String Shoving: Dynamics in High Energy Collisions

• String shoving is a non-perturbative mechanism in Lund models that converts overlapping string energy into transverse pressure before hadronization.
• It operates through a Gaussian transverse field profile, repelling nearby flux tubes and dynamically transferring momentum in dense events.
• In collider simulations, string shoving moderates the effects of rope hadronization, impacting strangeness production and flow-like observables.

String shoving (SS) is a non-perturbative pre-hadronization mechanism in Lund-string-based event generators in which nearby color strings or flux tubes repel one another in transverse space when dense events generate substantial string overlap. In the modern collider-phenomenology usage, SS is primarily a microscopic model for collective transverse dynamics and ridge-like correlations, while rope hadronization addresses hadrochemistry and strangeness enhancement; in PACIAE 4.0 for $p$Pb at $\sqrt{s_{NN}}=5.02~\mathrm{TeV}$, SS is treated specifically as a regulator that acts before flavor ropes, pushes strings apart, reduces overlap, and thereby tempers rope-induced strange-baryon enhancement rather than acting as an independent strange-quark source (Bierlich et al., 2016, Bierlich, 2018, Lei et al., 16 Sep 2025). The abbreviation “SS” is field-dependent, but in this literature it denotes string shoving rather than unrelated uses such as shortest superstring or string stability.

1. Historical emergence and conceptual scope

String shoving entered the literature as a dynamical extension of rope hadronization. The 2016 formulation by Bierlich, Gustafson, and Lönnblad introduced the idea that overlap energy among many nearby Lund strings should not only increase an effective string tension at hadronization, but should also generate a transverse pressure before hadronization, producing flow-like signatures without assuming a thermalized medium (Bierlich et al., 2016). The 2018 PYTHIA 8 and DIPSY formulation then placed SS and rope hadronization into a single microscopic framework of string–string interactions: shoving at intermediate proper times, ropes at hadronization time (Bierlich, 2018).

Subsequent work broadened the mechanism both technically and phenomenologically. A new “parallel frame” formalism generalized SS from beam-parallel toy geometries to arbitrary string pairs and made AA applications in Angantyr tractable (Bierlich et al., 2020). Later studies examined how shoving alters the space-time distribution of primary hadron vertices, how it affects jet-like observables, and how it behaves in pp, pPb, Xe-Xe, and PbPb systems (Chakraborty, 2020, Chakraborty, 2021, Chakraborty et al., 2022).

Milestone	Main addition	Platform or system
2016 (Bierlich et al., 2016)	Dynamical transverse pressure from overlapping strings	DIPSY + PYTHIA8, pp
2018 (Bierlich, 2018)	Unified shoving-plus-ropes collectivity framework	PYTHIA 8 and DIPSY
2020 (Bierlich et al., 2020)	Pairwise “parallel frame” and AA flow studies	PYTHIA8 + Angantyr
2022 (Chakraborty et al., 2022)	Space-time hadron-vertex consequences of string interactions	pp and PbPb
2025 (Lei et al., 16 Sep 2025)	PACIAE 4.0 treatment of SS inside rope hadronization	$p$Pb at $5.02$ TeV

This development established a stable division of labor within the string-interaction picture. Rope hadronization modifies local hadronization probabilities through larger effective tension; string shoving modifies the pre-hadronic space-time evolution by generating transverse repulsion.

2. Microscopic picture and formal structure

The basic microscopic assumption is that a confining string has a finite transverse field profile rather than being an infinitely thin line. In the Gaussian ansatz used in the PYTHIA/DIPSY formulation, the transverse field is written as

$$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$

where $R$ is the string width and $C$ is a normalization constant. Two string pieces separated by $d_\perp$ then repel with force per unit length

$$f(d_\perp) = \frac{g\kappa d_\perp}{R^2}\exp\left(-\frac{d_\perp^2(t)}{4R^2} \right),$$

with $\kappa$ the ordinary string tension and $\sqrt{s_{NN}}=5.02~\mathrm{TeV}$0 a dimensionless strength parameter (Bierlich, 2018).

The intended chronology is explicitly pre-hadronic. At $\sqrt{s_{NN}}=5.02~\mathrm{TeV}$1, strings are taken to have no transverse extension and therefore do not interact. Around $\sqrt{s_{NN}}=5.02~\mathrm{TeV}$2, after the parton shower has ended, strings can begin to shove each other; at $\sqrt{s_{NN}}=5.02~\mathrm{TeV}$3 they have reached full transverse extension and the shoving effect is maximal; at roughly $\sqrt{s_{NN}}=5.02~\mathrm{TeV}$4, hadronization occurs, and remaining overlap can instead be treated as rope formation (Bierlich, 2018).

The 2020 reformulation expressed the dynamics in hyperbolic coordinates,

$\sqrt{s_{NN}}=5.02~\mathrm{TeV}$5

and wrote the local evolution schematically as

$\sqrt{s_{NN}}=5.02~\mathrm{TeV}$6

thereby emphasizing that SS is a dynamical response of overlapping boost-extended string pieces rather than a static modification of fragmentation parameters (Bierlich et al., 2020).

This formal structure makes the physical content of SS distinct from ordinary color reconnection. Shoving is not a purely topological rewiring of strings; it is a coordinate-space interaction among finite-width flux tubes that converts overlap into transverse momentum.

3. Relation to rope hadronization and other non-perturbative mechanisms

Within the broader string-interaction framework, rope hadronization and string shoving are complementary but not interchangeable. Rope hadronization acts when strings still overlap at hadronization and increases the effective string tension, thereby enhancing tunneling production of heavier flavors and diquarks. In the 2018 formulation this was tied directly to the Schwinger-like relation

$\sqrt{s_{NN}}=5.02~\mathrm{TeV}$7

so increasing $\sqrt{s_{NN}}=5.02~\mathrm{TeV}$8 reduces strangeness suppression (Bierlich, 2018). SS, by contrast, modifies the geometry and transverse kinematics before hadronization.

The distinction is especially explicit in PACIAE 4.0. There, rope hadronization (RH) contains two mechanisms, flavor ropes (FR) and string shoving (SS). FR turns overlapping strings into ropes with a larger effective string tension and therefore directly boosts $\sqrt{s_{NN}}=5.02~\mathrm{TeV}$9 and diquark–antidiquark production. SS acts earlier, pushes adjacent strings outward, reduces their overlap, and thus diminishes the FR effect; accordingly, FR+SS yields a slightly smaller string tension than FR alone (Lei et al., 16 Sep 2025).

A further distinction is needed between SS and the reconnection and rescattering mechanisms used in the same PACIAE study (Lei et al., 16 Sep 2025).

Mechanism	Stage or framework	Primary role
MPI-CR	Color reconnection	Reduce total string length by MPI-based merging
QCD-CR	Color reconnection	Generate junction topologies and enhance baryons
FR	Rope hadronization	Increase effective string tension and enhance strange and baryonic production
SS	Rope hadronization, pre-FR	Push strings apart and reduce overlap
PRS	Partonic rescatterings	Weak or negligible effect on multi-strange baryons
HRS	Hadronic rescatterings	Promote $p$0 and $p$1 production

This separation is central to interpreting phenomenology. In the PACIAE logic, QCD-CR supplies baryon topology through junctions, FR supplies direct tension-driven strangeness enhancement, SS regulates FR by reducing overlap, and HRS can further increase multi-strange baryon yields. A common misconception is therefore to treat SS itself as the direct source of enhanced strangeness; in this setup its function is chiefly corrective and geometric rather than an independent flavor-production channel (Lei et al., 16 Sep 2025).

4. Collectivity, ridge observables, and small-system phenomenology

The original phenomenological target of string shoving was the near-side ridge and related flow-like observables in small systems. In the 2018 PYTHIA 8 implementation, enabling SS in high-multiplicity pp events generated a ridge-like long-range near-side structure and produced a $p$2 contribution that partly survived a pseudorapidity gap, unlike the mostly short-range contribution from ordinary color reconnection (Bierlich, 2018). In dense toy AA systems studied with the later parallel-frame formalism, the generated $p$3 increased with density and tracked the initial eccentricity closely enough to resemble a hydrodynamic response to geometry, although the realistic AA implementation still underpredicted measured flow (Bierlich et al., 2020).

More differential pp studies exposed the model’s limits. A dedicated 13 TeV PYTHIA8 analysis found that SS qualitatively reproduces a ridge-like long-range near-side correlation in high-multiplicity pp events, but also produces a substantial long-range near-side correlation in low-multiplicity events, leading to overestimated associated yields there and to over-subtraction when low-multiplicity templates are used to extract $p$4 (Kim et al., 2021). This established that the model can generate collectivity-like correlations, but not always with the experimentally observed multiplicity systematics.

A later ultra-long-range study at $p$5 TeV sharpened the point. In PYTHIA8 with shoving, the extracted $p$6 was finite in low-multiplicity pp collisions but decreased with increasing $p$7, whereas ALICE reported only weak multiplicity dependence within uncertainties. The same study found that the strongest SS sensitivity appeared in events with very low $p$8, roughly dijet-like topologies, and argued that flattenicity may be a better event-activity estimator than $p$9 for suppressing non-flow biases (Ortiz et al., 9 Dec 2025). This supports a restricted interpretation in which SS may explain part of the low-multiplicity limit, while hydrodynamic behavior becomes more relevant at high multiplicity.

The mechanism has also been extended toward jet observables. In a new PYTHIA8 implementation designed for jet studies, an enhanced repulsion strength $5.02$0 produced a suppressed near-side charged-hadron peak and a broadened away-side peak in pp correlation observables, suggesting that interacting strings may offer a common non-perturbative contribution to both collectivity and jet modification in denser $5.02$1-A or A-A environments (Chakraborty, 2020). The claim there is qualitative and explicitly exploratory.

5. Strangeness production and the PACIAE $5.02$2Pb interpretation

In hadrochemical applications, SS acquires a different role from its original ridge motivation. The PACIAE 4.0 study of $5.02$3Pb collisions at $5.02$4 examined multiplicity-dependent integrated yields $5.02$5 for $5.02$6, $5.02$7, $5.02$8, $5.02$9, and $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$0, the multiplicity variable $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$1, and the ratios $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$2, $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$3, $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$4, and $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$5. No flow observables or $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$6-differential spectra were used to assess SS in that work (Lei et al., 16 Sep 2025).

The central result is highly species-dependent. The studied mechanisms have little effect on kaon production. The substantial differences occur for strange baryons. MPI-CR systematically underestimates strange-baryon production. QCD-CR improves the description because junction structures enhance baryon formation, but remains below the integrated strange-baryon yields. Adding flavor ropes in QCD-CR+FR overshoots strange-baryon production across multiplicities. Adding SS to obtain QCD-CR+FR+SS reduces the excessive FR enhancement and gives the best overall agreement with ALICE data (Lei et al., 16 Sep 2025).

This preferred role of SS is explicitly moderating rather than directly enhancing. In the PACIAE interpretation, SS occurs before FR, pushes strings outward, reduces overlap, and therefore lowers the effective string tension relative to QCD-CR+FR. That is why QCD-CR+FR+SS performs better than QCD-CR+FR for $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$7, $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$8, and $$\mathcal{E}(r_\perp)=C\,\exp\left(-\frac{r_\perp^2}{2R^2}\right),$$9. The data placement is also species dependent: for $R$0, the data lie between QCD-CR and QCD-CR+FR+SS; for $R$1 and $R$2, they lie between QCD-CR+FR+SS and QCD-CR+FR. The residual undershoot of QCD-CR+FR+SS grows from $R$3 to $R$4, which the authors interpret as indicating that heavier strange baryons require stronger effective string tension (Lei et al., 16 Sep 2025).

Broader PYTHIA8 and Angantyr studies are consistent with this division of labor. Proceedings results on pp, p-Pb, and Xe-Xe found that ropes, with or without shoving, raise $R$5 with multiplicity and system size, while fragmentation retuning remains necessary because some kaon-to-pion observables are overshot in the untuned setup (Chakraborty, 2021). Taken together, these studies suggest that SS is often necessary for a realistic dense-string picture, but that the dominant direct mechanism for strangeness enhancement remains rope-induced tension বৃদ্ধি rather than shoving itself.

6. Implementations, parameters, and unresolved issues

String shoving has been implemented in several generator environments, including DIPSY + PYTHIA8, standalone PYTHIA8, PYTHIA8 + Angantyr for nuclear collisions, and PACIAE 4.0 (Bierlich, 2018, Bierlich et al., 2020, Lei et al., 16 Sep 2025). The technical realization differs across these platforms. PYTHIA-based implementations moved from earlier beam-parallel approximations to a pairwise “parallel frame” in which arbitrary string pieces can be treated symmetrically. The accumulated repulsion is implemented not as a continuous analytic deformation of the full string worldsheet, but through discrete transverse nudges mapped onto hadrons after hadronization, which keeps the computation tractable in high-multiplicity environments (Bierlich et al., 2020).

The parameterization is correspondingly model dependent. Early phenomenology described physically relevant shoving strengths as $R$6 values not too far from unity (Bierlich, 2018). The 2020 AA study used canonical values $R$7, $R$8, and $R$9, and slightly retuned $C$0 from 2.28 to 2.4 in pp to preserve basic observables after shoving (Bierlich et al., 2020). The 2022 space-time study used $C$1 and $C$2 and emphasized that shoving broadens primary-hadron vertex distributions and reduces local string density, thereby damping subsequent rope formation (Chakraborty et al., 2022). By contrast, the PACIAE $C$3Pb study did not provide explicit SS force laws, overlap functions, kick formulas, or SS-specific tuned coefficients; it only listed global MPI and fragmentation parameters such as $C$4, $C$5, and several $C$6 settings (Lei et al., 16 Sep 2025).

Several limitations recur across the literature. First, many published SS results are explicitly qualitative rather than precision quantitative; the 2018 ridge study already noted the simplicity of its MPI geometry model and the expectation of improvement from more realistic proton substructure (Bierlich, 2018). Second, realistic AA applications remain incomplete because soft-gluon-rich string topologies, early-time interactions before full transverse expansion, and late-time interactions near or after breakup are not yet treated fully consistently (Bierlich et al., 2020). Third, the 2022 hadron-vertex study emphasized that current shoving and rope implementations are not fully self-consistent, because in practice shove momentum is transferred to primary hadrons after hadronization rather than dynamically updating string positions before rope overlap is evaluated (Chakraborty et al., 2022). Fourth, the 2025 pp ultra-long-range study showed that SS alone does not reproduce the observed weak multiplicity dependence of $C$7 across the full range and is best viewed as a partial, low-multiplicity explanation rather than a complete description of pp collectivity (Ortiz et al., 9 Dec 2025).

The resulting picture is technically coherent but not closed. String shoving has become a standard non-perturbative component of the Lund-based description of dense hadronic events, with a well-defined role as a pre-hadronic transverse-pressure mechanism. What remains unsettled is not its conceptual identity, but the degree to which a fully realistic treatment of arbitrary string topologies, consistent rope–shoving coupling, and experimentally unbiased event classification can make the framework quantitatively competitive across pp, pA, and AA observables.