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arXiv:2610.11121

Quantum Magic and the Strong Coupling Constant

Qiaofeng Liu, Ian Low, Zhewei Yin

hep-phhep-thnucl-thquant-phSubmitted Oct 8, 2026• Announced Oct 9, 2026
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Abstract

The three gauge couplings of the Standard Model (SM) can be parameterized as the fine structure constant $α$, the weak mixing angle $\sin^2θ_w$, and the strong coupling $α_s$. They are fundamental constants whose values remain unexplained. Previously we showed that minimizing magic production in charged-lepton scattering reproduces $\sin^2θ_w$ with great precision. Here we study magic production in the six flavor-diagonal color-singlet channels $q\bar q\to q\bar q$ at tree level, including photon, gluon, $W/Z$, and Higgs exchange. Varying only $α_s$, with all other inputs fixed at each center-of-mass energy, we find that the finite-$α_s$ magic minimum in the top channel reproduces the $\overline{\mathrm{MS}}$ value of $α_s$ to within $5\%$ from $1~$TeV to $10^9~$GeV. Higgs exchange involving the large top Yukawa coupling is essential for this agreement. All six channels produce near-minimal magic at the SM couplings, and reducing $α_s$ toward $α$ increases their magic production, potentially explaining why the strong interaction is ``strong.'' To understand why minimizing magic reproduces both $\sin^2θ_w$ and $α_s$, we conjecture that the parameters of fundamental interactions among color singlets reflect a principle of quantum computational efficiency, with the physical universe emerging from a quantum simulation subject to constrained resources.

Comments
10 pages, 6 figures

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The three gauge couplings of the Standard Model (SM) can be parameterized as the fine structure constant ααα, the weak mixing angle sin⁡2θw\sin^2θ_wsin2θw​, and the strong coupling αsα_sαs​. They are fundamental constants whose values remain unexplained. Previously we showed that minimizing magic production in charged-lepton scattering reproduces sin⁡2θw\sin^2θ_wsin2θw​ with great precision. Here we study magic production in the six flavor-diagonal color-singlet channels qqˉ→qqˉq\bar q\to q\bar qqqˉ​→qqˉ​ at tree level, including photon, gluon, W/ZW/ZW/Z, and Higgs exchange. Varying only αsα_sαs​, with all other inputs fixed at each center-of-mass energy, we find that the finite-αsα_sαs​ magic minimum in the top channel reproduces the MS‾\overline{\mathrm{MS}}MS value of αsα_sαs​ to within 5%5\%5% from 1 1~1 TeV to 109 10^9~109 GeV. Higgs exchange involving the large top Yukawa coupling is essential for this agreement. All six channels produce near-minimal magic at the SM couplings, and reducing αsα_sαs​ toward ααα increases their magic production, potentially explaining why the strong interaction is ``strong.'' To understand why minimizing magic reproduces both sin⁡2θw\sin^2θ_wsin2θw​ and αsα_sαs​, we conjecture that the parameters of fundamental interactions among color singlets reflect a principle of quantum computational efficiency, with the physical universe emerging from a quantum simulation subject to constrained resources.
The weak mixing angle sWs_WsW​ is a fundamental constant in the Standard Model (SM) and measured at the ZZZ boson mass to be s^W2(mZ)=0.23129±0.00004\widehat{s}^2_W(m_Z) = 0.23129 \pm 0.00004sW2​(mZ​)=0.23129±0.00004 in the MS‾\overline{\rm MS}MS renormalization scheme, where mZ=91.2 GeVm_Z=91.2\ \text{GeV}mZ​=91.2 GeV. On the other hand, non-stabilizerness - the magic - characterizes the computational advantage of a quantum system over classical computers. We consider the production of magic from stabilizer initial states, which carry zero magic, in the 2-to-2 scattering of charged leptons in the SM at the tree level, which is mediated by the photon and the ZZZ boson. Using the second order stabilizer Rényi entropy, and averaging over all 60 initial stabilizer states and the scattering angle, we compute and minimize the magic production as a function of sW2s^2_WsW2​ in the Møller scattering e−e−→e−e−e^-e^-\to e^-e^-e−e−→e−e−, which is free of kinematic thresholds. At the centre-of-mass energy s=mZ\sqrt{s}=m_Zs​=mZ​, there is a unique minimum in magic production at sW2(mZ)=0.2317\mathbf{s}^{2}_W(m_Z)=0.2317sW2​(mZ​)=0.2317, which agrees with the measured s^W2(mZ)\widehat{s}^2_W(m_Z)sW2​(mZ​) at the sub-percent level. At higher energies, the magic-minimizing sW2\mathbf{s}^{2}_WsW2​ continues to agree with the empirical value at the percent level or better, up to 10 TeV. The finding suggests the electroweak sector of the SM tends to generate minimal quantum resources from the computational viewpoint.
Quantum magic, or nonstabilizerness, is a quantum resource associated with computational advantage in quantum systems. In high energy collisions, Quantum Electrodynamics (QED) is inefficient at generating magic while the weak mixing angle, a fundamental constant of nature, sits near a value that minimizes magic production in charged-lepton scattering. These observations were made in the laboratory (lab) basis, in which spin is projected along the incoming beam axis. An alternative choice is the helicity basis, in which spin is projected along the direction of motion of each particle. The transformation between these two bases is, in general, not a Clifford operation and therefore can change the amount of magic. We present a detailed study of magic production in both bases for QED and electroweak processes, and compare these results with the basis-invariant non-local magic. In the ultra-relativistic limit, magic production is generally smaller in the helicity basis due to helicity selection rules, while the lab basis generally yields less magic in the non-relativistic regime. We provide circuit realizations of the ultra-relativistic Bhabha amplitudes using linear combinations of unitaries and show that the lab basis construction contains a larger TTT-gate count at generic scattering angles. Interestingly, in both bases the physical weak mixing angle lies close to the value that minimizes magic production.
The production of magic states is studied in two settings. The first is the electroweak (EW) sector of the Standard Model (SM). The second is an extension featuring a new broken U(1)U(1)U(1) gauge symmetry and a Dirac fermion charged under it. This setup resembles a dark U(1)U(1)U(1) scenario, with the additional fermion playing the role of a dark matter candidate that annihilates into SM particles through its coupling to the new gauge boson. In the EW sector, the low-energy regime reproduces earlier magic production results obtained for Quantum Electrodynamics, whereas the high-energy and ZZZ-resonance regimes generate new magic distribution functions and non-trivially reorganize the stabilizer state classes, with Bhabha scattering exhibiting the strongest sensitivity to electroweak effects. Also, a subset of fixed stabilizer states is identified, for which the magic distributions remain unchanged across the different energy regimes. In the dark sector, the main effect of the new massive mediator is the appearance of new magic distributions functions for Moller-like, Bhabha-like, and inverse pair-annihilation processes in the low-energy limit. These reach the maximal magic value at the SM-to-dark fermion mass ratios mf/mχ→0m_f/m_χ\to 0mf​/mχ​→0 and mf/mχ→1.83929m_f/m_χ\to 1.83929mf​/mχ​→1.83929.
Entanglement is a hallmark of quantum theory, yet it alone does not capture the full extent of quantum complexity: some highly entangled states can still be classically simulated. Non-classical behavior also requires magic, the non-Clifford component that enables universal quantum computation. Here, we investigate whether the interplay between entanglement and magic-state resources constrains the structure of fundamental interactions. We study gluon-gluon and graviton-graviton scattering at tree level. We focus on the high-energy limit, where mass-dependent terms are negligible, and conformal symmetry is preserved. In this regime, all particles behave as massless degrees of freedom, allowing to isolate their transverse helicities as two-qubit states. We explicitly break gauge and general covariance by modifying the quartic vertices and analyzing the resulting generation of entanglement and magic. We find that imposing maximal entanglement (MaxEnt) alone does not uniquely recover gauge-invariant and diffeomorphism invariant interactions, but adding the condition of minimal, but nonzero, magic-state generation singles it out. Our results indicate that nature favors MaxEnt and low magic: maximal quantum correlations with limited non-Cliffordness, sufficient for universal quantum computing but close to classical simulability. This dual informational principle may underlie the emergence of gauge invariance in fundamental physics.
We consider the quantum magic in systems of dense neutrinos undergoing coherent flavor transformations, relevant for supernova and neutron-star binary mergers. Mapping the three-flavor-neutrino system to qutrits, the evolution of quantum magic is explored in the single scattering angle limit for a selection of initial tensor-product pure states for Nν≤8N_ν\le 8Nν​≤8 neutrinos. For ∣νe⟩⊗Nν|ν_e\rangle^{\otimes N_ν}∣νe​⟩⊗Nν​ initial states, the magic, as measured by the α=2α=2α=2 stabilizer Renyi entropy M2M_2M2​, is found to decrease with radial distance from the neutrino sphere, reaching a value that lies below the maximum for tensor-product qutrit states. Further, the asymptotic magic per neutrino, M2/NνM_2/N_νM2​/Nν​, decreases with increasing NνN_νNν​. In contrast, the magic evolving from states containing all three flavors reaches values only possible with entanglement, with the asymptotic M2/NνM_2/N_νM2​/Nν​ increasing with NνN_νNν​. These results highlight the connection between the complexity in simulating quantum physical systems and the parameters of the Standard Model.
We consider the quantum magic in systems of dense neutrinos undergoing coherent flavor transformations, relevant for supernova and neutron-star binary mergers. Mapping the three-flavor-neutrino system to qutrits, the evolution of quantum magic is explored in the single scattering angle limit for a selection of initial tensor-product pure states for N ν ≤ 8 neutrinos. For | ν e 〉 ⊗ N ν initial states, the magic, as measured by the α = 2 stabilizer Renyi entropy M 2 , is found to decrease with radial distance from the neutrino sphere, reaching a value that lies below the maximum for tensor-product qutrit states. Further, the asymptotic magic per neutrino, M 2 / N ν , decreases with increasing N ν . In contrast, the magic evolving from states containing all three flavors reaches values only possible with entanglement, with the asymptotic M 2 / N ν increasing with N ν . These results highlight the connection between the complexity in simulating quantum physical systems and the parameters of the Standard Model.