Soft Drop: Jet Grooming & Soft Matter

• Soft Drop is a jet grooming procedure that iteratively removes soft, wide-angle radiation to reveal the hard collinear core in jets.
• It uses key parameters (z_cut and β) to control grooming aggressiveness and alter logarithmic and non-perturbative properties.
• In soft matter, 'Soft Drop' describes liquid droplet deformation on soft or non-wetting substrates governed by capillarity and elasticity.

Searching arXiv for the core "Soft Drop" papers to ground the article in the primary literature. arxiv_search({"query":"id:(Larkoski et al., 2014) OR ti:\"Soft Drop\"","max_results":5,"sort_by":"relevance"}) arxiv_search({"query":"id:(Marzani et al., 2017) OR ti:\"The jet mass distribution after Soft Drop\"","max_results":5,"sort_by":"relevance"}) arxiv_search({"query":"id:(Cal et al., 2021) OR ti:\"The soft drop momentum sharing fraction z_g beyond leading-logarithmic accuracy\"","max_results":5,"sort_by":"relevance"}) arxiv_search({"query":"id:(Chevy et al., 2012) OR ti:\"Liquid Hertz Contact\" OR ti:\"Drops on soft solids\"","max_results":10,"sort_by":"relevance"}) arxiv_search({"query":"(\"Soft-drop thrust\" OR \"Recursive Soft Drop\" OR \"soft-drop jet observables\" OR \"A measurement of soft-drop jet observables\")","max_results":10,"sort_by":"relevance"}) Soft Drop denotes two unrelated concepts in contemporary physics. In collider QCD, it is a recursive grooming and declustering procedure introduced to remove soft, wide-angle radiation from jets while retaining the hard collinear core, with the algorithm controlled by a softness threshold $z_{\rm cut}$ and an angular exponent $\beta$; the limit $\beta=0$ corresponds roughly to the modified Mass Drop Tagger (Larkoski et al., 2014). In soft-matter and wetting literature, the phrase “soft drop” is used for deformable liquid droplets, either on non-wetting substrates where capillary restoring forces acquire logarithmic corrections or on elastic substrates where elastocapillary coupling alters contact angles and dynamics (Chevy et al., 2012, Lubbers et al., 2013). In the modern literature, the collider-physics meaning is the one associated with a named algorithm and the broadest derived formalism.

1. Algorithmic definition and parameter space

In hadron-collider applications, Soft Drop is applied to a jet that has typically been found with anti-$k_t$ and then reclustered with Cambridge/Aachen (C/A), so that the clustering tree is angular ordered. One undoes the last clustering step $j\to j_1,j_2$, tests the pair against the Soft Drop condition,

$$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$

and, if the condition fails, discards the softer branch and continues along the harder branch until a passing splitting is found or the tree is exhausted (Larkoski et al., 2014). The first passing splitting defines the groomed jet.

The $e^+e^-$ version is formulated on hemisphere jets obtained with the $e^+e^-$ Cambridge–Aachen algorithm and uses the corresponding energy-sharing criterion

$$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$

with grooming applied separately to the two hemispheres before constructing groomed event shapes such as soft-drop thrust (Marzani et al., 2019). A related hemisphere-based normalization was introduced to make the resulting thrust observable collinear safe for $\beta=0$ (Baron et al., 2018).

The two parameters play distinct roles. Larger $\beta$0 makes grooming more aggressive by tightening the momentum-sharing threshold. The exponent $\beta$1 controls angular dependence: $\beta$2 retains some soft-collinear radiation, $\beta$3 removes soft radiation independent of angle and yields the mMDT-like limit, and $\beta$4 makes the procedure sufficiently aggressive that it behaves more like a tagger than a groomer (Larkoski et al., 2014). In the soft limit, the grooming boundary is a straight line in the Lund plane,

$\beta$5

which is the geometric origin of many of the algorithm’s logarithmic properties (Larkoski et al., 2014).

A persistent misconception is that Soft Drop and mMDT are interchangeable. The papers are explicit that $\beta$6 or $\beta$7 gives the mMDT-like limit, but $\beta$8 changes the retained phase space and therefore changes both the logarithmic structure and the non-perturbative sensitivity (Marzani et al., 2017).

2. Perturbative structure, IRC safety, and factorization

Soft Drop was designed not only to suppress contamination from pileup, underlying event, and hadronization, but also to simplify analytic QCD calculations. The original resummed analysis already showed that for mass-like observables the coefficient of the double logarithm is reduced in the groomed region, and that in the $\beta$9 limit only single logarithms survive; non-global logarithms are suppressed for finite $\beta=0$0 and vanish at $\beta=0$1 for the energy correlation observable studied there (Larkoski et al., 2014). This simplification is the principal reason Soft Drop became a precision-substructure benchmark.

For groomed jet mass, the first-principles calculation of the mass distribution after Soft Drop established NLL resummation matched to exact NLO fixed order, with non-perturbative corrections assessed analytically and by Monte Carlo envelopes (Marzani et al., 2017). In that framework the cumulative distribution is organized through a flavor sum with radiators $\beta=0$2, while the perturbative endpoint is corrected to avoid boundary artifacts. The paper emphasized that moving from NLL+LO to NLL+NLO significantly shifts the central prediction and substantially reduces the uncertainty (Marzani et al., 2017).

For the momentum-sharing fraction $\beta=0$3, the crucial point is that IRC safety depends on $\beta=0$4. For $\beta=0$5, $\beta=0$6 is IRC safe; for $\beta=0$7, it is not IRC safe in the usual sense and instead is Sudakov safe, with the divergence regulated by the Sudakov suppression from vetoed emissions (Cal et al., 2021). The beyond-LL treatment was formulated in SCET as

$\beta=0$8

with hard, collinear, soft, and non-global structures separated and RG evolved to a common scale (Cal et al., 2021).

For the groomed radius $\beta=0$9, the central theoretical result is an all-order equivalence between Soft Drop declustering and a veto on emissions in the annulus between the groomed and ungroomed jet boundaries, valid in the small-$k_t$0, small-$k_t$1 regime (Kang et al., 2019). That equivalence leads to a factorization theorem and NLL resummation including non-global logarithms and Abelian clustering logarithms due to C/A clustering. The same theme reappears in the EFT treatment of the joint $k_t$2–$k_t$3 distribution, where the effective theory changes across large-, intermediate-, and small-$k_t$4 regimes because the radiation that stops grooming and the radiation that sets the jet mass can coincide or separate into distinct modes (Pathak et al., 2020).

The most important conceptual correction to naive intuition is that Soft Drop does not trivially make every observable IRC safe. Some observables remain Sudakov safe rather than IRC safe, and some require genuinely joint resummation because the stopping angle and the measured quantity are kinematically entangled (Cal et al., 2021, Cal et al., 2020).

3. Canonical observables induced by Soft Drop

The first passing splitting defines a family of observables that have become standard benchmarks for groomed substructure.

Observable	Definition	Principal theoretical feature
Groomed jet mass $k_t$5	$k_t$6	NLL resummation matched to NLO; non-perturbative corrections grow with $k_t$7 (Marzani et al., 2017)
Momentum sharing $k_t$8	$k_t$9 of the first passing splitting	Direct sensitivity to Altarelli–Parisi splitting functions; Sudakov safe for $j\to j_1,j_2$0 (Cal et al., 2021)
Groomed angle $j\to j_1,j_2$1, $j\to j_1,j_2$2	$j\to j_1,j_2$3, $j\to j_1,j_2$4	NLL factorization via veto equivalence; NGLs and clustering effects enter (Kang et al., 2019)
Energy drop $j\to j_1,j_2$5	$j\to j_1,j_2$6	Requires joint resummation with $j\to j_1,j_2$7; $j\to j_1,j_2$8 case is Sudakov safe (Cal et al., 2020)

The groomed jet mass is the most mature precision observable. Its distribution is amenable to high-order resummation and fixed-order matching, and its hadronization and underlying-event corrections can be estimated analytically or through generator envelopes (Marzani et al., 2017). The field-theoretic non-perturbative analysis later sharpened this picture: in the Soft Drop operator expansion region, the leading power corrections are governed by three universal non-perturbative parameters per initiating flavor and cannot be reduced to a standard normalized shape function (Hoang et al., 2019). This was pushed further with NNLL perturbative inputs, allowing model-independent universality tests of hadronization for quark and gluon jets in both $j\to j_1,j_2$9 and $$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$0 collisions (Ferdinand et al., 2023).

The $$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$1 observable is distinctive because it directly samples the first collinear splitting that survives grooming. At LL it exposes only the color factor $$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$2, while at NLL$$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$3 the full splitting functions and matching terms induce sensitivity not only to the initiating parton’s color representation but also to its spin, producing quark–gluon differences at the 10% level in the calculations reported there (Cal et al., 2021). This makes $$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$4 one of the clearest perturbative probes of QCD branching dynamics after grooming.

The groomed radius $$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$5 is both a substructure variable and a proxy for the effective angular support of the groomed jet. Its active area scales like $$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$6, so it directly quantifies how much of the original jet survives grooming (Kang et al., 2019). By contrast, the energy drop $$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$7 is a measure of the cumulative soft radiation removed by grooming. Its analysis showed that, for ordinary Soft Drop, the correct all-order structure requires a double-differential treatment in $$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$8 and $$\frac{\min(p_{T1},p_{T2})}{p_{T1}+p_{T2}} > z_{\rm cut}\left(\frac{\Delta R_{12}}{R_0}\right)^\beta,$$9, especially in the $e^+e^-$0 Sudakov-safe case (Cal et al., 2020).

4. Event-shape formulations and precision $e^+e^-$1 programs

Soft Drop was adapted from single-jet substructure to $e^+e^-$2 event shapes by grooming hemispheres defined relative to the thrust axis. In soft-drop thrust, one first finds the ungroomed thrust axis, divides the event into hemispheres, grooms each hemisphere separately, and then computes thrust from the groomed hemispheres with an explicit denominator rescaling so that the observable is collinear safe for $e^+e^-$3 (Marzani et al., 2019). The underlying motivation is that ungroomed thrust suffers from large hadronization corrections at small $e^+e^-$4, which destabilize extractions of $e^+e^-$5.

The fixed-order and resummed studies of soft-drop thrust and related event shapes reached complementary conclusions. The dedicated thrust fit study used NLO+NLL predictions matched to Sherpa pseudo-data and found that grooming reduces the hadronization-induced shift in the fitted $e^+e^-$6, stabilizes the fits under changes of the lower fit bound, and makes it possible to extend the fit range below the default interval $e^+e^-$7 (Marzani et al., 2019). In that analysis, the most favorable cases were $e^+e^-$8 and $e^+e^-$9 with $e^+e^-$0 or $e^+e^-$1 (Marzani et al., 2019).

At fixed NNLO in $e^+e^-$2, soft-drop thrust, hemisphere mass, and narrow-jet mass were computed with the CoLoRFulNNLO subtraction method (Kardos et al., 2018). The main result was that grooming generally improves perturbative convergence, with $e^+e^-$3 giving the best stability in the parameter scan for soft-drop thrust and hemisphere mass (Kardos et al., 2018). However, the earlier SCET analysis of soft-drop thrust also stressed a limitation: the transition region around $e^+e^-$4 is not as theoretically benign as the deep two-jet region, and the interplay between grooming and observable definition can be nontrivial (Baron et al., 2018). That study found groomed small-$e^+e^-$5 jet mass to be better behaved than groomed thrust in the intermediate region (Baron et al., 2018).

The same logic has been extended to hadronic event shapes. Soft-drop groomed transverse thrust in dijet events was formulated by splitting the event into hemispheres using the transverse-thrust axis, grooming each hemisphere, and then constructing a groomed event shape with a final factor that preserves collinear safety (Baron et al., 2020). The resulting predictions reached NLO+NLL$e^+e^-$6 accuracy within a CAESAR/Sherpa implementation, and the phenomenological study found that Soft Drop is very efficient in removing the underlying event (Baron et al., 2020). This suggests that Soft Drop is not merely a jet-substructure prescription but also an event-level perturbative filter.

5. Variants, measurements, and specialized applications

Recursive Soft Drop generalizes the original algorithm by enforcing the Soft Drop condition $e^+e^-$7 times rather than stopping at the first passing splitting (Dreyer et al., 2018). In the top-down formulation, $e^+e^-$8 reproduces ordinary Soft Drop, $e^+e^-$9 gives no grooming, and $$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$0 yields a fully recursive limit in which groomed jets formally have zero catchment area (Dreyer et al., 2018). In simulations, this improved jet-mass resolution for boosted $$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$1, top, and Higgs jets, with the benefit saturating around $$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$2 for $$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$3 jets but extending to larger $$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$4 for more complex multi-prong topologies (Dreyer et al., 2018). The same paper introduced Bottom-Up Soft Drop, including an event-wide global form of grooming (Dreyer et al., 2018).

Heavy-flavor jets provide a specialized testing ground for Soft Drop because the grooming observables probe the dead cone and threshold structure of the running coupling. An NLL treatment of Soft Drop observables for $$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$5- and $$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$6-tagged jets showed that the groomed opening angle

$$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$7

is sensitive to the dead cone, while $$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$8 can be treated through a conditional-probability formalism; in that study, $$\frac{\min(E_i,E_j)}{E_i+E_j} > z_{\rm cut}\left(1-\cos\theta_{ij}\right)^{\beta/2},$$9 emerged as both theoretically cleaner and more sensitive to mass effects (Caletti et al., 2023).

Experimental soft-drop measurements have validated much of the perturbative framework. ATLAS measured the groomed jet mass, momentum sharing $\beta=0$0, and groomed opening angle $\beta=0$1 in dijet events at $\beta=0$2 TeV using $\beta=0$3, for both all-particle and track-based definitions (Collaboration, 2019). The unfolded data were compared with resummed calculations and hadron-level Monte Carlo, and the paper reported that analytical and shower-based predictions provide an excellent description of the data in most regions of phase space (Collaboration, 2019). That work also gave the first comparison between an analytical prediction and an unfolded measurement of $\beta=0$4 (Collaboration, 2019).

A second misconception is that Soft Drop removes all non-perturbative effects to negligible levels. The literature is more precise: grooming substantially reduces these effects and often makes them more universal or more controllable, but the low-mass, small-angle, or strongly Sudakov-safe regions remain sensitive to hadronization, boundary effects, and, in hadronic collisions, residual initial-state or underlying-event structure (Marzani et al., 2017, Collaboration, 2019, Ferdinand et al., 2023).

6. “Soft drop” in soft-matter and wetting physics

In soft-matter physics, the phrase refers not to a grooming algorithm but to a liquid droplet whose shape and mechanics are governed by capillarity, gravity, and, in some cases, substrate elasticity. For a droplet of radius $\beta=0$5, surface tension $\beta=0$6, and density $\beta=0$7 resting on a non-wetting substrate in the weak-deformation regime $\beta=0$8, with capillary length $\beta=0$9, the naive scaling $\beta$00 for the sag is corrected to

$\beta$01

and the corresponding effective stiffness

$\beta$02

vanishes logarithmically as the deformation tends to zero (Chevy et al., 2012). The resulting “liquid Hertz contact” picture explains the increase of bounce contact time at low impact speed and the size dependence of low-frequency oscillations (Chevy et al., 2012).

A different but related soft-matter usage concerns droplets on elastic substrates. There the governing lengths are the drop size $\beta$03, the molecular cutoff $\beta$04, and the elastocapillary length $\beta$05, and the central result is a double transition in contact angles as the substrate is softened (Lubbers et al., 2013). The microscopic contact-line geometry departs from Young’s law when $\beta$06, while the apparent macroscopic contact angle changes only in the very soft limit $\beta$07 (Lubbers et al., 2013). A later gradient-dynamics model recast this elastocapillary problem in a long-wave form with fields $\beta$08 and $\beta$09, recovering the same double transition while also describing viscoelastic braking of spreading and a softness-dependent change in coarsening mode for interacting droplets (Henkel et al., 2021).

These soft-matter uses are historically and conceptually independent of the jet-grooming algorithm. The shared phrase “Soft Drop” therefore spans two distinct research programs: one in perturbative QCD, where declustering and vetoed phase space define groomed observables, and one in capillarity and wetting, where droplet compliance and elastocapillary coupling determine statics and dynamics.