SHRiMPS Minimum Bias Module Overview

• SHRiMPS Minimum Bias Module is a component in SHERPA that computes nucleon–nucleon cross sections and inelastic probabilities using a dynamic, impact-parameter dependent approach.
• It employs the KMR soft-QCD multi-channel eikonal model with Good–Walker state decomposition to capture both integrated cross-section and detailed geometric fluctuations.
• When integrated into Monte Carlo Glauber models, it enhances predictions of wounded-nucleon multiplicity and geometrical anisotropies, refining soft-QCD and diffraction-sensitive observables.

The SHRiMPS minimum-bias module is a component of the SHERPA event generator that, in the context discussed here, implements the KMR soft-QCD multi-channel eikonal model and provides a dynamically computed, fluctuating nucleon–nucleon cross section for minimum-bias and Glauber-type applications. Its defining feature is that it supplies both integrated hadronic cross sections and impact-parameter dependent inelastic probabilities, with cross-section fluctuations generated through a two-state Good–Walker decomposition of the proton. In the recent pA application, this allows the cross section to be used directly inside a Monte Carlo Glauber model without imposing an ad hoc transverse profile, and it leads to modified wounded-nucleon multiplicity tails and enhanced geometric anisotropies (Roux, 29 Jul 2025).

1. Definition and role within SHERPA

In the cited pA study, SHRiMPS is used as the implementation of the KMR soft-QCD multi-channel eikonal model in SHERPA. Its role is to compute integrated hadronic cross sections such as $\sigma_{\rm tot}$, $\sigma_{\rm el}$, $\sigma_{\rm inel}$, and diffractive contributions; impact-parameter dependent inelastic probabilities $d\sigma_{\rm inel}/d^2b$; and cross-section fluctuations through a two-state Good–Walker decomposition of the proton (Roux, 29 Jul 2025).

This combination distinguishes the module from prescriptions that provide only an average inelastic cross section. In the SHRiMPS construction, the transverse dependence is not imposed externally but is obtained from the dynamics itself. In the Glauber context, that property is operationally important because the interaction range in transverse space directly determines whether a projectile proton interacts with a target nucleon at a given separation, how many nucleons are wounded, and how broadly the wounded system extends in the transverse plane (Roux, 29 Jul 2025).

A recurring theme in minimum-bias phenomenology is that soft-QCD models are constrained not by a single inclusive rate but by a correlated set of observables involving multiplicity, low-$p_T$ production, diffraction-sensitive event classes, and underlying-event activity. The ATLAS minimum-bias literature treats exactly these observables as the natural validation space for a SHRiMPS-type model, even where SHRiMPS is not discussed explicitly (Kar, 2010).

2. Eikonal and Good–Walker structure

The KMR model underlying SHRiMPS is formulated in the eikonal approximation. For fixed impact parameter $b$, the elastic amplitude is written as

$$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$

where $\Omega(s,b)$ is the opacity or eikonal. The corresponding impact-parameter differential cross sections are

$$\frac{d\sigma_{\rm tot}}{d^2b}(s,b)=2\left(1-e^{-\Omega(s,b)/2}\right),$$

$$\frac{d\sigma_{\rm el}}{d^2b}(s,b)=\left(1-e^{-\Omega(s,b)/2}\right)^2,$$

and

$\sigma_{\rm el}$0

These expressions are the basis for treating the cross section as a probability profile in transverse space rather than merely an integrated number (Roux, 29 Jul 2025).

The model is extended to a multi-channel Good–Walker formalism by decomposing the proton into diffractive eigenstates $\sigma_{\rm el}$1, so that hadrons are superpositions of such states. The elastic amplitude becomes an average over channels,

$\sigma_{\rm el}$2

and for each channel $\sigma_{\rm el}$3 the inelastic non-diffractive probability is

$\sigma_{\rm el}$4

In SHRiMPS, this channel structure is the source of cross-section fluctuations (Roux, 29 Jul 2025).

The opacity itself is built from parton densities associated with the Good–Walker states:

$\sigma_{\rm el}$5

These component densities evolve with rapidity according to coupled emission/absorption equations,

$\sigma_{\rm el}$6

$\sigma_{\rm el}$7

with absorptive correction factor

$\sigma_{\rm el}$8

Here $\sigma_{\rm el}$9 governs parton emission, while $\sigma_{\rm inel}$0 governs recombination via the triple-pomeron vertex. The boundary conditions are supplied by modified dipole-like form factors,

$\sigma_{\rm inel}$1

which are Fourier transformed to obtain initial impact-parameter profiles (Roux, 29 Jul 2025).

The practical implication is that SHRiMPS computes the transverse extent of the interaction rather than assuming it. This suggests a more constrained treatment of geometry whenever the module is embedded into Glauber-like initial-state modeling.

3. Embedding in Monte Carlo Glauber calculations

The pA Monte Carlo Glauber model built around SHRiMPS follows a standard geometric workflow. Nucleons are distributed in the nucleus according to a Woods–Saxon distribution, with a minimum separation of $\sigma_{\rm inel}$2 fm. An impact parameter $\sigma_{\rm inel}$3 for the proton–nucleus collision is then sampled with $\sigma_{\rm inel}$4. The proton–nucleon inelastic cross section is converted into an interaction probability for each target nucleon, and the event-by-event number of wounded nucleons $\sigma_{\rm inel}$5 is counted, with wounded nucleons defined here as nucleons that undergo an inelastic non-diffractive interaction (Roux, 29 Jul 2025).

Three prescriptions for the pN cross section are compared:

Prescription	Core definition	Fluctuation content
BD	$\sigma_{\rm inel}$6	No fluctuation; fixed interaction range
GG	Fluctuating black disk with sampled $\sigma_{\rm inel}$7 from $\sigma_{\rm inel}$8	Fluctuates only the integrated cross section
SH	KMR/SHRiMPS cross section with Good–Walker state dependence	Fluctuates both integrated cross section and detailed $\sigma_{\rm inel}$9-profile

For the BD model, $d\sigma_{\rm inel}/d^2b$0 is chosen to reproduce the same average inelastic cross section as SHRiMPS. In the GG model, the integrated inelastic cross section is sampled from

$d\sigma_{\rm inel}/d^2b$1

The SH prescription instead depends on the Good–Walker state combination of the colliding nucleons; each nucleon is randomly assigned one of two Good–Walker states with probability $d\sigma_{\rm inel}/d^2b$2, and the resulting $d\sigma_{\rm inel}/d^2b$3-dependent probability comes directly from SHRiMPS (Roux, 29 Jul 2025).

The study uses two SHRiMPS tunes with similar average cross sections: tune 1 has $d\sigma_{\rm inel}/d^2b$4 fm$d\sigma_{\rm inel}/d^2b$5 and $d\sigma_{\rm inel}/d^2b$6 fm$d\sigma_{\rm inel}/d^2b$7, while tune 2 has $d\sigma_{\rm inel}/d^2b$8 fm$d\sigma_{\rm inel}/d^2b$9 and $p_T$0 fm$p_T$1. Tune 1 is used for the main results (Roux, 29 Jul 2025).

4. Effects on wounded-nucleon multiplicity and geometry

The comparison between BD, GG, and SH isolates three distinct fluctuation mechanisms. BD has no fluctuation and therefore a fixed interaction range. GG fluctuates only the integrated cross section and retains a trivial black-disk geometry once the total cross section is sampled. SH fluctuates both the integrated cross section and the detailed impact-parameter profile through the Good–Walker channel structure. The cited study notes that GG can fluctuate to more extreme integrated cross sections than SH, whereas SH introduces a nontrivial transverse probability profile that becomes especially important for geometry (Roux, 29 Jul 2025).

For pPb events with $p_T$2 fm, both SH and GG produce much longer tails to large $p_T$3 than BD. SH extends to larger interaction radii than BD, but GG still reaches even more extreme $p_T$4 values because it can fluctuate to larger effective cross sections. SH therefore does not always exceed GG in the large-$p_T$5 tail, since its interaction profile is probabilistic rather than a sharp black disk. At fixed $p_T$6, position fluctuations are reduced: BD becomes sharply peaked, GG remains peaked but still fluctuates through cross-section sampling, and SH becomes doubly peaked, reflecting the two possible Good–Walker states of the projectile and target (Roux, 29 Jul 2025).

The central physical separation made in the study is that integrated cross-section fluctuations control the overall width and tails of the $p_T$7 distribution, whereas the detailed impact-parameter dependence adds stochasticity but is less effective than large cross-section fluctuations at generating extreme $p_T$8. Since charged-particle multiplicity is strongly correlated with the number of participants, the paper argues that improving the wounded-nucleon distribution improves multiplicity distributions as well, and specifically that the SHRiMPS-based model can give a better description of the small and large tails of multiplicity distributions in pA collisions (Roux, 29 Jul 2025).

Geometric effects are quantified through the eccentricities

$p_T$9

with participant-plane angle

$b$0

BD and GG give similar eccentricity distributions, whereas SH yields systematically broader $b$1 distributions and larger mean values; the differences can reach about $b$2 in the mean eccentricities. One of the main conclusions is therefore that nontrivial impact-parameter dependence affects geometry much more strongly than it affects the $b$3 distribution, making the SHRiMPS $b$4-profile fluctuations especially relevant for initial-state geometry and hydrodynamic modeling of anisotropic flow (Roux, 29 Jul 2025).

5. Minimum-bias observables and empirical benchmark space

Minimum-bias measurements at the LHC define the observable space against which a SHRiMPS-type module is naturally tested. Early ATLAS analyses define minimum bias as events selected with a loose, nearly inclusive trigger that accepts a large fraction of the inelastic cross section, using a Minimum Bias Trigger Scintillator single-arm trigger and corrections back to particle level through efficiency corrections and unfolding (Kar, 2010). Subsequent ATLAS summaries emphasize that the standard observables are charged-particle multiplicity distributions, charged-particle $b$5 spectra, pseudorapidity distributions, and average transverse momentum as a function of charged-particle multiplicity, supplemented by diffraction-enhanced or diffraction-suppressed samples and underlying-event observables (Leyton, 2012).

The modern ATLAS review of minimum-bias charged-particle distributions covers $b$6 collisions at $b$7 TeV, with pseudorapidity acceptance $b$8, thresholds $b$9 MeV and $$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$0 MeV, and event selections such as $$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$1 or $$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$2 depending on the phase space (Kulchitsky, 2023). The broader LHC phenomenology review frames these observables as the main constraints on soft particle production, energy evolution of inelastic activity, multiplicity fluctuations, multiplicity–momentum correlations, and the onset or saturation of multi-parton interactions or minijet activity (Grosse-Oetringhaus, 2018).

Across these studies, a consistent empirical pattern emerges. Pre-LHC phenomenological tunes and generators generally failed in the soft regime, especially at low $$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$3 and low multiplicity where diffraction matters; no single tune gives uniform agreement across all energies and phase spaces; and multiplicity tails remain difficult for generators when diffraction and MPI both matter (Kar, 2010). ATLAS tune developments such as AMBT, AMBT1, and AMBT2b improved agreement in specific phase-space regions, but the same body of work also shows that a tune successful in one subset of observables can fail in another, particularly when moving between inclusive minimum-bias, diffraction-suppressed, and underlying-event selections (Leyton, 2012).

For SHRiMPS, this benchmark space implies more than reproducing an average inelastic cross section. A plausible implication is that a successful minimum-bias implementation must simultaneously describe inclusive soft multiplicities, low-$$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$4 spectra, diffraction-sensitive low-multiplicity regions, multiplicity-dependent $$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$5, and the enhanced activity characteristic of underlying-event plateaus. The ATLAS literature explicitly treats these as the observables that constrain a minimum-bias module of the SHRiMPS type (Kulchitsky, 2023).

6. Extension to A+A, limits of simplified prescriptions, and broader theoretical context

The pA study states that the generalization of the SHRiMPS-based Glauber construction to A+A collisions is straightforward. Since the framework uses a nucleon–nucleon inelastic probability as input, the same cross-section prescription can be applied to every nucleon–nucleon encounter in nucleus–nucleus collisions. The extension is therefore described as conceptual rather than formal: one replaces the proton projectile by a nucleus–nucleus geometry and uses the same SHRiMPS-based fluctuating interaction probabilities for all binary encounters (Roux, 29 Jul 2025).

The comparison with BD and GG also clarifies what is lost in more schematic prescriptions. GG is stronger for producing very large or very small $$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$6 tails because it can fluctuate to more extreme integrated cross sections, while SH is stronger for modifying spatial eccentricities because it carries a nontrivial transverse probability profile. This addresses a common simplification in Glauber practice: a fluctuating total cross section is not equivalent to a fluctuating impact-parameter profile, and the two affect multiplicity tails and initial-state geometry differently (Roux, 29 Jul 2025).

A broader minimum-bias theory literature has also proposed detector-level, observable-based descriptions in which the natural expansion parameter is the observed multiplicity $$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$7 rather than an impact parameter or participant count. In that framework, the cross section is expanded in symmetric polynomials of phase-space coordinates, the leading behavior approaches flat phase space as $$S_{el}(s,b)=1+iT_{el}(s,b), \qquad \mathrm{Im}\,T_{el}(s,b)=1-e^{-\Omega(s,b)/2},$$8, and the formalism is explicitly stated not to predict the multiplicity distribution itself (Larkoski et al., 2021). Relative to that viewpoint, SHRiMPS occupies a different position: it supplies a dynamical, impact-parameter dependent nucleon–nucleon interaction model with Good–Walker fluctuations, and it has been used directly as Glauber input for pA geometry and multiplicity studies (Roux, 29 Jul 2025).

Taken together, these developments locate the SHRiMPS minimum-bias module at the intersection of soft-QCD phenomenology, fluctuating cross-section modeling, and initial-state geometry. Its specific contribution is not merely the provision of an average hadronic cross section, but the calculation of a channel-dependent transverse interaction profile that can be propagated into minimum-bias and heavy-ion initial-state simulations while remaining answerable to the standard LHC minimum-bias observable set.