Regge Ansatz for Intrinsic Charm

• The paper introduces a Regge-based forward production model of intrinsic charm, detailing normalized energy-fraction distributions for charm hadrons in nucleon–air collisions.
• It demonstrates that the hadron-level formulation amplifies leading hadrons carrying a large fraction of the projectile energy, which significantly affects atmospheric lepton fluxes.
• The study contrasts the Regge ansatz with BHPS, meson–baryon, and fitted-charm approaches, clarifying distinct nonperturbative representations of intrinsic charm.

Searching arXiv for papers on intrinsic charm and Regge-based ansätze. The Regge ansatz for intrinsic charm is a Regge-inspired description of nonperturbative charm that is used to model very forward charm production, especially when the phenomenology is driven by hadrons carrying a large fraction of the projectile energy. In the most explicit recent implementation, the ansatz is formulated directly at the hadron-production level in nucleon–air collisions, with normalized energy-fraction distributions for $\bar D^0$, $D^-$, and $\Lambda_c$, and with an overall normalization $w_{\rm intr}^c$ left free (Das et al., 8 Aug 2025). It belongs to the broader intrinsic-charm program, where intrinsic charm denotes a nonperturbative charm component of the proton wave function rather than charm generated radiatively by perturbative QCD, but it is distinct from the more common BHPS, meson-baryon, and fitted-charm constructions, which are generally not formulated in Regge language (Hobbs, 2016).

1. Definition within intrinsic-charm phenomenology

Within modern PDF language, intrinsic charm can be defined operationally by flavor-scheme matching: if 4FNS PDFs are transformed back to 3FNS, then “the 3FNS charm PDF is purely intrinsic,” and the absence of intrinsic charm corresponds to the vanishing of that 3FNS charm distribution (Ball et al., 2022). In that setup, the 3FNS intrinsic charm PDF is scale independent, whereas the 4FNS charm PDF contains both intrinsic and perturbative radiative pieces. This definition is conceptually different from the Regge ansatz, which is not, in its contemporary atmospheric implementation, a fitted 3FNS charm PDF at all, but a forward hadron-level production model.

The same distinction appears in general-mass DIS factorization. The formal DIS framework places any intrinsic charm contribution in the boundary conditions for perturbative evolution, so an arbitrary nonperturbative charm input can be accommodated in ACOT or FONLL, while the zero-intrinsic-charm limit reduces to S-ACOT or FONLL-zic (Ball et al., 2015). This means that a Regge ansatz is best understood as one possible nonperturbative input or forward source term, not as the definition of intrinsic charm itself.

A central point follows from this broader setting. “Intrinsic charm” is a physical hypothesis about the proton wave function or about forward charm production, whereas “Regge ansatz” is a specific modeling choice for how that nonperturbative component is parameterized. Confusing the two obscures the fact that most intrinsic-charm studies use non-Regge constructions.

2. Explicit hadron-level formulation

In the explicit atmospheric implementation, intrinsic charm is introduced through the inclusive production spectrum

$$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$

with

$x_{h_c}=\frac{E_{h_c}}{E_p}\,.$

The Regge-shaped hadron distribution is

$$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$

with normalization

$$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$

The function is normalized to unity, the adopted parameters are

$a_\psi=-2.0,\qquad a_N=-0.5\,,$

and the overall factor $w_{\rm intr}^c$ is “not predicted by first principles” (Das et al., 8 Aug 2025).

The model is formulated directly in terms of hadron species. For proton and neutron projectiles the assumed channels are

$D^-$0

The baryon distribution is imposed through complementarity,

$D^-$1

together with

$D^-$2

As a result, hadronization is built into the ansatz itself rather than introduced through a separate fragmentation function.

With the stated parameter values, this implies a mesonic shape proportional to $D^-$3. By the complementarity relation, the corresponding $D^-$4 spectrum is reflected and therefore becomes much harder. This suggests that the Regge ansatz is designed to amplify the part of intrinsic charm phenomenology that matters most in atmospheric cascades: leading hadrons carrying a very large fraction of the projectile energy.

The formulation is also manifestly not symmetric at the inclusive hadron level. It enhances $D^-$5, $D^-$6, and $D^-$7, and therefore encodes projectile-remnant structure rather than a generic perturbative $D^-$8 pair-production mechanism.

3. Regge and QGSM lineage

The hadron-level ansatz has clear antecedents in earlier Regge-based and QGSM treatments of forward charm. In one Regge-based analysis of large-$D^-$9 nuclear dependence, the endpoint behavior was written in triple-Regge form as

$\Lambda_c$0

so that, after $\Lambda_c$1-integration, one recovers a standard endpoint power $\Lambda_c$2. In the same framework the intrinsic-charm contribution to open charm was modeled as

$\Lambda_c$3

and the nuclear dependence was promoted to

$\Lambda_c$4

Representative endpoint exponents were taken as $\Lambda_c$5, $\Lambda_c$6, and $\Lambda_c$7 (Kopeliovich et al., 2010).

QGSM treatments made the Regge content even more explicit. In that framework, quark distributions and fragmentation functions are expressed in terms of Regge intercepts, and ordinary sea charm contributes only to multi-Pomeron graphs with $\Lambda_c$8, whereas intrinsic charm is treated as valence-like and should therefore be inserted into the one-Pomeron graph with $\Lambda_c$9 (Artemenkov et al., 2010). A later QGSM-based discussion wrote a valence-like charm form

$w_{\rm intr}^c$0

which is the closest explicit PDF-like Regge form for intrinsic charm appearing in that literature (Lykasov et al., 5 Jan 2025).

These antecedents clarify an important structural point. The Regge ansatz for intrinsic charm did not historically emerge as a universal charm PDF of the proton. It emerged as a forward-production description in which endpoint powers, Regge intercepts, Pomeron topology, and leading-particle effects determine how a nonperturbative heavy component manifests itself in hadronic observables.

4. Atmospheric-cascade implementation

In atmospheric-lepton calculations, the Regge ansatz is implemented in MCEq as an additional inclusive forward charm-hadron source term, using the H3a all-nucleon flux and the hadron-level spectra described above. The computation adds intrinsic-charm production for proton–air and neutron–air collisions, propagates the resulting $w_{\rm intr}^c$1, $w_{\rm intr}^c$2, and $w_{\rm intr}^c$3 channels through the cascade, and then follows their decays into prompt $w_{\rm intr}^c$4 and $w_{\rm intr}^c$5. In this formulation, intrinsic charm enters through inclusive hadron production distributions rather than through modified charm PDFs (Das et al., 8 Aug 2025).

The phenomenological outcome is highly constrained. Fitting the angle-averaged atmospheric muon flux for zenith angles $w_{\rm intr}^c$6–$w_{\rm intr}^c$7 yields

$w_{\rm intr}^c$8

whereas saturating the prompt atmospheric $w_{\rm intr}^c$9 bound gives

$$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$0

Using the muon-preferred normalization causes the predicted prompt $$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$1 flux to exceed the IceCube upper limit by a factor of about $$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$2 at

$$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$3

By contrast, the IceCube upper bound at the same energy lies only about $$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$4 above the default MCEq pQCD prediction. When the neutrino bound is imposed, the intrinsic-charm contribution becomes too small to account for the muon excess, and the fit instead requires an unflavored-meson scaling parameter $$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$5, compared with $$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$6 in the no-intrinsic-charm case.

The reason small values of $$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$7 can matter at all is kinematic. Because the intrinsic-charm hadrons are so forward, the daughter leptons inherit a large fraction of the incident nucleon energy. The same feature that makes the Regge ansatz efficient for prompt atmospheric fluxes also makes it strongly constrained by neutrino limits.

5. Contrast with BHPS, meson-baryon, and fitted-charm approaches

Most intrinsic-charm studies do not use a Regge ansatz. A prominent example is the atmospheric-neutrino calculation in which intrinsic charm is imported from the NLO CTEQ 6.5C PDFs using the BHPS model with

$$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$8

at the input scale. That work states explicitly that it does not formulate intrinsic charm in Regge language, and attributes the hard large-$$\frac{d\sigma_{p-{\rm air}}^{h_c{\rm (intr)}}(E,x_{h_c})}{dx_{h_c}} = w_{\rm intr}^c \,\sigma_{p-{\rm air}}(E)\, f_{h_c/p}^{\rm (intr)} (x_{h_c})\,,$$9 shape to the inverse invariant mass squared of the $x_{h_c}=\frac{E_{h_c}}{E_p}\,.$0 light-cone Fock state; within that setup, the prompt neutrino flux can be enhanced by a factor $x_{h_c}=\frac{E_{h_c}}{E_p}\,.$1 at large neutrino energies once the charm-initiated channel is included (Giannini et al., 2018).

The 2016 review of intrinsic charm organizes the subject around three classes of shape assumptions: BHPS or light-front Fock-state models, meson-baryon models, and global-fit ansätze including BHPS-like and low-$x_{h_c}=\frac{E_{h_c}}{E_p}\,.$2-dominated “sealike” charm. It explicitly notes that it does not present a Regge-inspired parametrization of the form $x_{h_c}=\frac{E_{h_c}}{E_p}\,.$3, even though the sealike category is the closest in spirit to Regge small-$x_{h_c}=\frac{E_{h_c}}{E_p}\,.$4 intuition (Hobbs, 2016).

Meson-baryon models likewise generate intrinsic charm without invoking Regge theory. In a comprehensive meson-baryon analysis, the proton’s intrinsic $x_{h_c}=\frac{E_{h_c}}{E_p}\,.$5 and $x_{h_c}=\frac{E_{h_c}}{E_p}\,.$6 distributions arise from convolutions of meson-baryon splitting functions with charm distributions inside charmed hadrons, and the resulting proton-level shapes can be parameterized as

$x_{h_c}=\frac{E_{h_c}}{E_p}\,.$7

For the preferred confining model at $x_{h_c}=\frac{E_{h_c}}{E_p}\,.$8 GeV, the fitted exponents are

$x_{h_c}=\frac{E_{h_c}}{E_p}\,.$9

so $$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$0 is harder than $$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$1 at large $$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$2 (Hobbs et al., 2013). These forms are structurally similar to beta-function ansätze and therefore adjacent to Regge-style parametrizations, but the underlying dynamics is hadronic fluctuation, not Regge intercept matching.

By contrast, the modern fitted-charm extraction does not impose a fixed functional form for intrinsic charm at all. The intrinsic component is isolated by inverse matching to 3FNS and exhibits a valence-like structure peaking at

$$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$3

with the signal localized in

$$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$4

while the result is compatible with zero for $$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$5 once missing-higher-order uncertainties are included (Ball et al., 2022). This suggests that a strongly rising small-$$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$6 Regge term is not what is currently favored by the fitted-charm evidence. In formal DIS terms, however, such a term could still be tested as an input boundary condition, because the ACOT/FONLL framework is flexible enough to accommodate arbitrary fitted charm shapes (Ball et al., 2015).

A common misconception is therefore that any hard or forward intrinsic-charm model is already “Regge.” The literature does not support that identification. BHPS, meson-baryon, sealike global-fit, and fitted-charm approaches are distinct constructions, and only a subset of forward-production models makes Regge structure explicit.

6. Discriminating observables and open issues

The phenomenology most relevant to a Regge ansatz is the phenomenology of forward charm. In proton–proton collisions at large rapidity or large Feynman-$$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$7, intrinsic charm can dominate over the standard extrinsic $$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$8-fusion mechanism, and the most sensitive observables are precisely those that probe the large-$$f_{h_c/p}^{\rm (intr)} (x) = N_{\rm intr}\, x^{-a_\psi}(1-x)^{-a_\psi + 2(1-a_N)}\,,$$9 projectile region (Maciula et al., 2020). This is why atmospheric muons and neutrinos, forward fixed-target charm production, and hadron–antihadron asymmetries recur throughout the intrinsic-charm literature.

Beam-dump and fixed-target neutrino experiments provide especially clean tests. In the SHiP study based on BHPS intrinsic charm, the forward approximation

$$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$0

was used to evaluate prompt neutrino fluxes, and the paper argued that measurements of the $$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$1-neutrino event rate and of the muon charge asymmetry as a function of energy can constrain intrinsic-charm models (Bai et al., 2018). SeaQuest is described as being in an ideal kinematic region for intrinsic-charm $$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$2 production, with acceptance

$$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$3

overlapping a BHPS-like peak near $$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$4, so it provides a benchmark forward setup against which alternative large-$$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$5 models, including Regge-inspired ones, can be compared (Vogt, 2021).

Other forward measurements show that sensitivity alone is not enough; the detailed normalization and shape matter. In LHCb SMOG fixed-target kinematics, a BHPS $$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$6 contribution is predicted to be small and to decrease with center-of-mass energy (Vogt, 2023). In contrast, charged $$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$7-meson asymmetries can remain highly sensitive: one recent QGSM-based analysis reports that the maximal deviation of the $$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$8 asymmetry relative to zero intrinsic charm can be about $$N_{\rm intr} = \frac{\Gamma (-2a_\psi+4-2a_N)} {\Gamma(-a_\psi+1)\Gamma(-a_\psi+3-2a_N)}\,.$$9 at

$a_\psi=-2.0,\qquad a_N=-0.5\,,$0

for

$a_\psi=-2.0,\qquad a_N=-0.5\,,$1

and that asymmetric intrinsic-charm inputs can even change the sign of the $a_\psi=-2.0,\qquad a_N=-0.5\,,$2 and $a_\psi=-2.0,\qquad a_N=-0.5\,,$3 asymmetries (Lykasov et al., 5 Jan 2025).

The current status is therefore mixed. The Regge ansatz is a physically targeted and computationally simple model for very forward intrinsic charm, and its efficiency in atmospheric cascades demonstrates why forward charm is so constraining. At the same time, its contemporary atmospheric implementation is in tension with the requirement of fitting both the IceCube atmospheric muon excess and the prompt atmospheric neutrino bound (Das et al., 8 Aug 2025). More broadly, the intrinsic-charm field increasingly points toward valence-like, possibly asymmetric large-$a_\psi=-2.0,\qquad a_N=-0.5\,,$4 structures, but not toward a single universal Regge form. The principal unresolved question is not whether the large-$a_\psi=-2.0,\qquad a_N=-0.5\,,$5 tail matters; it is which nonperturbative representation of that tail survives simultaneous constraints from DIS, forward hadroproduction, atmospheric leptons, and charm-hadron asymmetries.