Pure Decay Modes: Single-Mechanism Processes

• Pure decay modes are decay processes characterized by the isolation of one dominant mechanism or exponential law, reducing interference from competing channels.
• They appear in nuclear, particle, few-body, and non-Hermitian systems, offering clear distinctions between direct and sequential decay mechanisms.
• Empirical and theoretical studies show that identifying pure decay modes streamlines modeling by focusing on a single decay constant or topology.

Searching arXiv for recent and relevant papers on “pure decay modes” and closely related usages across nuclear, particle, few-body, and non-Hermitian contexts. “Pure decay modes” is a context-dependent term used across several research domains to denote decay channels or decay kinetics that are effectively unmixed, single-topology, or single-rate. In nuclear and particle-decay studies, it can refer to a decay proceeding through one dominant physical mechanism, such as a pure \textit{W}-boson-exchange topology in a charmed-baryon transition or a direct ground-state emission without an initiating weak process (Collaboration et al., 2023, Pfützner et al., 2011). In resonance theory, it can denote a survival law governed by a single exponential associated with a Gamow state (Madrid, 2015). In open quantum few-body systems, it can describe a probability sector controlled by one exponential decay rate (Kim et al., 2011). In recent non-Hermitian network theory, it denotes eigenmodes with smooth exponential spatial attenuation and no oscillatory component (Liu et al., 15 Jul 2025). The shared conceptual core is the isolation of one decay channel, one decay topology, or one decay constant from competing processes.

1. Conceptual scope and definitions

The term has no single universal definition across fields. Instead, the literature uses it in several technically distinct but structurally related senses.

In the review of radioactive decays near the limits of nuclear stability, “pure” decay modes are direct, ground-state particle emissions driven by the internal instability of the parent nucleus, without an initiating weak transition (Pfützner et al., 2011). Within that convention, proton radioactivity, true two-proton radioactivity, ground-state neutron or two-neutron emission, and alpha decay are pure modes, whereas beta-delayed particle emission is explicitly not pure because a beta transition first populates excited daughter states (Pfützner et al., 2011).

In charmed-baryon weak decays, a pure mode is one whose amplitude arises exclusively from a single topological class. The decay $\Lambda_c^+ \to \Xi^0 K^+$ is described as a pure \textit{W}-boson-exchange decay because it “can only be produced via a W-boson-exchange process,” with external and internal \textit{W}-emission topologies excluded by quark-content and hadronization constraints (Collaboration et al., 2023).

In resonance theory, the pure object is not the channel but the temporal law. A Gamow state evolves as

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$

so its survival probability is purely exponential,

$p_s(t)=e^{-\Gamma_R t/\hbar},$

with lifetime $\tau=\hbar/\Gamma_R$ (Madrid, 2015). In this usage, purity means absence of nonexponential background corrections.

In few-body tunneling, the term refers to a probability component dominated by one exponential rate. For two repulsively interacting bosons escaping a one-dimensional trap, a “pure decay mode” is a probability component with

$P(t)\simeq A e^{-\gamma t},$

over a substantial time window, with negligible multirate interference (Kim et al., 2011).

In non-Hermitian graph theory, the phrase is spatial rather than temporal. Pure decay modes are eigenstates whose amplitudes decay monotonically as a pure exponential without oscillatory wave patterns:

$|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$

with no position-dependent phase winding (Liu et al., 15 Jul 2025).

This distribution of meanings suggests that “pure decay modes” is best understood as a family-resemblance concept: decay governed by one dominant mechanism, topology, or exponential law rather than by interference among several comparably important ones.

2. Pure kinetics: exponential decay and Gamow-state formulations

A foundational usage of purity concerns the time dependence of unstable systems. In the Gamow-state formalism, resonances are represented by kets $|z_R\rangle$ satisfying

$$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$

which directly produce exponential survival (Madrid, 2015). This identifies a sharp distinction between the pole width $\Gamma_R$, which governs the temporal law, and other width-like quantities constructed from energy-resolved decay probabilities (Madrid, 2015).

The same formalism introduces energy-differential decay probabilities into continuum channels $|E,i\rangle$. The energy-resolved probability density at time $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$0 is

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$1

which leads to the differential decay width

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$2

and corresponding partial widths and branching fractions (Madrid, 2015). In the sharp-resonance approximation, these expressions reduce to Golden-Rule-like formulas, and the branching fractions coincide with standard Golden Rule branching fractions (Madrid, 2015).

A closely related experimental problem is whether an observed decay law is actually pure exponential. In the ESR single-ion spectroscopy of hydrogen-like $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$3Pm$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$4, the electron-capture decay-time distribution was found to be well described by

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$5

with no statistically significant time modulation (Ozturk et al., 2019). The fitted laboratory-frame decay constants were $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$6 and $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$7, corresponding to half-lives $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$8 and $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$9, respectively (Ozturk et al., 2019). A modulated model,

$p_s(t)=e^{-\Gamma_R t/\hbar},$0

yielded $p_s(t)=e^{-\Gamma_R t/\hbar},$1, compatible with zero, and a conservative Gaussian $p_s(t)=e^{-\Gamma_R t/\hbar},$2 confidence upper bound $p_s(t)=e^{-\Gamma_R t/\hbar},$3 (Ozturk et al., 2019). The result was therefore interpreted as “pure” decay kinetics under ESR conditions, excluding the previously reported $p_s(t)=e^{-\Gamma_R t/\hbar},$4 oscillations in the same system (Ozturk et al., 2019).

This usage emphasizes that purity can be an empirical statement about the absence of measurable departures from a single exponential, rather than a statement about microscopic exclusivity.

3. Few-body tunneling and single-rate sectors

In open quantum few-body systems, purity often attaches to a particular sector of configuration space rather than to the total decay process. For two repulsively interacting bosons tunneling through a delta barrier, the configuration space is partitioned into “in-in,” “in-out,” and “out-out” sectors according to whether both, one, or neither particle remains in the trap (Kim et al., 2011). The associated probabilities satisfy

$p_s(t)=e^{-\Gamma_R t/\hbar},$5

(Kim et al., 2011).

The main result is that the in-in probability is a pure single-exponential decay,

$p_s(t)=e^{-\Gamma_R t/\hbar},$6

with

$p_s(t)=e^{-\Gamma_R t/\hbar},$7

where $p_s(t)=e^{-\Gamma_R t/\hbar},$8 and $p_s(t)=e^{-\Gamma_R t/\hbar},$9 are complex wave numbers determined by outgoing boundary conditions at the delta barrier (Kim et al., 2011). This rate is interpreted as the decay rate of a two-body Gamow resonance. The paper reports that for all interaction strengths studied, $\tau=\hbar/\Gamma_R$0 is linear over the relevant time window, and the slope agrees with $\tau=\hbar/\Gamma_R$1 extracted from the outgoing-boundary-condition problem (Kim et al., 2011).

The in-out sector is more subtle. For weak to moderate repulsion, $\tau=\hbar/\Gamma_R$2, the in-out probability is well captured by a sequential single-particle model in which one boson escapes from the in-in sector at rate $\tau=\hbar/\Gamma_R$3, and the remaining boson decays at essentially the lowest one-particle Gamow rate $\tau=\hbar/\Gamma_R$4 (Kim et al., 2011). In that regime, the single-mode description remains accurate with relative error below $\tau=\hbar/\Gamma_R$5 (Kim et al., 2011). For larger $\tau=\hbar/\Gamma_R$6, however, the in-out sector crosses over to a two-mode description,

$\tau=\hbar/\Gamma_R$7

and the paper identifies a crossover in model superiority around $\tau=\hbar/\Gamma_R$8 (Kim et al., 2011).

This provides a useful technical distinction. A pure decay mode is not necessarily the full decay process. Rather, a specific sector may be pure, while another sector of the same physical problem requires a multirate description. The paper explicitly states that purity is excellent for $\tau=\hbar/\Gamma_R$9 across $P(t)\simeq A e^{-\gamma t},$0, whereas for $P(t)\simeq A e^{-\gamma t},$1 it only holds for $P(t)\simeq A e^{-\gamma t},$2 (Kim et al., 2011).

4. Pure channels at the limits of nuclear stability

Near the drip lines, the term “pure decay mode” is commonly used to distinguish direct ground-state emission from mixed or delayed mechanisms. The review of radioactive decays at the limits of stability makes this distinction explicit through separation-energy criteria (Pfützner et al., 2011).

For proton emission, ground-state proton radioactivity occurs when the proton separation energy is negative,

$P(t)\simeq A e^{-\gamma t},$3

and the decay is sufficiently hindered by the Coulomb and centrifugal barriers to have a radioactive timescale (Pfützner et al., 2011). True two-proton radioactivity is characterized by

$P(t)\simeq A e^{-\gamma t},$4

so sequential one-proton emission is energetically closed and the decay must proceed as a genuine three-body process (Pfützner et al., 2011). The review treats this as a paradigmatic pure mode, sharply separated from beta-delayed two-proton emission, which is not pure because it proceeds through a prior beta decay (Pfützner et al., 2011).

The distinction matters experimentally and theoretically. True $P(t)\simeq A e^{-\gamma t},$5 emitters such as $P(t)\simeq A e^{-\gamma t},$6Fe, $P(t)\simeq A e^{-\gamma t},$7Zn, $P(t)\simeq A e^{-\gamma t},$8Ni, and the short-lived case $P(t)\simeq A e^{-\gamma t},$9Mg are described through three-body models, hyperspherical-harmonics treatments, and correlation observables in Jacobi coordinates; simple diproton or purely sequential pictures are found inadequate in the true-$|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$0 regime (Pfützner et al., 2011). The paper’s criterion therefore attaches purity to the mechanism: direct ground-state three-body decay with no intermediate allowed one-proton channel.

A different nuclear usage concerns alpha versus weak decay in neutron-rich heavy and superheavy nuclei. In neutron-rich thorium and uranium isotopes studied with axially deformed RMF plus BCS pairing, alpha emission is reported as energetically and dynamically suppressed, whereas beta-minus decay is predicted to dominate, with half-lives of “tens of seconds” for $|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$1Th and $|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$2U (Kumar et al., 2016). Quantitatively, the calculated alpha-decay penetrabilities for these isotopes are tiny: for $|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$3Th, $|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$4 with $|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$5; for $|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$6U, $|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$7 with $|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$8 (Kumar et al., 2016). The paper interprets this as suppression of pure alpha emission and dominance of pure $|\psi_{i_n}|=|\psi_{i_0}|e^{-\kappa n},$9 decay in the neutron-rich window (Kumar et al., 2016).

Related RMF studies of even-even superheavy isotopes with $|z_R\rangle$0 and $|z_R\rangle$1 use the local asymmetry parameter

$|z_R\rangle$2

and its surface-to-center ratio

$|z_R\rangle$3

as structural diagnostics of likely dominant modes (Bhuyan, 2018, Bhuyan et al., 2018). These papers do not compute all competing half-lives explicitly, but argue that increasing surface neutron richness biases the upper ends of the isotopic chains toward $|z_R\rangle$4 decay, whereas mid-chain nuclei nearer the valley of stability remain alpha dominated (Bhuyan, 2018, Bhuyan et al., 2018). A plausible implication is that “purity” here is inferred from density structure and asymmetry trends rather than from direct width calculations.

A more explicit competition study appears in the Skyrme-HF+BCS+pnQRPA analysis of alpha versus weak decay along the decay chains from hypothetical $|z_R\rangle$5 and $|z_R\rangle$6 nuclei (Sarriguren, 2022). That paper proposes an operational classification:

• pure alpha if $|z_R\rangle$7, equivalently $|z_R\rangle$8;
• pure weak if $|z_R\rangle$9, equivalently $$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$0;
• mixed otherwise (Sarriguren, 2022).

Under this definition, high-$$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$1 members of the chains are mostly pure alpha, lower-$$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$2 actinide members can become pure weak, and an intermediate region exhibits genuine competition (Sarriguren, 2022). This is one of the clearest formal branch-based uses of “pure decay mode.”

5. Pure topology in weak and strong interaction decays

In hadronic weak decays, purity may refer to a unique topological amplitude. The BESIII measurement of the decay asymmetry in $$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$3 is a precise example (Collaboration et al., 2023). In the topological classification of charmed-baryon nonleptonic decays, amplitudes are organized into external \textit{W}-emission, internal \textit{W}-emission, and \textit{W}-exchange classes. $$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$4 is classified as pure \textit{W}-exchange because only the exchange topology contributes (Collaboration et al., 2023).

Experimentally, BESIII performed a full angular analysis using $$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$5 of $$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$6 data collected between $$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$7 and $$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$8 and measured

$$H|z_R\rangle = z_R |z_R\rangle,\qquad z_R=E_R-i\Gamma_R/2,$$9

with two solutions for the relative $\Gamma_R$0–$\Gamma_R$1 phase shift,

$\Gamma_R$2

or

$\Gamma_R$3

(Collaboration et al., 2023). Because

$\Gamma_R$4

a phase difference near $\Gamma_R$5 implies $\Gamma_R$6, giving a “non-interference effect” even if both partial waves are present (Collaboration et al., 2023). In this setting, purity concerns the underlying quark-level topology, not the absence of partial-wave structure.

A different topological usage appears in searches for Majorana neutrinos in $\Gamma_R$7 decays. The proposed lepton-number-violating channels

$\Gamma_R$8

are described as pure $\Gamma_R$9 probes because they proceed only via Majorana-neutrino exchange, with no competing Standard Model amplitudes at tree level (Mandal et al., 2016). For the CKM-favored $|E,i\rangle$0 transition, only the $|E,i\rangle$1-channel Majorana-neutrino exchange contributes, and in the narrow-width approximation the branching ratio factorizes into production and decay of the on-shell heavy neutrino (Mandal et al., 2016). Here again, purity is exclusivity of the microscopic mechanism.

The phrase is also used for strong decays. In the molecular model of $|E,i\rangle$2, the decays

$|E,i\rangle$3

are treated as pure strong decays, with electromagnetic and weak effects neglected (Dong et al., 2013). The model predicts a favored value

$|E,i\rangle$4

and favored three-body widths

$|E,i\rangle$5

for $|E,i\rangle$6 (Dong et al., 2013). The paper’s use of “pure” means purely strong-interaction mediated within the adopted effective theory.

These cases show that in particle physics the term often marks topological cleanliness: one amplitude class, one interaction type, or one mediating particle.

6. Direct versus sequential decay and the problem of mechanism purity

A recurrent theme in nuclear-decay work is the distinction between direct and sequential decay. The decay of the Hoyle state in $|E,i\rangle$7C provides a particularly clear example (Zheng et al., 2018).

The paper distinguishes:

• direct or “pure” $|E,i\rangle$8 decay, in which the Hoyle state tunnels directly into three alpha particles without forming an intermediate $|E,i\rangle$9Be;
• sequential decay, in which $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$00C$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$01Be$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$02, followed by $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$03Be$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$04 (Zheng et al., 2018).

Using one- and two-dimensional tunneling calculations, the authors estimate the direct-to-sequential ratio for the Hoyle state to be more than an order of magnitude below the experimental upper limit of $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$05 at $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$06 confidence (Zheng et al., 2018). Specifically, they report a direct equal-energy component

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$07

and, after accounting for indistinguishability, approximately

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$08

still far below the experimental limit (Zheng et al., 2018). They further state that their total direct-to-sequential ratio is about 40 times below the $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$09 limit, अर्थात್ of order $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$10 (Zheng et al., 2018).

The significance is twofold. First, the “pure” direct mechanism is found to be strongly suppressed by barrier physics near threshold. Second, the paper argues against a common simplification: equal-energy three-alpha events do not uniquely diagnose direct decay. A hypothetical Efimov state at $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$11 would, if it existed, mainly decay sequentially while still yielding three alphas of equal energies, a “counterintuitive result of tunneling” (Zheng et al., 2018). This directly illustrates that purity of kinematic signature does not imply purity of mechanism.

The same direct-versus-sequential distinction underlies true two-proton radioactivity in the drip-line review (Pfützner et al., 2011). There, a mode is “pure” only if sequential one-proton emission is energetically forbidden and the observed correlations are consistent with genuine three-body decay. In both cases, purity is inseparable from exclusion of sequential pathways.

7. Extensions beyond conventional decay theory

Recent work has extended the phrase “pure decay modes” into areas not traditionally associated with radioactive or unstable-particle decay. The most explicit example is non-Hermitian graph physics (Liu et al., 15 Jul 2025).

In that setting, pure decay modes are eigenstates of a directed non-Hermitian network whose spatial amplitudes obey a strictly exponential law along the graph, without the oscillatory structure characteristic of conventional non-Hermitian skin modes (Liu et al., 15 Jul 2025). The paper formulates a pathwise condition

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$12

and defines a decay charge at node $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$13 by

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$14

A central result is the combinatorial expression

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$15

where $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$16 and $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$17 are the numbers of outgoing and incoming edges, respectively (Liu et al., 15 Jul 2025). For odd numbers of nodes the charges are integers; for even numbers they can be half-integers; and they satisfy the neutrality condition

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$18

(Liu et al., 15 Jul 2025).

The paper also gives a cycle-consistency condition

$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$19

for every directed cycle and a sufficient pairing rule $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$20 for connectivity on a cycle (Liu et al., 15 Jul 2025). Experimental microwave-circuit realizations confirmed the predicted pure exponential profiles (Liu et al., 15 Jul 2025).

This usage is formally remote from nuclear and particle decay, yet it preserves the same abstract structure: purity is the exclusion of oscillatory admixtures, leaving a single exponential law. A plausible implication is that “pure decay mode” has become a transferable mathematical descriptor for single-exponential behavior, whether in time, space, or channel space.

8. Comparative interpretation and recurring misconceptions

Across the surveyed literature, several distinctions recur and help prevent category mistakes.

First, a pure decay mode is not always a pure final state. In $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$21, the decay is pure in topology, but the observed amplitude still contains both $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$22- and $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$23-wave contributions (Collaboration et al., 2023). Conversely, in the Hoyle-state problem, an equal-energy final state does not guarantee direct decay, because sequential decay can mimic that signature under specific conditions (Zheng et al., 2018).

Second, pure exponential kinetics does not imply a unique microscopic model. The $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$24Pm$$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$25 ESR result shows that the observed electron-capture decay is purely exponential within sensitivity, but that statement concerns the measured time distribution and excludes only sizable modulations under those conditions (Ozturk et al., 2019). It does not, by itself, establish a unique microscopic explanation beyond that empirical constraint.

Third, purity can be sector-specific. In the two-boson tunneling model, the in-in probability is pure single-exponential over all interaction strengths studied, while the in-out sector ceases to be pure at stronger coupling and must be described by two modes (Kim et al., 2011).

Fourth, direct and pure are not always synonymous. In several contexts they coincide, as in pure direct $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$26 decay or true $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$27 radioactivity (Zheng et al., 2018, Pfützner et al., 2011). But in particle-physics topological language, a process can be “pure” because only one diagrammatic topology contributes, even though the decay may still involve internal interference among partial waves or resonant subchannels (Collaboration et al., 2023, Mandal et al., 2016).

Finally, the literature also shows a pragmatic use of purity as effective dominance. In the competition study of superheavy nuclei, “pure alpha” and “pure weak” are defined operationally through branching thresholds such as $$e^{-iHt/\hbar}|z_R\rangle = e^{-iE_R t/\hbar} e^{-\Gamma_R t/(2\hbar)} |z_R\rangle,$$28 (Sarriguren, 2022). This suggests a useful editorial shorthand: purity is often an asymptotic or phenomenological notion, meaning that one mode dominates so strongly that others can be neglected.

Taken together, these works indicate that “pure decay modes” is less a single doctrine than a cross-disciplinary organizing idea. It captures the limiting cases in which decay can be described by one exponential law, one open channel, one decay topology, or one direct mechanism, thereby simplifying both interpretation and inference (Madrid, 2015, Pfützner et al., 2011, Collaboration et al., 2023, Liu et al., 15 Jul 2025).