Finite Volume Analysis of Relativistic Field Theories Using Lattice Regularization

量子染色体动力学（QCD）是描述自然界中一种基本相互作用，即强相互作用的理论。基本的自由度是夸克（粒子）和胶子（力介质）。耦合到无质量费米子（夸克）的规范场（胶子）没有自由参数，也没有内禀标度。根据规范群的性质和费米子的含量，这些理论在量子化后分为两类：一类保持标度不变性，只包含无质量的激发，另一类打破标度不变性，产生丰富的大质量激发谱。无论量子理论是否是标度不变的，它总是强相互作用的，非微扰的形式是必要的，以研究为什么有些理论是标度不变的，而另一些不是。研究从一个理论到另一个理论的流动，并试图理解一个尺度是如何出现的，这也是必要的。PI将提取这两类理论的属性（标度不变和标度破坏），并研究标度不变理论到破坏标度不变性理论的流程。这将使我们能够理解强相互作用中基本尺度的出现。耦合到无质量费米子的规范场的格点形式主义是提取这些理论的基本非微扰性质的一种行之有效的方法。打破尺度不变性的理论已经使用这种方法进行了非常详细的研究。PI将使用格形式主义，并在有限周期盒中研究潜在的尺度不变理论。盒子的大小设定了一个外部尺度，PI将能够通过研究作为外部尺度函数的渐近行为来提取理论的尺度不变性质。通过固定规范群，并在临界费米子味数周围改变一个单位的费米子味数，PI将从一个理论（更多的费米子味）开始，并为其中一个费米子味添加质量项。通过将质量参数从零变到无穷大，PI将能够在两个理论之间流动，这两个理论在经典水平上都是标度不变的，但其中一个（更少数量的费米子口味）在量子水平上打破了标度不变性。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Quantum Chromo Dynamics (QCD) is a theory that describes one of the fundamental interactions in nature, namely, strong interactions. The fundamental degrees of freedom are quarks (particles) and gluons (force mediators). Gauge fields (gluons) coupled to massless fermions (quarks) have no free parameters and they have no intrinsic scale. Depending on the nature of the gauge group and the fermionic content, these theories fall into two categories upon quantization: One that maintains scale invariance and contains only massless excitations and one that breaks scale invariance and produces a rich spectrum of massive excitations. Whether the quantum theory is scale invariant or not, it is always strongly interacting and a non-perturbative formalism is necessary to study why some theories are scale invariant and the others are not. It is also necessary to study the flow from one theory to another and try to understand how a scale emerges. The PI will extract properties of theories in both classes (scale invariant and scale breaking) and study the flow of a scale invariant theory to a theory that breaks scale invariance. This will enable us to understand the emergence of the fundamental scale in strong interactions.Lattice formalism of gauge fields coupled to massless fermions is a well tested approach to the extraction of the fundamental non-perturbative properties of these theories. Theories that break scale invariance have been studied in great detail using this method. The PI will use the lattice formalism and study potentially scale invariant theories in a finite periodic box. The size of the box sets an external scale and the PI will be able to extract the scale invariant properties of a theory by studying the asymptotic behavior as a function of the external scale. By fixing the gauge group and varying the number of fermion flavors by one unit around a critical number of fermion flavors the PI will start with one theory (larger number of fermion flavors) and add a mass term for one of the fermion flavors. By varying the mass parameter from zero to infinity, the PI will be able to flow between two theories where both of them are scale invariant at the classical level but one (fewer number of fermion flavors) breaks the scale invariance at the quantum level.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.