Conservation laws and effective hadronization models

TL;DR

This work reframes hadronization as a conditioned stochastic diffusion to reconcile local string-breaking dynamics with global conservation laws. By employing the Doob $h$-transform, it shows that global constraints renormalize local fragmentation through an emergent drift tied to the future viability of the cascade, enabling exact Markovian updates in a conditioned ensemble. An EFT tower organized by the remaining string mass reveals a UV fixed point, an intermediate running regime, and an IR boundary layer, with non-local tail operators capturing rare events near termination. The approach yields a clean factorization between universal microscopic fragmentation and infrared constraint effects, supported by analytic results and numerical validation, and points to practical hybrid sampling strategies for efficient hadronization simulations. This framework provides a principled path toward systematically improvable hadronization models with clear RG-like structure and observable predictions across kinematic regimes.

Abstract

Hadronization models based on local string-breaking dynamics are typically Markovian by construction, yet the physical ensemble of final states is shaped by global constraints that couple the entire fragmentation trajectory. Recasting hadronization as a conditioned stochastic diffusion process provides a precise mathematical resolution to this tension. In particular, this language reveals explicitly that constraints stemming from conservation laws induce non-Markovian correlations between otherwise independent fragmentation steps, and that these correlations can be absorbed exactly into a renormalization of the local dynamics through a Doob $h$-transform. We develop this formalism for a $q\bar{q}$ string in the chiral limit, where the longitudinal-transverse factorization of the Lund kernel becomes exact, enabling systematic power counting and clean ultraviolet (UV)/infrared (IR) separation. The dynamics organize naturally into a tower of effective theories distinguished by the remaining string mass, spanning a UV fixed point with scale-invariant transport coefficients, an intermediate regime where transverse phase space induces controlled running, and an IR boundary layer where non-local effects enter at leading order. The tower exhibits genuine Wilsonian structure, including $β$-functions, anomalous dimensions, and systematic matching conditions. The resulting framework achieves a clean factorization of universal microscopic fragmentation dynamics from infrared constraint effects, and opens new directions for both the theoretical analysis and practical simulation of hadronization.

Figures (9)

• Figure 1: Two complementary depictions of string hadronization. Left: The Lund plane representation showing string breaks in light-cone coordinates, where each hadron carries a fraction $z$ of the available light-cone momentum and transverse mass $m_\perp^2$. The worldsheet area element controls the probability for string breaking via tunneling. Right: The stochastic mass evolution representation, where hadronization is viewed as a discrete Markov chain governing the evolution of the remaining string mass $M_n \to M_{n+1}$, with each step corresponding to a single hadron emission. The cascade terminates when the mass enters the termination band $\mathcal{S} = [M_\star, M_{\rm cut}]$ or fails by undershooting into $\mathcal{F} = (0, M_\star)$.
• Figure 2: Transport coefficients governing the continuum limit of string mass evolution. Left: Drift coefficient $\mu(M)$ (mean mass loss per step) and its UV fixed-point $\mu_{\rm UV} = c_1 M$ with $c_1 = -1/(2a+3) \approx -0.23$. Right: Diffusion coefficient $\sigma^2(M)$ (variance of mass fluctuations) and UV scaling $\sigma^2_{\rm UV} = d_1 M^2$. Insets show ratios approaching unity for $M \gtrsim 10$ GeV, confirming scale-invariant UV behavior. Shaded regions denote the EFT tower: termination band $\mathcal{S}$ ($M_\star < M < M_{\rm cut}$ with $M_\star = 0.62$ GeV, $M_{\rm cut} = 1.0$ GeV), IR boundary layer ($1.0$ -- $1.5$ GeV), running regime ($1.5$ -- $10$ GeV), and UV fixed point ($M > 10$ GeV). Monash tune: $a = 0.68$, $b = 0.98$ GeV$^{-2}$.
• Figure 3: Non-local tail operators encoding single-step transition probabilities. Left: Failure operator $\Gamma_{\mathcal{F}}(M)$, the probability of undershooting into $\mathcal{F} = (0, M_\star)$ with $M_\star = 0.62$ GeV. Right: Termination operator $\Gamma_{\mathcal{S}}(M)$, the probability of landing in $\mathcal{S} = [M_\star, M_{\rm cut}]$, shown for $M_{\rm cut} = 0.7, 1.0, 1.5$ GeV. Both scale as $(M_{\rm scale}/M)^{2(a+1)} \approx (M_{\rm scale}/M)^{3.4}$ in the UV and are power-suppressed at large $M$, but become $\mathcal{O}(1)$ near $M_{\rm cut}$ where they enter the EFT at leading order.
• Figure 4: Success probability $h(M)$ for cascade termination in $\mathcal{S}$ rather than $\mathcal{F}$, shown for $M_{\rm cut} = 0.9, 1.2, 1.7, 3.0$ GeV with asymptotic plateaus $h_\infty \approx 0.51, 0.78, 0.94, 0.99$ respectively. Solid curves: matched composite EFT interpolating between the boundary, running, and UV approximations. Open circles: exact solution from matrix inversion of the integral equation \ref{['eq:h_integral_eq']}. Broad translucent bands: the truncated Neumann series \ref{['eq:neumann_series']} at order $N=8$, shown in the boundary-layer region within a few multiples of $\ell = 1/|r_-|$ above each $M_{\rm cut}$. The rapid rise near each $M_{\rm cut}$ marks the boundary layer where tail operators enter at leading order. The survival probability formula in \ref{['eq:h_inf_complete']} achieves $\lesssim 1\%$ accuracy for $h_\infty$ across all cases.
• Figure 5: Doob $h$-transformed fragmentation kernel $K_h(M \to M') = K_{\rm bare}(M \to M') \cdot h(M')/h(M)$ for $M_{\rm cut} = 1.0$ GeV ($h_\infty = 0.62$). Each panel compares the bare kernel (solid black) to the conditioned kernel (dashed) at $M = 1.1, 1.4, 2.0, 3.0, 7.0, 10.0$ GeV. Near $M_{\rm cut}$, the $h$-transform suppresses large downward jumps that risk undershooting into $\mathcal{F}$ while enhancing transitions that land safely in $\mathcal{S}$. At large $M$, where $h(M') \approx h(M) \approx h_\infty$, the conditioning effect vanishes and $K_h \to K_{\rm bare}$.