[Paper Review] On the phase diagram of the Higgs SU(2) model

Claudio Bonati, Guido Cossu+3 more|ArXiv.org|Jan 28, 2009

Quantum Chromodynamics and Particle Interactions3 references4 citations

TL;DR

This study reevaluates the phase diagram of the Higgs $SU(2)$ model with infinite Higgs self-coupling ($\lambda = \infty$), showing via large-scale Monte Carlo simulations on lattices up to $45^4$ that regions previously believed to host first-order transitions actually exhibit only smooth crossovers. The apparent first-order behavior at smaller lattices is an artifact of finite-size effects, with true thermodynamic behavior emerging only at much larger volumes.

ABSTRACT

The Higgs SU(2) model with fixed Higgs length is usually believed to have two different phases at high gauge coupling (β), separated by a line of first order transitions but not distinuguished by any typical symmetry associated with a local order parameter, as first proved by Fradkin and Shenker. We show that in regions of the parameter space where it is usually supposed to be a first order phase transition only a smooth crossover is in fact present.

Motivation & Objective

• To resolve the long-standing ambiguity in the phase structure of the Higgs $SU(2)$ model at $\lambda = \infty$, particularly the nature of the transition at high gauge coupling $\beta$.
• To test whether the widely assumed line of first-order transitions in the phase diagram is genuine or an artifact of finite-size effects in lattice simulations.
• To determine the true infinite-volume behavior of key observables—such as the gauge-Higgs coupling, plaquettes, $Z_2$ monopoles, and Polyakov loops—across varying lattice sizes.
• To clarify the role of lattice size in detecting phase transitions, especially given that previous studies used small lattices ($L \leq 12$) that may mislead conclusions.

Proposed method

• Performed large-scale Monte Carlo simulations on lattices up to $45^4$ at $\beta = 2.5$ and $\beta = 2.725$, where first-order transitions were previously suspected.
• Monitored the gauge-Higgs coupling, plaquettes, $Z_2$ monopoles, and Polyakov loops as key observables to detect phase transition signatures.
• Used the susceptibility $\chi(O) = L^4(\langle O^2 \rangle - \langle O \rangle^2)$ and the Binder fourth-order cumulant $V_4(O) = 1 - \langle O^4 \rangle / (3\langle O^2 \rangle^2)$ to diagnose the nature of the transition.
• Fitted susceptibility data to $a + bL^4$ and Binder cumulant minima to asymptotic forms including $L^{-4}$ and $L^{-8}$ corrections to distinguish first-order from crossover behavior.
• Analyzed the convergence of the Polyakov loop to its infinite-volume limit using exponential fits to $a + b\exp(-cL)$, comparing small- and large-$L$ regimes.
• Employed the Fradkin-Shenker theorem as a theoretical anchor, noting that analyticity of all local observables in a region implies no local order parameter can distinguish phases.

Experimental results

Research questions

• RQ1 Is the transition at high $\beta$ in the $SU(2)$ Higgs model with $\lambda = \infty$ truly first-order, or is it a finite-size artifact?
• RQ2 What is the true nature of the phase transition in the region $\beta \approx 2.5$ to $\beta \approx 2.725$, as revealed by infinite-volume extrapolations?
• RQ3 How large must the lattice size be to reliably detect the true thermodynamic behavior of the Higgs $SU(2)$ model, especially for observables like the gauge-Higgs coupling and Polyakov loop?
• RQ4 Do the Binder cumulant minima and susceptibility scaling behavior converge to values consistent with a crossover ($V_4 \to 2/3$) or a first-order transition ($V_4 < 2/3$) in the thermodynamic limit?
• RQ5 Can the critical end-point of the first-order line be located, or is the line of first-order transitions absent in the $\lambda = \infty$ limit?

Key findings

• At $\beta = 2.5$, the Binder fourth-order cumulant minimum for the gauge-Higgs coupling converges to $B = 0.666666(1)$, consistent with a crossover and not a first-order transition.
• At $\beta = 2.725$, the Binder cumulant minimum converges to $B = 0.666667(1)$, indistinguishable from $2/3$, confirming a crossover behavior in the thermodynamic limit.
• Susceptibility peaks for the gauge-Higgs coupling scale as $a + bL^4$ for $L \leq 20$ at $\beta = 2.5$, but saturate at larger $L$, indicating a crossover rather than a first-order transition.
• The Polyakov loop extrapolation shows a clear discrepancy between small-$L$ fits ($a \approx 0.00244$) and large-$L$ fits ($a \approx 0.00085$), demonstrating that small-volume data misrepresent the true infinite-volume limit.
• All monitored observables—plaquettes, $Z_2$ monopoles, and Polyakov loops—exhibit the same two-stage behavior: first-order-like scaling at small $L$, crossover-like convergence at large $L$, confirming the universal nature of the finite-size artifact.
• The study concludes that the region around $\beta = 2.5$ to $\beta = 2.725$ does not host a first-order transition, but only a smooth crossover, challenging the conventional phase diagram of the Higgs $SU(2)$ model.