Universal Soft Behaviour (USB)

• Universal Soft Behaviour (USB) is a phenomenon where scattering amplitudes and particle production rates approach a momentum-independent limit as one characteristic momentum becomes soft.
• In high-energy collisions, USB manifests through coherent soft gluon bremsstrahlung and exhibits scaling with QCD color factors and participant nucleons.
• In exceptional EFTs on de Sitter space, USB acts as a unification principle by enforcing generalised energy conservation to uniquely determine soft limits of tree-level amplitudes.

Universal Soft Behaviour (USB) describes a distinctive pattern whereby scattering amplitudes or particle production rates in a wide class of theories approach a universal, momentum-independent limit as one of the characteristic momenta becomes soft (i.e., approaches zero). This phenomenon appears with particular clarity in two major contexts: the production of particles in high-energy collisions at low transverse momentum ($p_T \to 0$) and the structure of S-matrix elements in exceptional effective field theories (EFTs) on de Sitter (dS) backgrounds. USB provides a powerful organizing principle, connecting phase-space, symmetry, and stability constraints to physical observables in both perturbative QCD and in nontrivial gravitational backgrounds.

1. USB in High-Energy Collisions and Coherent QCD Radiation

In the limit $p, p_T \rightarrow 0$ across collider processes such as $e^+e^-$, $pp$, and nuclear ($AA$) collisions, particle production is dominated by coherent soft gluon bremsstrahlung. The large wavelength of such soft gluons precludes resolving the substructure of the emitting partons, so only the aggregate color charge determines radiation patterns. In perturbative QCD, the inclusive distribution for emitting a gluon of energy $k$ and angle $\Theta$ off a partonic ancestor ($a$) is described at leading (Born) order: $$\dfrac{dN_a}{dk\, d\Theta} = \frac{2}{\pi} \frac{C_a}{k\Theta} \alpha_s + \mathcal{O}(\alpha_s^2)$$ where $C_a$ is a QCD color factor ($C_F$ for quark jets, $C_A$ for gluon jets), and $\alpha_s$ is the strong coupling constant. This result highlights that the emission rate in the soft limit is inversely proportional to both $k$ and $\Theta$, directly proportional to the total color charge, and determined entirely by the minimum partonic configuration. Higher-order and incoherent corrections are suppressed.

2. Universal Limiting Behavior of Soft-Momentum Spectra

Experimental observations indicate that the invariant particle density,

$I_0 = E \frac{dN}{d^3p} \bigg|_{p,\, p_T \to 0}$

approaches a limiting value as $p_T \to 0$, which is independent of the primary collision energy. While overall multiplicity ($dN/dy$) may rise with energy, the zero-$p_T$ limit does not. This feature is a direct manifestation of the dominance of coherent soft gluon bremsstrahlung governed by the color structure of QCD, as higher-energy effects do not couple efficiently to the soft sector.

3. Color Factor Scaling and Its Experimental Verification

The proportionality of the limiting soft spectrum to the QCD color factors allows for robust, model-independent predictions: $$\frac{I_0^{(gg)}}{I_0^{(q\bar{q})}} = \frac{C_A}{C_F}$$ In SU(3) QCD, $C_A / C_F = 9/4 \approx 2.25$. Experimental measurements, particularly in $e^+e^-$ three-jet events (e.g., by the DELPHI Collaboration), confirm values close to this ratio, providing compelling empirical support for the universality of soft particle production as determined by underlying color algebra.

4. Participant Scaling in Nuclear Collisions

In nucleus–nucleus ($AA$) collisions, the “wounded nucleon”—or participant—model posits that each nucleon contributes once to soft particle production, irrespective of the number of its scatterings. If $N_\text{part}$ is the count of participating nucleons, data show: $$I_0^{(AA)} \approx \frac{N_\text{part}}{2} I_0^{(pp)}$$ This scaling is interpreted as a direct consequence of coherence: rescatterings within the nucleus preserve phase relations, resulting in linear scaling with the count of independent color emitters rather than the number of binary collisions.

5. Universality of Soft Particle Ratios

Across $e^+e^-$, $pp$, and $AA$ systems, ratios of low-momentum hadron yields (e.g., $\pi/K$, $\pi/p$) display convergence—a universal behavior of the soft sector. This insensitivity to the details of the initial state or hadronic re-interactions suggests that soft particles “decouple” promptly, and their distributions are a pristine imprint of the QCD bremsstrahlung pattern.

6. USB in Exceptional EFTs on de Sitter Space

In an apparently disparate context, USB arises as a fundamental organizing principle for the S-matrix in de Sitter space across all “exceptional” EFTs. The defining feature is that the tree-level amplitude $\mathcal{A}(k)$ satisfies: $$\lim_{k \rightarrow 0} \mathcal{A}(k) \sim \mathcal{O}(1)$$ for all such theories—a notable deviation from flat-space behavior where, for example, DBI and special Galileon theories enjoy improved soft-theorem scaling ($\mathcal{O}(p^2)$ or softer). The underlying mechanism invoking USB is the “Generalised Energy Conservation” (GEC) condition: in dS, time-translation invariance is broken, but imposing the analogue of energy conservation (demanding the vanishing of non-conserving contributions, $\mathcal{A}_{k_T \neq 0}=0$) removes nonanalytic and divergent terms in the soft limit. Imposing GEC is mathematically equivalent to enforcing USB, as demonstrated for DBI ($\Delta=4$), special Galileon ($\Delta=5$), and $SU(N)$ NLSM ($\Delta=3$) where the leading soft divergences cancel only when couplings are precisely tuned.

7. USB as a Unification and Selection Rule

The emergence of USB from mass spectrum and stability (through GEC) considerations leads to a conjecture: all exceptional EFTs in dS with adequate conformal dimension ($\Delta \geq 4$) are completely fixed—that is, their higher-point interactions are uniquely determined by the requirement of USB. This “soft bootstrap” makes USB a unification principle within dS EFTs, echoing the uniqueness of General Relativity being fixed by spectrum (massless spin-2) and stability requirements. In both cases, symmetry and infrared consistency alone suffice to fix the unique interacting theory.

Table: Manifestations of Universal Soft Behaviour

Context	Underlying Mechanism	Scaling in Soft Limit
QCD (colliders, $p_T\to0$)	Coherent soft gluon bremsstrahlung	$I_0 \propto C_a$ (momentum-independent)
Nuclear collisions	Participant (“wounded nucleon”) scaling	$I_0^{(AA)} \propto N_\text{part}$
dS Exceptional EFTs	Generalised energy conservation	$\lim_{k\to0} \mathcal{A}(k)\sim\mathcal{O}(1)$

USB in both QCD and exceptional dS EFTs thus emerges as a deep, model-independent consequence of coherence, color algebra, and the constraints arising from stability and symmetry. It not only enables robust predictions for particle spectra and ratios but also acts as a structural principle uniquely determining physically viable EFTs in curved space.