Abstract We develop Hybrid Monte Carlo (HMC) algorithms for constrained Hamiltonian systems of gauge–Higgs models and introduce a new observable for the constraint effective Higgs potential. We use an extension of the so-called Rattle algorithm to general Hamiltonians for constrained systems, which we adapt to the 4D Abelian–Higgs model and the 5D SU(2) gauge theory on the torus and on the orbifold. The derivative of the potential is measured via the expectation value of the Lagrange multiplier for the constraint condition and allows a much more precise determination of the effective potential than conventional histogram methods. With the new method, we can access the potential over the full domain of the Higgs variable, while the histogram method is restricted to a short region around the expectation value of the Higgs field in unconstrained simulations, and the statistical precision does not deteriorate when the volume is increased. We further verify our results by comparing to the one-loop Higgs potential of the 4D Abelian–Higgs model in unitary gauge and find good agreement. To our knowledge, this is the first time this problem has been addressed for theories with gauge fields. The algorithm can also be used in four dimensions to study finite temperature and density transitions via effective Polyakov loop actions.

中文翻译：

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规范-希格斯模型的约束混合蒙特卡罗算法

摘要 我们为规范-希格斯模型的约束哈密顿系统开发了混合蒙特卡罗 (HMC) 算法，并为约束有效希格斯势引入了一个新的可观测值。我们将所谓的 Rattle 算法扩展到约束系统的一般哈密顿量，我们适应了 4D Abelian-Higgs 模型和环面和轨道上的 5D SU(2) 规范理论。电势的导数是通过拉格朗日乘子的期望值测量的约束条件，并允许比传统的直方图方法更精确地确定有效电势。使用新方法，我们可以访问 Higgs 变量的整个域上的势能，而直方图方法在无约束模拟中被限制在希格斯场期望值附近的一个短区域内，并且随着体积的增加统计精度不会变差。我们通过与幺正规范中 4D Abelian-Higgs 模型的单环 Higgs 势进行比较来进一步验证我们的结果，并找到了很好的一致性。据我们所知，这是第一次针对规范场理论解决这个问题。该算法还可用于四个维度，以通过有效的 Polyakov 循环动作研究有限的温度和密度转变。据我们所知，这是第一次针对规范场理论解决这个问题。该算法还可用于四个维度，以通过有效的 Polyakov 循环动作研究有限的温度和密度转变。据我们所知，这是第一次针对规范场理论解决这个问题。该算法还可用于四个维度，以通过有效的 Polyakov 循环动作研究有限的温度和密度转变。