic component is described by the IceTop Double Logarithmic Parabola,

$$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$

in the 2023 study (Weyrauch et al., 2023), and equivalently by the same DLP structure in the later event-by-event formulations (Weyrauch, 13 Jun 2025, Weyrauch, 16 Sep 2025). Here $S_{\rm em,125}$ serves as the energy proxy, while $\beta_{\rm em}$ and $$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$0 control slope and curvature.

The muon component is described by a Greisen-type or modified NKG/Greisen-like profile. In the 2023 and 2025 event-by-event studies, it takes the form

$$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$1

with $$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$2 as the estimator for the low-energy muon content (Weyrauch et al., 2023, Weyrauch, 13 Jun 2025). In the later muon-number reconstruction paper, the corresponding reference distance is shifted to $$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$3,

$$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$4

with $$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$5 used as the muon proxy (Weyrauch, 16 Sep 2025).

The physical division is operationally sharp. The electromagnetic term dominates closer to the core and is primarily related to primary energy, whereas the low-energy muon term becomes relatively more important at larger lateral distances and is the quantity of interest for muon-content studies (Weyrauch et al., 2023). The reference distances are chosen to balance contamination and fluctuations: $$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$6 for energy estimation and $$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$7 or $$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$8 for the muon proxy (Weyrauch et al., 2023, Weyrauch, 13 Jun 2025, Weyrauch, 16 Sep 2025).

3. Probabilistic reconstruction and detector-level likelihoods

The IceTop two-component LDF is not only a sum of mean profiles; it is embedded in a detector-level probabilistic model. Rather than fitting mean signals alone, the method uses probability density functions for the electromagnetic and muon responses and combines them into a tank-signal likelihood (Weyrauch et al., 2023, Weyrauch, 13 Jun 2025, Weyrauch, 16 Sep 2025).

For the muon contribution, the response is derived from dedicated Geant4-based tank simulations. In the 2023 formulation, the muon signal PDF for a given zenith angle $$Q(R)= n+\frac{a}{\pi}\,\frac{1}{1+\left(\frac{R-\zeta}{\delta}\right)^2},$$9 and average expected number of muons $Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$0 is

$Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$1

with a Gaussian approximation for $Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$2 (Weyrauch et al., 2023). The later event-by-event IceCube paper states the same Poisson mixture structure and likewise notes that for large multiplicities the muon PDF is approximated by a Gaussian (Weyrauch, 13 Jun 2025). The 2025 muon-number study further specifies the effective tank area used to relate the muon LDF expectation to the expected muon count,

$Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$3

(Weyrauch, 16 Sep 2025).

The electromagnetic contribution is modeled as approximately log-normal or Gaussian in $Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$4 (Weyrauch et al., 2023), and the EM likelihood is attenuated by a snow factor $Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$5 (Weyrauch et al., 2023, Weyrauch, 13 Jun 2025, Weyrauch, 16 Sep 2025). The total tank-signal PDF is then built by convolution. In the 2023 paper the general combined form is

$Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$6

with special-case approximations when one component is much narrower than the other, and full convolution otherwise (Weyrauch et al., 2023). The 2025 event-by-event paper provides an explicit SLC convolution,

$Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$7

and an HLC likelihood split into large-signal, small-signal, and intermediate regimes (Weyrauch, 13 Jun 2025). The 2025 muon-number paper gives the analogous three-regime HLC expression and states that an 85% phase-space inclusion provides good performance for the threshold function $Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$8 (Weyrauch, 16 Sep 2025).

The full event likelihood includes signal, saturation, timing, and silent-detector terms. In the 2023 IceTop work,

$Q(R)= n + \frac{2a/\pi}{4(R-\zeta)^2+\delta^2},$9

with a quadratic penalty

$e^\pm$0

applied to stabilize the fit (Weyrauch et al., 2023). The later IceCube formulations retain the global likelihood structure while separating HLC and SLC contributions and including silent tanks through no-hit probabilities (Weyrauch, 13 Jun 2025, Weyrauch, 16 Sep 2025).

4. Reconstruction observables and reported performance

The principal outputs of the IceTop two-component LDF are a proxy for primary energy and a proxy for low-energy muon content. In the 2023 study these are $e^\pm$1 and $e^\pm$2 (Weyrauch et al., 2023). In the 2025 event-by-event IceCube paper the same pair is reconstructed and interpreted as proxies for primary energy and low-energy muon number, respectively (Weyrauch, 13 Jun 2025). In the 2025 muon-number study, the muon proxy is $e^\pm$3, while the low-energy muon number in simulation is defined for muons above 210 MeV and within 1 km lateral distance (Weyrauch, 16 Sep 2025).

The 2023 IceTop paper reports that $e^\pm$4 tracks primary energy nearly linearly, has small mass dependence, and yields energy resolution below $e^\pm$5 above $e^\pm$6 PeV, with some degradation near $e^\pm$7 PeV due to saturation (Weyrauch et al., 2023). The same study reports that $e^\pm$8 correlates approximately linearly with the true number of low-energy muons; its bias decreases from roughly $e^\pm$9 at lower $\mu^\pm$0 to around $\mu^\pm$1 at higher muon number, and the resolution becomes better than $\mu^\pm$2, though a noticeable mass dependence remains (Weyrauch et al., 2023).

The 2025 event-by-event IceCube paper reports Energy resolution around $\mu^\pm$3 above $\mu^\pm$4, improving to below $\mu^\pm$5 at higher energies, and Muon-number resolution about $\mu^\pm$6 at the highest energies considered (Weyrauch, 13 Jun 2025). It additionally states that the bias is small, with only a few-percent primary-mass dependence for energy reconstruction and no significant primary-mass dependence for the muon-number reconstruction (Weyrauch, 13 Jun 2025). The companion 2025 muon-number paper reports that the reconstruction reaches full efficiency for

$\mu^\pm$7

with energy resolution about 12% above $\mu^\pm$8 PeV, improving to well below 10% near 100 PeV, and muon-number resolution reaching below 20% for showers around 100 PeV (Weyrauch, 16 Sep 2025).

These results are model-dependent in the specific sense that the studies use simulation sets based on Sibyll 2.1, EPOS-LHC, and QGSJet-II.04, with small differences attributed to differences in $\mu^\pm$9, total muon number, and lateral particle distributions (Weyrauch, 16 Sep 2025). A plausible implication is that the two-component LDF is not only a reconstruction tool but also a diagnostic for hadronic-interaction modeling, because it yields event-level observables tied separately to EM and muonic shower content.

5. Two-component structure in radio emission

In radio-emission studies, the two-component concept has a different meaning. The 2010 MGMR paper treats the observed radio field as the coherent sum of geomagnetic and charge-excess contributions rather than as a detector-signal decomposition into electromagnetic and muonic particle populations (Vries et al., 2010). The effective formulation is

$r$0

with the observed intensity depending on

$r$1

The cross term introduces constructive or destructive interference depending on observer position (Vries et al., 2010).

The geomagnetic component is modeled as a transverse current induced by the Earth’s magnetic field, while the charge-excess component arises from the net negative charge in the shower front (Vries et al., 2010). Their different polarizations imply that the intensity pattern is not circularly symmetric. For the vertical-shower geometry discussed in the paper, the two components can interfere constructively on one side of the shower axis, destructively on the opposite side, and remain orthogonal along the perpendicular axis (Vries et al., 2010).

The same paper also distinguishes two radial regimes governed by different shower physics. At large observer distances, using the point-like approximation $r$2, the electric field is

$r$3

with approximate scaling $r$4, so the shower profile $r$5 controls the pulse shape (Vries et al., 2010). At small distances, a new near-axis expression yields

$r$6

and the first term, controlled by the derivative of the pancake function $r$7, dominates near the axis (Vries et al., 2010).

This is a different sense of “two-component” from the IceTop case. The two parts are not two radial LDF functions for two particle populations in a detector, but two coherent emission mechanisms whose superposition determines the field pattern. The paper also introduces a composition-sensitive ratio,

$r$8

and shows systematic differences between proton- and iron-induced showers, with proton showers exhibiting larger fluctuations (Vries et al., 2010).

6. Single-component approximations often mistaken for two-component models

A recurring source of confusion is that many lateral-profile studies describe data with flexible one-piece functions that encode inner-core and outer-tail behavior without constituting a true two-component LDF. Several papers in the present corpus fall into this category.

The LOPES radio analysis is explicit on this point. Although the physical radio emission includes a dominant geomagnetic contribution and a minor charge-excess contribution, the fitted lateral distribution remains a one-dimensional exponential (Apel et al., 2013). Its composition-sensitive observables are the scale parameter $r$9, the practical slope indicator

$S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$0

and the reconstructed $S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$1 obtained through

$S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$2

The paper reports that proton showers tend to have larger $S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$3 and larger $S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$4, corresponding to steeper LDFs, while iron showers have smaller $S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$5 and flatter LDFs (Apel et al., 2013). The dispersion around the $S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$6-slope fit corresponds to an $S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$7 uncertainty of about $S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$8 on average, rising to about $S_{\text{tot}}(r)=S_{\text{em}}(r)+S_{\mu}(r).$9 for the most inclined showers (Apel et al., 2013).

The Yakutsk Cherenkov study similarly uses a Lorentzian parameterization and states that it reproduces both the compact inner region and the more extended outer region reasonably well over 100–1000 m, but does not introduce separate near-core and far-core terms (Al-Zubaidi et al., 2021). The fit coefficients vary with primary type, zenith angle, and energy, and the goodness-of-fit is reported with $$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$0 values near 0.97–0.99 (Al-Zubaidi et al., 2021).

The Tunka-25 Cherenkov parameterization is another example. It employs one analytic expression with four parameters $$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$1, each expressed as cubic polynomials in $$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$2,

$$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$3

and reports approximation accuracy better than 25% for protons and gamma rays, about 20% for iron at distances $$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$4–$$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$5 m, and not less than 10% at other distances, with best agreement in the interval $$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$6–$$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$7 m (Al-Rubaiee et al., 2011). The function is explicitly a single smooth approximation, not a sum of core and halo terms.

The AIRES logistic study treats $$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$8 and $$S_{\rm em}(r)=S_{\rm em,125} \left(\frac{r}{r_{\rm em}}\right)^{-\beta_{\rm em}(S_{\rm em,125})-\kappa(S_{\rm em,125})\log_{10}(r/r_{\rm em})}, \qquad r_{\rm em}=125~\mathrm{m},$$9 secondaries separately, but again does not define a single additive two-component LDF. Its central construction is a sigmoidal function with parameters $S_{\rm em,125}$0, $S_{\rm em,125}$1, $S_{\rm em,125}$2, and $S_{\rm em,125}$3, whose energy dependence is given by

$S_{\rm em,125}$4

(Fadhel et al., 2022). The paper reports “a good agreement” with AGASA data for proton and iron primaries at $S_{\rm em,125}$5 eV for vertical showers initiated by charged muons (Fadhel et al., 2022).

These cases indicate that multi-parameter or multi-class modeling should not be conflated with a true two-component LDF. The former can mimic complex shape changes; the latter requires explicit additive structure or coherent superposition of distinct physical contributions.

7. Related developments, limitations, and interpretive boundaries

Beyond the explicit IceTop and MGMR radio cases, other LDF studies remain adjacent to the two-component theme without fully entering it. The Auger muon reconstruction paper uses a single MLDF shape,

$S_{\rm em,125}$6

with

$S_{\rm em,125}$7

and reconstructs it through a likelihood that simultaneously incorporates binary and ADC detector information at the same station (Supanitsky et al., 3 Oct 2025). The reference distance is

$S_{\rm em,125}$8

and the reconstructed observable is $S_{\rm em,125}$9 (Supanitsky et al., 3 Oct 2025). The paper does not define a dual-shape LDF, yet it does show that “combined” can refer to detector-response channels rather than physical components.

The HAWC LDF study likewise compares several single-form parameterizations, including a modified scaling formalism,

$\beta_{\rm em}$0

but does not interpret the extra factor as a two-component decomposition (Morales-Soto et al., 2019). Its fitted lateral age parameter $\beta_{\rm em}$1 exhibits mass sensitivity, with

$\beta_{\rm em}$2

reported as exceeding 1 above about $\beta_{\rm em}$3 GeV and reaching approximately 1.75 at $\beta_{\rm em}$4 GeV (Morales-Soto et al., 2019). This is a shape-complexity result, not evidence of an explicit two-component LDF.

The principal limitations of explicit two-component approaches are likewise paper-specific. In IceTop, assumptions include the use of the reconstructed primary zenith as the muon direction, the dependence on snow correction for the EM component, threshold choices in the HLC regime splitting, and the fact that performance was demonstrated mainly for quasi-vertical or nearly vertical contained events (Weyrauch et al., 2023, Weyrauch, 13 Jun 2025, Weyrauch, 16 Sep 2025). In radio, the two-component interference pattern is azimuth dependent and therefore cannot be reduced to a purely radial scalar LDF without loss of information (Vries et al., 2010). A plausible implication is that the term “LDF” becomes increasingly approximate when the observable has strong directional structure or when detector response entangles the two components.

In current usage, then, the two-component LDF has two well-defined forms. In surface arrays such as IceTop, it is an additive electromagnetic-plus-muon reconstruction ansatz fitted through detector-response likelihoods (Weyrauch et al., 2023, Weyrauch, 13 Jun 2025, Weyrauch, 16 Sep 2025). In radio-emission theory, it is the coherent superposition of geomagnetic and charge-excess fields whose interference shapes the lateral footprint (Vries et al., 2010). Many other studies investigate composition sensitivity, lateral age, or radial-profile flexibility with sophisticated single-form parameterizations, but they remain distinct from a true two-component LDF in the strict analytical sense [(Apel et al., 2013); (Al-Zubaidi et al., 2021); (Al-Rubaiee et al., 2011); (Fadhel et al., 2022); (Supanitsky et al., 3 Oct 2025); (Morales-Soto et al., 2019)].